Research ArticleDimensional Regularization Approach to the RenormalizationGroup Theory of the Generalized Sine-Gordon Model
Takashi Yanagisawa
Electronics and Photonics Research Institute National Institute of Advanced Industrial Science and Technology1-1-1 Umezono Tsukuba Ibaraki 305-8568 Japan
Correspondence should be addressed to Takashi Yanagisawa t-yanagisawaaistgojp
Received 7 February 2018 Accepted 18 July 2018 Published 6 September 2018
Academic Editor Emmanuel Lorin
Copyright copy 2018 Takashi YanagisawaThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present the dimensional regularization approach to the renormalization group theory of the generalized sine-Gordon modelThe generalized sine-Gordon model means the sine-Gordon model with high frequency cosine modes We derive renormalizationgroup equations for the generalized sine-Gordon model by regularizing the divergence based on the dimensional method Wediscuss the scaling property of renormalization group equations The generalized model would present a new class of scalingproperty
1 Introduction
The sine-Gordon model is an interesting model and plays animportant role in physics [1ndash13] There are many phenomenathat are related to the sine-Gordon model In this sense thesine-Gordon model has universality In the weak couplingphase the sine-Gordon model is perturbatively equivalent tothe massive Thirring model [1 14ndash16] The two-dimensional(2D) sine-Gordonmodel describes a crossover between weakcoupling region and strong coupling region The renormal-ization equations are the same as those for the Kosterlitz-Thouless transition of the 2D classical XY model [17ndash19]The 2D sine-Gordon model is mapped to the Coulomb gasmodel where particles interact with each other through loga-rithmic interaction [4 20 21] The Kondo problem belongsto the same universality class where the renormalizationgroup equations are given by the same equations for the2D sine-Gordon model [20ndash27] The renormalization groupequations in the Kondo problem was derived before thoseby Kosterlitz and Thouless The one-dimensional Hubbardmodel is mapped to the 2D sine-Gordon model by using abosonization method [28ndash31] where the Hubbard model isan important model that describes the metal-insulator tran-sition and high-temperature superconductivity [32ndash39] Thesine-Gordon model appears in a multiband superconductor
where the Nambu-Goldstone modes become massive dueto the Josephson couplings [40ndash47] The Josephson plasmaoscillation in layered high-temperature superconductors wasanalyzed based on the sine-Gordon model [48] In a series ofpapers [41ndash43 45 46] we introduced the sine-Gordon modelinto the study of superconductivity and examined significantexcitation modes in superconductors A generalization fromU(1) to a compact continuous group G for the sine-Gordonmodel was also investigated [49] where the sine-Gordonmodel considered in this paper and in references cited aboveis a model with U(1) group
In this paper we investigate the renormalization grouptheory for the 2D generalized sine-Gordon model by usingthe dimensional regularization method to regularize thedivergence [50ndash52] Here the generalized sine-Gordonmodelis a sine-Gordon model that includes high frequency cosinepotential terms such as cos(119899120601) for an integer n Therenormalization of the generalized sine-Gordon model wasinvestigated [53] by the Wegner-Houghton method [54] andby the functional renormalization groupmethod [55]We usethe dimensional regularization method in deriving the renor-malization group equation for the generalized sine-Gordonmodel The divergence is regularized near two dimensionsby putting the dimension 119889 = 2 + 120598 The divergent part ofintegral is evaluated as a pole in the form 1120598 This is called
HindawiAdvances in Mathematical PhysicsVolume 2018 Article ID 9238280 7 pageshttpsdoiorg10115520189238280
2 Advances in Mathematical Physics
the minimal subtraction method Then the beta function forthe coupling constant is derived
2 Lagrangian
Let us consider a real scalar field 120601 The Lagrangian of thegeneralized sine-Gordon model is given by
L = 121199050 (120597120583120601119861)
2 + 11199050sum119899 1205721198990 cos (119899120601119861) (1)
where 120601119861 is a bare real scalar field and 1199050 and 1205720119899 are barecoupling constants The second term indicates the potentialenergy of the scalar field 120601 The generalized sine-Gordonmodel contains high frequency terms such as cos(119899120601) (n =1 2 sdot sdot sdot ) We write the renormalized coupling constants as119905 and 120572119899 respectively We adopt that 119905 gt 0 and 120572119899 ge 0120572119898 for some 119898 may be zero but at least one 120572119899 should bepositive (nonzero) The dimensions of 119905 and 120572119899 are given as[119905] = 1205832minus119889 and [120572119899] = 1205832 where 120583 is a parameter representingthe energy scale The scalar field 120601 is dimensionless Therelations between bare and renormalized quantities are givenby
where119885119905 and119885120572119899 are renormalization constants 119905 and 120572119899 aredimensionless constants by virtue of the energy scale 120583 Wedefine the renormalized field 120601119877 by
120601119861 = radic119885120601120601119877 (3)
where 119885120601 is the renormalization constant for the field 120601 TheLagrangian with renormalized quantities is written as
L = 120583119889minus21198851206012119905119885119905 (120599120583120601)2 + sum
119899
120583119889120572119899119885120572119899119905119885119905 cos (119899radic119885120601120601) (4)
where 120601 denotes the renormalized field 120601119877 The secondterm represents the interaction of the field 120601 as seen byexpanding cos(119899radic119885120601120601) as a power series There is the otherrepresentation of interaction parameters We can absorb theparameter 119905 in the definition of the field 120601 and the parameter120572119899 In this case field 120601 in the interaction term includes theparameter in the form cos(119899radic119885120601120573120601) where 120573 = radic119905 Wewill obtain the same result since it does not depend on therepresentation
+minus minus
Figure 1 One-loop contributions to the renormalization of 120572119899
3 Renormalization of 120572119899We consider the renormalization of 120572119899 up to the lowest orderof 120572119899 By considering tadpole diagrams in Figure 1 the cosinefunction is renormalized to
cos (119899radic119885120601120601)997888rarr (1 minus 1
21198992119885120601 ⟨1206012⟩ + sdot sdot sdot) cos (119899radic119885120601120601)= exp (minus1
21198992119885120601 ⟨1206012⟩) cos (119899radic119885120601120601) (5)
Since the expectation value ⟨1206012⟩ diverges we regularize itusing the dimensional regularization method
119885120601 ⟨1206012⟩ = 1199051205832minus119889119885119905 int 119889119889119896(2120587)119889
11198962 + 11989820 = minus 119905
120598Ω119889
(2120587)119889 (6)
for 119889 = 2 + 120598 where 1198980 is introduced to avoid infrareddivergence Ω119889 is the solid angle in 119889 dimensions and 119885119905 wasput as 1 In order to remove the divergence the constant 119885120572119899is determined as follows
119885120572119899 = 1 minus 11989922 1199051120598
Ω119889(2120587)119889 (7)
Since the bare coupling constant 1205721198990 is independent of 120583 wehave 120583 1205971205721198990120597120583 = 0 This results in
120583120597120572119899120597120583 = minus2120572119899 minus 120572119899120583120597 ln119885120572119899120597120583 (8)
We set 119885119905 = 1 up to the lowest order of 120572119899 so that we have120583 120597119905120597120583 = (119889 minus 2)119905 The beta function for 120572119899 at the lowestorder in 120572119899 is given by
120573(120572119899) has a zero at 119905 = 119905119888119899119905119888119899 = 8120587
1198992 (10)
for 119889 = 2 There is a fixed point of 119905 for each 1198994 Renormalization of 119905There is an effect of renormalization on the coupling constant119905 that is the correction to the kinetic term Let us consider
Advances in Mathematical Physics 3
the two-point function Γ(119901)The bare lowest order two-pointfunction is given by
Γ(0)119861 (119901) = 111990501199012 = 1
1199051205832minus119889119885119905 1199012 (11)
This corresponds to the kinetic part of the bare Lagrangian
L(0)119861 = 120583119889minus2119885120601
2119905119885119905 (120597120583120601)2 (12)
41 Real Space Formulation The lowest order correction tothe two-point function is given by a second-order term for 120572ℓ
(ℓ = 1 2 sdot sdot sdot ) such as120572119899120572119898 cos(119899radic119885120601120601(119909)) cos(119898radic119885120601120601(1199091015840))From the formula cos 1205791 cos 1205792 = (12)(cos(1205791+1205792)+cos(1205791minus1205792)) the correction to the action comes from
(2120587)1198891198700 (11989801003816100381610038161003816119909 minus 1199101003816100381610038161003816)
(15)
where 1198700 is the 0th modified Bessel function Because1198700(119909) increases divergently as 119909 approaches zero 119868119899119898 isapproximated as
cos (radic119885120601 (119899 minus 119898) 120601 (119909)) (1 minus 121198982119885120601 (nabla119909120601
sdot 119903)2)
(16)
where we put 1199091015840 = 119909 + 119903 The cosine functioncos(radic119885120601(119899 minus 119898)120601(119909)) would oscillate as a function of 119909 thecontribution for 119899 = 119898 will be small Thus we consider onlythe contributions with 119899 = 119898
We extract the divergent term in 119868119899119899 There may be two waysto do this We discuss these methods in the following(1) In the first method we regularize ⟨1206012⟩ by introducinga cutoff 119886 in the real space
by replacing 1198700(1198980119903) with 1198700(1198980radic1199032 + 1198862) By using theasymptotic relation 1198700(119911) asymp minus120574 minus ln(1199112) with the Eulerconstant 120574 the integral with respect to 119903 is performed asfollows [49]
near 119889 = 2 where we set 119888 = (1198901205742)2 We consider the casewhere 119905 is close to the critical value 119905119888119899 = 81205871198992
11989921199058120587 = 1 + V119899 (20)
4 Advances in Mathematical Physics
where V119899 represents the deviation from the critical point Inthe lowest order of V119899 we have
119869119899 = Ω119889 (11988811989820)minus2 12 minus 119889 + 119874 (V119899)
asymp minus 1198992321205722119899120583119889+21198864
1120598 int 119889119889119909120583119889minus2119885120601
2119905119885119905 1198992 (120597120583120601)2
+ 119874 (V119899)
(22)
The constant 119885120572119899 was absorbed for the renormalization of 120572119899Then by taking the sum from each term the kinetic partL(0)119861is renormalized to
L(2)119861 = 120583119889minus2119885120601
2119905119885119905 [1 minus sum119899
1198994321205722119899120583119889+21198864
1120598 ] (120597120583120601)2 (23)
This indicates that we choose
119885120601119885119905 = 1 + sum
119899
1198994321205722119899120583119889+21198864
1120598 (24)
119885120601 and 119885119905 appear as a ratio 119885120601119885119905 in this order and then thecoupling constant 119905 is renormalized as 1199050 = 1199051205832minus119889119885119905119885120601 or wecan choose 119885120601 = 1 The equation 120583 1205971199050120597120583 = 0 results in
The numerical coefficient is not important and this dependson the choice of the cutoff 119886(2) In the second way the divergence comes from ⟨1206012⟩where we adopt that the integral with respect to 119903 is finiteThis treatment is similar to that in [31] where the Wilson
+ +
+ ++
Figure 2The contributions to the two-point function Γ(2)(119901) up tothe order of 120572119899120572119898
renormalization group method was used The correction 119868119899119899is written as
by introducing a cutoff in the integral Then we have
119868119899119899 cong minus 1198601198991612058711989941205722119899120583119889+2119886119889+2 1120598 int 119889119889119909120583119889minus21198851206012119905119885119905 (120597120583120601)2 (29)
This results in the same beta function 120573(119905)with the numericalfactor being slightly different
120573 (119905) = (119889 minus 2) 119905 + sum119899
42MomentumSpace Formulation In themomentum spacewe evaluate the two-point function by calculating the dia-grams in Figure 2 [6] This set of diagrams gives the self-energy Σ(119901) Σ(119901) is written as a sum of Σ119899119898(119901) that comesfrom the interaction term cos(119899120601) cos(119898120601) The diagrams inFigure 2 are summed up to give
sdot int 119889119889119909 [119890119894119901sdot119909 (sinh (1198991198981198660 (119909)) minus 1198991198981198660 (119909))minus (cosh (1198991198981198660 (119909)) minus 1)]
sdot int 119889119889119909 (119890119894119901sdot119909 minus 1) exp (1198991198981198660 (119909)) (33)
Using the expansion 119890119894119901sdot119909 = 1 + 119894119901 sdot 119909 minus (12)(119901 sdot 119909)2 + sdot sdot sdot wekeep the 1199012 term By using the formula1198700(119909) asymp minus120574minus log(1199092)for small 119909 Σ119899119898(119901) is written as
The integral diverges when 119905 = 119905119888119899 = 81205871198992 and 119899 = 119898 Thenwe consider the case 119899 = 119898 which gives the correction to thetwo-point function Γ(2)119899119861 = minusΣ119899119899 when 119905 asymp 119905119888119899 as follows
where we set 119905 asymp (81205871198992) sdot (1 + V119899) This term mainly comesfrom the region where 119905 asymp 119905119888119899 The two-point function up tothis order is
Γ(2)119861 (119901)= 1
1199051205832minus119889119885119905 [1199012 minus 1199012sum
The renormalized two-point function is given as Γ(2)119877 (119901) =119885120601Γ(2)119861 (119901) The renormalization constants are determined asshown above and thus we obtain the same renormalizationgroup equation
5 Renormalization Group Flow
Let us consider the case with two parameters 1205721 and 1205722 Therenormalization group equations are
1205831205971205721120597120583 = minus21205721 (1 minus 18120587119905) (37)
1205831205971205722120597120583 = minus21205722 (1 minus 12120587119905) (38)
120583 120597119905120597120583 = (119889 minus 2) 119905 + 1
3211990512057221 +1811990512057222 (39)
We have the critical value 119905 = 1199051198881 = 8120587 for 1205721 and 119905 = 1199051198882 =2120587 for 1205722 The parameter 119905 is an increasing function of 120583 intwo dimensions 119889 = 2 The space (119905 1205721 1205722) may be dividedinto four regions which are classified by the values to whichthe pair (1205721 1205722) is renormalized as 120583 increases We call themregions I II III and IV
I (1205721 1205722) 997888rarr (infininfin) (40)
II (1205721 1205722) 997888rarr (0infin) (41)
III (1205721 1205722) 997888rarr (infin 0) (42)
IV (1205721 1205722) 997888rarr (0 0) (43)
In region 119905 asymp 8120587 we put 119905 = 8120587(1 + V1) 1199091 = 2V1 1199101 =12057214and 1199102 = 12057222 The equations read
In this region 1199101 acts as a perturbation to the scalingequation of the conventional sine-Gordon model We showthe renormalization group flow as 120583 increases in Figure 3
6 Advances in Mathematical Physics
Ｃ2
1
t＝2 t＝1 t
Figure 3 Renormalization group flow as 120583 997888rarr infin for 1205721 and 1205722
6 Summary
We have discussed the dimensional regularization approachto the renormalization group theory of the generalized sine-Gordon model There are multiple critical points for thecoupling constant 119905 given as 119905119888119899 = 81205871198992 In the case where 119905 isclose to 119905119888119899 for some 119899 the renormalization group equationsare approximated by those for the sine-Gordon model withsingle-cosine potential (conventional sine-Gordon model)A nontrivial simple generalized model is the sine-Gordonmodel with 1205721 and 1205722 When 119905 is 1199051198881 = 8120587 1205722 acts as aperturbation for 119905 and 1205721 The renormalization flow as 120583 997888rarrinfin or 120583 997888rarr 0 depends on an initial set of parameters 119905 1205721and 1205722 This can be viewed as a competition between twointeractions1205721 and1205722 Thismay lead to a generalization of theKosterlitz-Thouless transition the crossover phenomenonin the Kondo effect and other phenomena In the Kondoproblem the appearance of logarithmic singularity [22 5657] suggested the renormalizability of the model Inmaterialswith many magnetic impurities the interaction betweenmagnetic impurities called the RKKY interaction [58ndash60]should be considered In this case the renormalization groupflow is drawn on a two-dimensional plane of two parametersThere may be a relation to the generalized sine-Gordonmodel
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
The author expresses his sincere thanks to K Odagiri foruseful discussionsThis workwas supported in part byGrant-in-Aid from the Ministry of Education Culture SportsScience and Technology (MEXT) of Japan (no 17K05559)
References
[1] S Coleman ldquoQuantum sine-Gordon equation as the massiveThirring modelrdquo Physical Review vol 11 p 2088 1975
[2] R F Dashen B Hasslacher and A Neveu ldquoParticle spectrumin model field theories from semiclassical functional integraltechniquesrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 11 no 12 pp 3424ndash3450 1975
[3] J V Jose L P Kadanoff S Kirkpatrick and D R NelsonldquoRenormalization vortices and symmetry-breaking perturba-tions in the two-dimensional planar modelrdquo Physical Review BCondensed Matter andMaterials Physics vol 16 no 3 pp 1217ndash1241 1977
[4] S Samuel ldquoGrand partition function in field theory withapplications to sine-Gordon field theoryrdquo Physical Review vol18 p 1916 1978
[5] A B Zamolodchikov and A B Zamolodchikov ldquoFactorized S-matrices in two dimensions as the exact solutions of certainrelativistic quantum field theory modelsrdquoAnnals of Physics vol120 no 2 pp 253ndash291 1979
[6] D J Amit Y Y Goldschmidt and G Grinstein ldquoRenormalisa-tion group analysis of the phase transition in the 2D Coulombgas sine-Gordon theory and XY-modelrdquo Journal of Physics AMathematical and General vol 13 no 2 pp 585ndash620 1980
[7] P Weigman ldquoOne-dimensional Fermi system and plane xymodelrdquo Journal of Physics C Solid State Physics vol 11 no 8p 1583 1987
[8] J Balog and A Hegedus ldquoTwo-loop beta functions of the Sine-Gordonmodelrdquo Journal of Physics AMathematical andGeneralvol 33 p 6543 2000
[9] R Rajaraman Solitons and Instantons North-Holland Publish-ing Company Amsterdam The Netherlands 1982
[10] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2004
[11] S Coleman Aspects of Symmetry Cambridge University PressCambridge UK 1985
[12] E C Marino Quantum Field Theory Approach to CondensedMatter Physics Cambridge University Press Cambridge UK2017
[13] E Weinberg Classical Solutions in Quantum Field TheoryCambridge University Press Cambbridge UK 2015
[14] S Mandelstam ldquoSoliton operators for the quantized sine-Gordon equationrdquo Physical Review vol 11 p 3026 1975
[15] B Schroer and T Truong ldquoEquivalence of the sine-Gordon andThirring models and cumulative mass effectsrdquo Physical Reviewvol 15 p 1684 1977
[16] M Faber and A N Ivanov ldquoOn the equivalence between sine-Gordonmodel andThirring model in the chirally broken phaseof theThirringmodelrdquoEuropean Physical Journal vol 20 p 7232001
[17] V I Berezinski ldquoDestruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Con-tinuous Symmetry Group II Quantum Systemsrdquo Journal ofExperimental andTheoretical Physics vol 34 p 610 1972
[18] J M Kosterlitz and D Thouless ldquoOrdering metastabilityand phase transitions in two-dimensional systemsrdquo Journal ofPhysics vol 6 p 1181 1973
[19] J M Kosterlitz ldquoCritical Exponents of the Two-DimensionalXY Modelrdquo Journal of Physics vol 7 p 1046 1974
[20] J Jose ldquoSine-Gordon Theory and the Classical Two-Dimensional xy Modelrdquo Physical Review vol 14 p 28261976
Advances in Mathematical Physics 7
[21] J Zinn-Justin Quantum Field Theory and Critical PhenomenaOxford University Press Oxford UK 1989
[22] J Kondo ldquoResistance Minimum in Dilute Magnetic AlloysrdquoProgress of Theoretical Physics vol 32 no 1 p 34 1964
[23] J Kondo The Physics of Dilute Magnetic Alloys CambridgeUniversity Press Cambridge UK 2012
[24] P W Anderson ldquoA poor manrsquos derivation of scaling laws for theKondo problemrdquo Journal of Physics C Solid State Physics vol 3p 2436 1970
[25] P W Anderson and G Yuval Physical Review Letters vol 23 p89 1969
[26] G Yuval and P W Anderson ldquoExact Results for the KondoProblem One-Body Theory and Extension to Finite Temper-aturerdquo Physical Review vol 1 p 1522 1970
[27] P W Anderson G Yuval and D R Hamann ldquoExact Resultsin the Kondo Problem II ScalingTheory Qualitatively CorrectSolution and SomeNewResults onOne-Dimensional ClassicalStatistical Modelsrdquo Physical Review B vol 1 p 4464 1970
[28] J Solyom ldquoThe Fermi gas model of one-dimensional conduc-torsrdquo Advances in Physics vol 28 no 2 pp 201ndash303 1979
[29] F D N Haldane ldquorsquoLuttinger liquid theoryrsquo of one-dimensionalquantum fluids I Properties of the Luttinger model and theirextension to the general 1D interacting spinless Fermi gasrdquoJournal of Physics vol 14 pp 2585ndash2609 1981
[30] S-T Chui and P A Lee ldquoEquivalence of a One-DimensionalFermion Model and the Two-Dimensional Coulomb GasrdquoPhysical Review Letters vol 35 p 315 1975
[31] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979
[32] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society of London Series A Mathemati-cal and Physical vol 276 no 1365 pp 238ndash257 1963
[33] M C Gutzwiller ldquo Correlation of Electrons in a Narrow rdquoPhysical Review A Atomic Molecular and Optical Physics vol137 no 6A pp A1726ndashA1735 1965
[34] J E Hirsch ldquoMonte Carlo Study of the Two-DimensionalHubbard Modelrdquo Physical Review Letters vol 51 p 1900 1983
[35] S Sorella S Baroni R Car and M Parrinello ldquoA noveltechnique for the simulation of interacting fermion systemsrdquoEurophysics Letters vol 8 p 663 1989
[36] S R White D J Scalapino R L Sugar E Y Loh J EGubernatis and R T Scalettar ldquoNumerical study of the two-dimensional Hubbard modelrdquo Physical Review vol 40 p 5061989
[37] K Yamaji T Yanagisawa T Nakanishi and S Koike ldquoVaria-tional Monte Carlo study on the superconductivity in the two-dimensionalHubbardmodelrdquoPhysica C Superconductivity vol304 p 225 1988
[38] T Yanagisawa and Y Shimoi ldquoExact results in strongly cor-related electrons - Spin-reflection positivity and the Perron-Frobenius theoremrdquo International Journal of Modern Physicsvol 10 p 3383 1996
[39] S Koikegami ldquoVariational Monte Carlo study on the super-conductivity in the two-dimensional Hubbard modelrdquo PhysicalReview vol 67 p 134517 2003
[40] A J Leggett ldquoNumber-Phase Fluctuations in Two-Band Super-conductorsrdquo Progress of Theoretical Physics vol 36 p 901 1966
[41] Y Tanaka and T Yanagisawa ldquoChiral Ground State in Three-Band Superconductorsrdquo Journal of the Physical Society of Japanvol 79 p 114706 2010
[42] Y Tanaka and T Yanagisawa ldquoChiral state in three-gap super-conductorsrdquo Solid State Communications vol 150 no 41-42 pp1980ndash1982 2010
[43] T Yanagisawa Y Tanaka I Hase and K Yamaji ldquoChiral statein three-gap superconductors Solid State Communrdquo Journal ofthe Physical Society of Japan vol 81 p 024712 2012
[44] V Stanev and Z Tesanovic ldquoThree-band superconductivityand the order parameter that breaks time-reversal symmetryrdquoPhysical Review B Condensed Matter andMaterials Physics vol81 Article ID 134522 2010
[45] T Yanagisawa and I Hase ldquoMasslessModes and AbelianGaugeFields in Multi-Band Superconductorsrdquo Journal of the PhysicalSociety of Japan vol 82 p 124704 2013
[46] T Yanagisawa and Y Tanaka ldquoFluctuation-inducedNambundashGoldstone bosons in a HiggsndashJosephson modelrdquoNew Journal of Physics vol 16 p 123014 2014
[47] T Yanagisawa ldquoNambundashGoldstone Bosons Characterized bythe Order Parameter in Spontaneous Symmetry BreakingrdquoJournal of the Physical Society of Japan vol 86 p 104711 2017
[48] T Koyama andMTachikiPhysical ReviewB CondensedMatterand Materials Physics vol 54 no 22 pp 16183ndash16191 1996
[49] T Yanagisawa ldquoChiral sine-GordonmodelrdquoEurophysics Lettersvol 113 p 41001 2016
[50] G rsquotHooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquo Nuclear Physics vol 44 p 189 1972
[51] D Gross ldquoMethods in field theoryrdquo in Les Houches LectureNotes R Balian and J Zinn-Justin Eds North-Holland Pub-lishing Company Amsterdam Netherlands 1976
[52] T Yanagisawa Recent Studies in Perturbation Theory D IUzunov Ed InTech Open Publisher 2017
[53] I Nandori U Jentschura K Sailer and G SoffldquoRenormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for dgtsim3 dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 69 no 2 2004
[54] F J Wegner and A Houghton ldquoRenormalization GroupEquation for Critical Phenomenardquo Physical Review A AtomicMolecular and Optical Physics vol 8 no 1 pp 401ndash412 1973
[55] S Nagy I Nandori J Polonyi and K Sailer ldquoFunctionalRenormalization Group Approach to the Sine-Gordon ModelrdquoPhysical Review Letters vol 102 no 24 2009
[56] Y Nagaoka ldquoSelf-Consistent Treatment of Kondorsquos Effect inDilute AlloysrdquoPhysical ReviewA AtomicMolecular andOpticalPhysics vol 138 no 4A pp A1112ndashA1120 1965
[57] D R Hamann ldquoNewSolution for Exchange Scattering inDiluteAlloysrdquo Physical Review vol 158 p 570 1967
[58] M A Ruderman and C Kittel ldquoIndirect Exchange Coupling ofNuclearMagneticMoments by Conduction Electronsrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 96 no 1pp 99ndash102 1954
[59] T Kasuya ldquoA Theory of Metallic Ferro- and Antiferromag-netism on Zenerrsquos Modelrdquo Progress of Theoretical Physics vol16 p 45 1956
[60] K Yosida ldquoMagnetic Properties of Cu-Mn Alloysrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 106 no5 pp 893ndash898 1957
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2 Advances in Mathematical Physics
the minimal subtraction method Then the beta function forthe coupling constant is derived
2 Lagrangian
Let us consider a real scalar field 120601 The Lagrangian of thegeneralized sine-Gordon model is given by
L = 121199050 (120597120583120601119861)
2 + 11199050sum119899 1205721198990 cos (119899120601119861) (1)
where 120601119861 is a bare real scalar field and 1199050 and 1205720119899 are barecoupling constants The second term indicates the potentialenergy of the scalar field 120601 The generalized sine-Gordonmodel contains high frequency terms such as cos(119899120601) (n =1 2 sdot sdot sdot ) We write the renormalized coupling constants as119905 and 120572119899 respectively We adopt that 119905 gt 0 and 120572119899 ge 0120572119898 for some 119898 may be zero but at least one 120572119899 should bepositive (nonzero) The dimensions of 119905 and 120572119899 are given as[119905] = 1205832minus119889 and [120572119899] = 1205832 where 120583 is a parameter representingthe energy scale The scalar field 120601 is dimensionless Therelations between bare and renormalized quantities are givenby
where119885119905 and119885120572119899 are renormalization constants 119905 and 120572119899 aredimensionless constants by virtue of the energy scale 120583 Wedefine the renormalized field 120601119877 by
120601119861 = radic119885120601120601119877 (3)
where 119885120601 is the renormalization constant for the field 120601 TheLagrangian with renormalized quantities is written as
L = 120583119889minus21198851206012119905119885119905 (120599120583120601)2 + sum
119899
120583119889120572119899119885120572119899119905119885119905 cos (119899radic119885120601120601) (4)
where 120601 denotes the renormalized field 120601119877 The secondterm represents the interaction of the field 120601 as seen byexpanding cos(119899radic119885120601120601) as a power series There is the otherrepresentation of interaction parameters We can absorb theparameter 119905 in the definition of the field 120601 and the parameter120572119899 In this case field 120601 in the interaction term includes theparameter in the form cos(119899radic119885120601120573120601) where 120573 = radic119905 Wewill obtain the same result since it does not depend on therepresentation
+minus minus
Figure 1 One-loop contributions to the renormalization of 120572119899
3 Renormalization of 120572119899We consider the renormalization of 120572119899 up to the lowest orderof 120572119899 By considering tadpole diagrams in Figure 1 the cosinefunction is renormalized to
cos (119899radic119885120601120601)997888rarr (1 minus 1
21198992119885120601 ⟨1206012⟩ + sdot sdot sdot) cos (119899radic119885120601120601)= exp (minus1
21198992119885120601 ⟨1206012⟩) cos (119899radic119885120601120601) (5)
Since the expectation value ⟨1206012⟩ diverges we regularize itusing the dimensional regularization method
119885120601 ⟨1206012⟩ = 1199051205832minus119889119885119905 int 119889119889119896(2120587)119889
11198962 + 11989820 = minus 119905
120598Ω119889
(2120587)119889 (6)
for 119889 = 2 + 120598 where 1198980 is introduced to avoid infrareddivergence Ω119889 is the solid angle in 119889 dimensions and 119885119905 wasput as 1 In order to remove the divergence the constant 119885120572119899is determined as follows
119885120572119899 = 1 minus 11989922 1199051120598
Ω119889(2120587)119889 (7)
Since the bare coupling constant 1205721198990 is independent of 120583 wehave 120583 1205971205721198990120597120583 = 0 This results in
120583120597120572119899120597120583 = minus2120572119899 minus 120572119899120583120597 ln119885120572119899120597120583 (8)
We set 119885119905 = 1 up to the lowest order of 120572119899 so that we have120583 120597119905120597120583 = (119889 minus 2)119905 The beta function for 120572119899 at the lowestorder in 120572119899 is given by
120573(120572119899) has a zero at 119905 = 119905119888119899119905119888119899 = 8120587
1198992 (10)
for 119889 = 2 There is a fixed point of 119905 for each 1198994 Renormalization of 119905There is an effect of renormalization on the coupling constant119905 that is the correction to the kinetic term Let us consider
Advances in Mathematical Physics 3
the two-point function Γ(119901)The bare lowest order two-pointfunction is given by
Γ(0)119861 (119901) = 111990501199012 = 1
1199051205832minus119889119885119905 1199012 (11)
This corresponds to the kinetic part of the bare Lagrangian
L(0)119861 = 120583119889minus2119885120601
2119905119885119905 (120597120583120601)2 (12)
41 Real Space Formulation The lowest order correction tothe two-point function is given by a second-order term for 120572ℓ
(ℓ = 1 2 sdot sdot sdot ) such as120572119899120572119898 cos(119899radic119885120601120601(119909)) cos(119898radic119885120601120601(1199091015840))From the formula cos 1205791 cos 1205792 = (12)(cos(1205791+1205792)+cos(1205791minus1205792)) the correction to the action comes from
(2120587)1198891198700 (11989801003816100381610038161003816119909 minus 1199101003816100381610038161003816)
(15)
where 1198700 is the 0th modified Bessel function Because1198700(119909) increases divergently as 119909 approaches zero 119868119899119898 isapproximated as
cos (radic119885120601 (119899 minus 119898) 120601 (119909)) (1 minus 121198982119885120601 (nabla119909120601
sdot 119903)2)
(16)
where we put 1199091015840 = 119909 + 119903 The cosine functioncos(radic119885120601(119899 minus 119898)120601(119909)) would oscillate as a function of 119909 thecontribution for 119899 = 119898 will be small Thus we consider onlythe contributions with 119899 = 119898
We extract the divergent term in 119868119899119899 There may be two waysto do this We discuss these methods in the following(1) In the first method we regularize ⟨1206012⟩ by introducinga cutoff 119886 in the real space
by replacing 1198700(1198980119903) with 1198700(1198980radic1199032 + 1198862) By using theasymptotic relation 1198700(119911) asymp minus120574 minus ln(1199112) with the Eulerconstant 120574 the integral with respect to 119903 is performed asfollows [49]
near 119889 = 2 where we set 119888 = (1198901205742)2 We consider the casewhere 119905 is close to the critical value 119905119888119899 = 81205871198992
11989921199058120587 = 1 + V119899 (20)
4 Advances in Mathematical Physics
where V119899 represents the deviation from the critical point Inthe lowest order of V119899 we have
119869119899 = Ω119889 (11988811989820)minus2 12 minus 119889 + 119874 (V119899)
asymp minus 1198992321205722119899120583119889+21198864
1120598 int 119889119889119909120583119889minus2119885120601
2119905119885119905 1198992 (120597120583120601)2
+ 119874 (V119899)
(22)
The constant 119885120572119899 was absorbed for the renormalization of 120572119899Then by taking the sum from each term the kinetic partL(0)119861is renormalized to
L(2)119861 = 120583119889minus2119885120601
2119905119885119905 [1 minus sum119899
1198994321205722119899120583119889+21198864
1120598 ] (120597120583120601)2 (23)
This indicates that we choose
119885120601119885119905 = 1 + sum
119899
1198994321205722119899120583119889+21198864
1120598 (24)
119885120601 and 119885119905 appear as a ratio 119885120601119885119905 in this order and then thecoupling constant 119905 is renormalized as 1199050 = 1199051205832minus119889119885119905119885120601 or wecan choose 119885120601 = 1 The equation 120583 1205971199050120597120583 = 0 results in
The numerical coefficient is not important and this dependson the choice of the cutoff 119886(2) In the second way the divergence comes from ⟨1206012⟩where we adopt that the integral with respect to 119903 is finiteThis treatment is similar to that in [31] where the Wilson
+ +
+ ++
Figure 2The contributions to the two-point function Γ(2)(119901) up tothe order of 120572119899120572119898
renormalization group method was used The correction 119868119899119899is written as
by introducing a cutoff in the integral Then we have
119868119899119899 cong minus 1198601198991612058711989941205722119899120583119889+2119886119889+2 1120598 int 119889119889119909120583119889minus21198851206012119905119885119905 (120597120583120601)2 (29)
This results in the same beta function 120573(119905)with the numericalfactor being slightly different
120573 (119905) = (119889 minus 2) 119905 + sum119899
42MomentumSpace Formulation In themomentum spacewe evaluate the two-point function by calculating the dia-grams in Figure 2 [6] This set of diagrams gives the self-energy Σ(119901) Σ(119901) is written as a sum of Σ119899119898(119901) that comesfrom the interaction term cos(119899120601) cos(119898120601) The diagrams inFigure 2 are summed up to give
sdot int 119889119889119909 [119890119894119901sdot119909 (sinh (1198991198981198660 (119909)) minus 1198991198981198660 (119909))minus (cosh (1198991198981198660 (119909)) minus 1)]
sdot int 119889119889119909 (119890119894119901sdot119909 minus 1) exp (1198991198981198660 (119909)) (33)
Using the expansion 119890119894119901sdot119909 = 1 + 119894119901 sdot 119909 minus (12)(119901 sdot 119909)2 + sdot sdot sdot wekeep the 1199012 term By using the formula1198700(119909) asymp minus120574minus log(1199092)for small 119909 Σ119899119898(119901) is written as
The integral diverges when 119905 = 119905119888119899 = 81205871198992 and 119899 = 119898 Thenwe consider the case 119899 = 119898 which gives the correction to thetwo-point function Γ(2)119899119861 = minusΣ119899119899 when 119905 asymp 119905119888119899 as follows
where we set 119905 asymp (81205871198992) sdot (1 + V119899) This term mainly comesfrom the region where 119905 asymp 119905119888119899 The two-point function up tothis order is
Γ(2)119861 (119901)= 1
1199051205832minus119889119885119905 [1199012 minus 1199012sum
The renormalized two-point function is given as Γ(2)119877 (119901) =119885120601Γ(2)119861 (119901) The renormalization constants are determined asshown above and thus we obtain the same renormalizationgroup equation
5 Renormalization Group Flow
Let us consider the case with two parameters 1205721 and 1205722 Therenormalization group equations are
1205831205971205721120597120583 = minus21205721 (1 minus 18120587119905) (37)
1205831205971205722120597120583 = minus21205722 (1 minus 12120587119905) (38)
120583 120597119905120597120583 = (119889 minus 2) 119905 + 1
3211990512057221 +1811990512057222 (39)
We have the critical value 119905 = 1199051198881 = 8120587 for 1205721 and 119905 = 1199051198882 =2120587 for 1205722 The parameter 119905 is an increasing function of 120583 intwo dimensions 119889 = 2 The space (119905 1205721 1205722) may be dividedinto four regions which are classified by the values to whichthe pair (1205721 1205722) is renormalized as 120583 increases We call themregions I II III and IV
I (1205721 1205722) 997888rarr (infininfin) (40)
II (1205721 1205722) 997888rarr (0infin) (41)
III (1205721 1205722) 997888rarr (infin 0) (42)
IV (1205721 1205722) 997888rarr (0 0) (43)
In region 119905 asymp 8120587 we put 119905 = 8120587(1 + V1) 1199091 = 2V1 1199101 =12057214and 1199102 = 12057222 The equations read
In this region 1199101 acts as a perturbation to the scalingequation of the conventional sine-Gordon model We showthe renormalization group flow as 120583 increases in Figure 3
6 Advances in Mathematical Physics
Ｃ2
1
t＝2 t＝1 t
Figure 3 Renormalization group flow as 120583 997888rarr infin for 1205721 and 1205722
6 Summary
We have discussed the dimensional regularization approachto the renormalization group theory of the generalized sine-Gordon model There are multiple critical points for thecoupling constant 119905 given as 119905119888119899 = 81205871198992 In the case where 119905 isclose to 119905119888119899 for some 119899 the renormalization group equationsare approximated by those for the sine-Gordon model withsingle-cosine potential (conventional sine-Gordon model)A nontrivial simple generalized model is the sine-Gordonmodel with 1205721 and 1205722 When 119905 is 1199051198881 = 8120587 1205722 acts as aperturbation for 119905 and 1205721 The renormalization flow as 120583 997888rarrinfin or 120583 997888rarr 0 depends on an initial set of parameters 119905 1205721and 1205722 This can be viewed as a competition between twointeractions1205721 and1205722 Thismay lead to a generalization of theKosterlitz-Thouless transition the crossover phenomenonin the Kondo effect and other phenomena In the Kondoproblem the appearance of logarithmic singularity [22 5657] suggested the renormalizability of the model Inmaterialswith many magnetic impurities the interaction betweenmagnetic impurities called the RKKY interaction [58ndash60]should be considered In this case the renormalization groupflow is drawn on a two-dimensional plane of two parametersThere may be a relation to the generalized sine-Gordonmodel
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
The author expresses his sincere thanks to K Odagiri foruseful discussionsThis workwas supported in part byGrant-in-Aid from the Ministry of Education Culture SportsScience and Technology (MEXT) of Japan (no 17K05559)
References
[1] S Coleman ldquoQuantum sine-Gordon equation as the massiveThirring modelrdquo Physical Review vol 11 p 2088 1975
[2] R F Dashen B Hasslacher and A Neveu ldquoParticle spectrumin model field theories from semiclassical functional integraltechniquesrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 11 no 12 pp 3424ndash3450 1975
[3] J V Jose L P Kadanoff S Kirkpatrick and D R NelsonldquoRenormalization vortices and symmetry-breaking perturba-tions in the two-dimensional planar modelrdquo Physical Review BCondensed Matter andMaterials Physics vol 16 no 3 pp 1217ndash1241 1977
[4] S Samuel ldquoGrand partition function in field theory withapplications to sine-Gordon field theoryrdquo Physical Review vol18 p 1916 1978
[5] A B Zamolodchikov and A B Zamolodchikov ldquoFactorized S-matrices in two dimensions as the exact solutions of certainrelativistic quantum field theory modelsrdquoAnnals of Physics vol120 no 2 pp 253ndash291 1979
[6] D J Amit Y Y Goldschmidt and G Grinstein ldquoRenormalisa-tion group analysis of the phase transition in the 2D Coulombgas sine-Gordon theory and XY-modelrdquo Journal of Physics AMathematical and General vol 13 no 2 pp 585ndash620 1980
[7] P Weigman ldquoOne-dimensional Fermi system and plane xymodelrdquo Journal of Physics C Solid State Physics vol 11 no 8p 1583 1987
[8] J Balog and A Hegedus ldquoTwo-loop beta functions of the Sine-Gordonmodelrdquo Journal of Physics AMathematical andGeneralvol 33 p 6543 2000
[9] R Rajaraman Solitons and Instantons North-Holland Publish-ing Company Amsterdam The Netherlands 1982
[10] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2004
[11] S Coleman Aspects of Symmetry Cambridge University PressCambridge UK 1985
[12] E C Marino Quantum Field Theory Approach to CondensedMatter Physics Cambridge University Press Cambridge UK2017
[13] E Weinberg Classical Solutions in Quantum Field TheoryCambridge University Press Cambbridge UK 2015
[14] S Mandelstam ldquoSoliton operators for the quantized sine-Gordon equationrdquo Physical Review vol 11 p 3026 1975
[15] B Schroer and T Truong ldquoEquivalence of the sine-Gordon andThirring models and cumulative mass effectsrdquo Physical Reviewvol 15 p 1684 1977
[16] M Faber and A N Ivanov ldquoOn the equivalence between sine-Gordonmodel andThirring model in the chirally broken phaseof theThirringmodelrdquoEuropean Physical Journal vol 20 p 7232001
[17] V I Berezinski ldquoDestruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Con-tinuous Symmetry Group II Quantum Systemsrdquo Journal ofExperimental andTheoretical Physics vol 34 p 610 1972
[18] J M Kosterlitz and D Thouless ldquoOrdering metastabilityand phase transitions in two-dimensional systemsrdquo Journal ofPhysics vol 6 p 1181 1973
[19] J M Kosterlitz ldquoCritical Exponents of the Two-DimensionalXY Modelrdquo Journal of Physics vol 7 p 1046 1974
[20] J Jose ldquoSine-Gordon Theory and the Classical Two-Dimensional xy Modelrdquo Physical Review vol 14 p 28261976
Advances in Mathematical Physics 7
[21] J Zinn-Justin Quantum Field Theory and Critical PhenomenaOxford University Press Oxford UK 1989
[22] J Kondo ldquoResistance Minimum in Dilute Magnetic AlloysrdquoProgress of Theoretical Physics vol 32 no 1 p 34 1964
[23] J Kondo The Physics of Dilute Magnetic Alloys CambridgeUniversity Press Cambridge UK 2012
[24] P W Anderson ldquoA poor manrsquos derivation of scaling laws for theKondo problemrdquo Journal of Physics C Solid State Physics vol 3p 2436 1970
[25] P W Anderson and G Yuval Physical Review Letters vol 23 p89 1969
[26] G Yuval and P W Anderson ldquoExact Results for the KondoProblem One-Body Theory and Extension to Finite Temper-aturerdquo Physical Review vol 1 p 1522 1970
[27] P W Anderson G Yuval and D R Hamann ldquoExact Resultsin the Kondo Problem II ScalingTheory Qualitatively CorrectSolution and SomeNewResults onOne-Dimensional ClassicalStatistical Modelsrdquo Physical Review B vol 1 p 4464 1970
[28] J Solyom ldquoThe Fermi gas model of one-dimensional conduc-torsrdquo Advances in Physics vol 28 no 2 pp 201ndash303 1979
[29] F D N Haldane ldquorsquoLuttinger liquid theoryrsquo of one-dimensionalquantum fluids I Properties of the Luttinger model and theirextension to the general 1D interacting spinless Fermi gasrdquoJournal of Physics vol 14 pp 2585ndash2609 1981
[30] S-T Chui and P A Lee ldquoEquivalence of a One-DimensionalFermion Model and the Two-Dimensional Coulomb GasrdquoPhysical Review Letters vol 35 p 315 1975
[31] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979
[32] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society of London Series A Mathemati-cal and Physical vol 276 no 1365 pp 238ndash257 1963
[33] M C Gutzwiller ldquo Correlation of Electrons in a Narrow rdquoPhysical Review A Atomic Molecular and Optical Physics vol137 no 6A pp A1726ndashA1735 1965
[34] J E Hirsch ldquoMonte Carlo Study of the Two-DimensionalHubbard Modelrdquo Physical Review Letters vol 51 p 1900 1983
[35] S Sorella S Baroni R Car and M Parrinello ldquoA noveltechnique for the simulation of interacting fermion systemsrdquoEurophysics Letters vol 8 p 663 1989
[36] S R White D J Scalapino R L Sugar E Y Loh J EGubernatis and R T Scalettar ldquoNumerical study of the two-dimensional Hubbard modelrdquo Physical Review vol 40 p 5061989
[37] K Yamaji T Yanagisawa T Nakanishi and S Koike ldquoVaria-tional Monte Carlo study on the superconductivity in the two-dimensionalHubbardmodelrdquoPhysica C Superconductivity vol304 p 225 1988
[38] T Yanagisawa and Y Shimoi ldquoExact results in strongly cor-related electrons - Spin-reflection positivity and the Perron-Frobenius theoremrdquo International Journal of Modern Physicsvol 10 p 3383 1996
[39] S Koikegami ldquoVariational Monte Carlo study on the super-conductivity in the two-dimensional Hubbard modelrdquo PhysicalReview vol 67 p 134517 2003
[40] A J Leggett ldquoNumber-Phase Fluctuations in Two-Band Super-conductorsrdquo Progress of Theoretical Physics vol 36 p 901 1966
[41] Y Tanaka and T Yanagisawa ldquoChiral Ground State in Three-Band Superconductorsrdquo Journal of the Physical Society of Japanvol 79 p 114706 2010
[42] Y Tanaka and T Yanagisawa ldquoChiral state in three-gap super-conductorsrdquo Solid State Communications vol 150 no 41-42 pp1980ndash1982 2010
[43] T Yanagisawa Y Tanaka I Hase and K Yamaji ldquoChiral statein three-gap superconductors Solid State Communrdquo Journal ofthe Physical Society of Japan vol 81 p 024712 2012
[44] V Stanev and Z Tesanovic ldquoThree-band superconductivityand the order parameter that breaks time-reversal symmetryrdquoPhysical Review B Condensed Matter andMaterials Physics vol81 Article ID 134522 2010
[45] T Yanagisawa and I Hase ldquoMasslessModes and AbelianGaugeFields in Multi-Band Superconductorsrdquo Journal of the PhysicalSociety of Japan vol 82 p 124704 2013
[46] T Yanagisawa and Y Tanaka ldquoFluctuation-inducedNambundashGoldstone bosons in a HiggsndashJosephson modelrdquoNew Journal of Physics vol 16 p 123014 2014
[47] T Yanagisawa ldquoNambundashGoldstone Bosons Characterized bythe Order Parameter in Spontaneous Symmetry BreakingrdquoJournal of the Physical Society of Japan vol 86 p 104711 2017
[48] T Koyama andMTachikiPhysical ReviewB CondensedMatterand Materials Physics vol 54 no 22 pp 16183ndash16191 1996
[49] T Yanagisawa ldquoChiral sine-GordonmodelrdquoEurophysics Lettersvol 113 p 41001 2016
[50] G rsquotHooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquo Nuclear Physics vol 44 p 189 1972
[51] D Gross ldquoMethods in field theoryrdquo in Les Houches LectureNotes R Balian and J Zinn-Justin Eds North-Holland Pub-lishing Company Amsterdam Netherlands 1976
[52] T Yanagisawa Recent Studies in Perturbation Theory D IUzunov Ed InTech Open Publisher 2017
[53] I Nandori U Jentschura K Sailer and G SoffldquoRenormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for dgtsim3 dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 69 no 2 2004
[54] F J Wegner and A Houghton ldquoRenormalization GroupEquation for Critical Phenomenardquo Physical Review A AtomicMolecular and Optical Physics vol 8 no 1 pp 401ndash412 1973
[55] S Nagy I Nandori J Polonyi and K Sailer ldquoFunctionalRenormalization Group Approach to the Sine-Gordon ModelrdquoPhysical Review Letters vol 102 no 24 2009
[56] Y Nagaoka ldquoSelf-Consistent Treatment of Kondorsquos Effect inDilute AlloysrdquoPhysical ReviewA AtomicMolecular andOpticalPhysics vol 138 no 4A pp A1112ndashA1120 1965
[57] D R Hamann ldquoNewSolution for Exchange Scattering inDiluteAlloysrdquo Physical Review vol 158 p 570 1967
[58] M A Ruderman and C Kittel ldquoIndirect Exchange Coupling ofNuclearMagneticMoments by Conduction Electronsrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 96 no 1pp 99ndash102 1954
[59] T Kasuya ldquoA Theory of Metallic Ferro- and Antiferromag-netism on Zenerrsquos Modelrdquo Progress of Theoretical Physics vol16 p 45 1956
[60] K Yosida ldquoMagnetic Properties of Cu-Mn Alloysrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 106 no5 pp 893ndash898 1957
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Advances in Mathematical Physics 3
the two-point function Γ(119901)The bare lowest order two-pointfunction is given by
Γ(0)119861 (119901) = 111990501199012 = 1
1199051205832minus119889119885119905 1199012 (11)
This corresponds to the kinetic part of the bare Lagrangian
L(0)119861 = 120583119889minus2119885120601
2119905119885119905 (120597120583120601)2 (12)
41 Real Space Formulation The lowest order correction tothe two-point function is given by a second-order term for 120572ℓ
(ℓ = 1 2 sdot sdot sdot ) such as120572119899120572119898 cos(119899radic119885120601120601(119909)) cos(119898radic119885120601120601(1199091015840))From the formula cos 1205791 cos 1205792 = (12)(cos(1205791+1205792)+cos(1205791minus1205792)) the correction to the action comes from
(2120587)1198891198700 (11989801003816100381610038161003816119909 minus 1199101003816100381610038161003816)
(15)
where 1198700 is the 0th modified Bessel function Because1198700(119909) increases divergently as 119909 approaches zero 119868119899119898 isapproximated as
cos (radic119885120601 (119899 minus 119898) 120601 (119909)) (1 minus 121198982119885120601 (nabla119909120601
sdot 119903)2)
(16)
where we put 1199091015840 = 119909 + 119903 The cosine functioncos(radic119885120601(119899 minus 119898)120601(119909)) would oscillate as a function of 119909 thecontribution for 119899 = 119898 will be small Thus we consider onlythe contributions with 119899 = 119898
We extract the divergent term in 119868119899119899 There may be two waysto do this We discuss these methods in the following(1) In the first method we regularize ⟨1206012⟩ by introducinga cutoff 119886 in the real space
by replacing 1198700(1198980119903) with 1198700(1198980radic1199032 + 1198862) By using theasymptotic relation 1198700(119911) asymp minus120574 minus ln(1199112) with the Eulerconstant 120574 the integral with respect to 119903 is performed asfollows [49]
near 119889 = 2 where we set 119888 = (1198901205742)2 We consider the casewhere 119905 is close to the critical value 119905119888119899 = 81205871198992
11989921199058120587 = 1 + V119899 (20)
4 Advances in Mathematical Physics
where V119899 represents the deviation from the critical point Inthe lowest order of V119899 we have
119869119899 = Ω119889 (11988811989820)minus2 12 minus 119889 + 119874 (V119899)
asymp minus 1198992321205722119899120583119889+21198864
1120598 int 119889119889119909120583119889minus2119885120601
2119905119885119905 1198992 (120597120583120601)2
+ 119874 (V119899)
(22)
The constant 119885120572119899 was absorbed for the renormalization of 120572119899Then by taking the sum from each term the kinetic partL(0)119861is renormalized to
L(2)119861 = 120583119889minus2119885120601
2119905119885119905 [1 minus sum119899
1198994321205722119899120583119889+21198864
1120598 ] (120597120583120601)2 (23)
This indicates that we choose
119885120601119885119905 = 1 + sum
119899
1198994321205722119899120583119889+21198864
1120598 (24)
119885120601 and 119885119905 appear as a ratio 119885120601119885119905 in this order and then thecoupling constant 119905 is renormalized as 1199050 = 1199051205832minus119889119885119905119885120601 or wecan choose 119885120601 = 1 The equation 120583 1205971199050120597120583 = 0 results in
The numerical coefficient is not important and this dependson the choice of the cutoff 119886(2) In the second way the divergence comes from ⟨1206012⟩where we adopt that the integral with respect to 119903 is finiteThis treatment is similar to that in [31] where the Wilson
+ +
+ ++
Figure 2The contributions to the two-point function Γ(2)(119901) up tothe order of 120572119899120572119898
renormalization group method was used The correction 119868119899119899is written as
by introducing a cutoff in the integral Then we have
119868119899119899 cong minus 1198601198991612058711989941205722119899120583119889+2119886119889+2 1120598 int 119889119889119909120583119889minus21198851206012119905119885119905 (120597120583120601)2 (29)
This results in the same beta function 120573(119905)with the numericalfactor being slightly different
120573 (119905) = (119889 minus 2) 119905 + sum119899
42MomentumSpace Formulation In themomentum spacewe evaluate the two-point function by calculating the dia-grams in Figure 2 [6] This set of diagrams gives the self-energy Σ(119901) Σ(119901) is written as a sum of Σ119899119898(119901) that comesfrom the interaction term cos(119899120601) cos(119898120601) The diagrams inFigure 2 are summed up to give
sdot int 119889119889119909 [119890119894119901sdot119909 (sinh (1198991198981198660 (119909)) minus 1198991198981198660 (119909))minus (cosh (1198991198981198660 (119909)) minus 1)]
sdot int 119889119889119909 (119890119894119901sdot119909 minus 1) exp (1198991198981198660 (119909)) (33)
Using the expansion 119890119894119901sdot119909 = 1 + 119894119901 sdot 119909 minus (12)(119901 sdot 119909)2 + sdot sdot sdot wekeep the 1199012 term By using the formula1198700(119909) asymp minus120574minus log(1199092)for small 119909 Σ119899119898(119901) is written as
The integral diverges when 119905 = 119905119888119899 = 81205871198992 and 119899 = 119898 Thenwe consider the case 119899 = 119898 which gives the correction to thetwo-point function Γ(2)119899119861 = minusΣ119899119899 when 119905 asymp 119905119888119899 as follows
where we set 119905 asymp (81205871198992) sdot (1 + V119899) This term mainly comesfrom the region where 119905 asymp 119905119888119899 The two-point function up tothis order is
Γ(2)119861 (119901)= 1
1199051205832minus119889119885119905 [1199012 minus 1199012sum
The renormalized two-point function is given as Γ(2)119877 (119901) =119885120601Γ(2)119861 (119901) The renormalization constants are determined asshown above and thus we obtain the same renormalizationgroup equation
5 Renormalization Group Flow
Let us consider the case with two parameters 1205721 and 1205722 Therenormalization group equations are
1205831205971205721120597120583 = minus21205721 (1 minus 18120587119905) (37)
1205831205971205722120597120583 = minus21205722 (1 minus 12120587119905) (38)
120583 120597119905120597120583 = (119889 minus 2) 119905 + 1
3211990512057221 +1811990512057222 (39)
We have the critical value 119905 = 1199051198881 = 8120587 for 1205721 and 119905 = 1199051198882 =2120587 for 1205722 The parameter 119905 is an increasing function of 120583 intwo dimensions 119889 = 2 The space (119905 1205721 1205722) may be dividedinto four regions which are classified by the values to whichthe pair (1205721 1205722) is renormalized as 120583 increases We call themregions I II III and IV
I (1205721 1205722) 997888rarr (infininfin) (40)
II (1205721 1205722) 997888rarr (0infin) (41)
III (1205721 1205722) 997888rarr (infin 0) (42)
IV (1205721 1205722) 997888rarr (0 0) (43)
In region 119905 asymp 8120587 we put 119905 = 8120587(1 + V1) 1199091 = 2V1 1199101 =12057214and 1199102 = 12057222 The equations read
In this region 1199101 acts as a perturbation to the scalingequation of the conventional sine-Gordon model We showthe renormalization group flow as 120583 increases in Figure 3
6 Advances in Mathematical Physics
Ｃ2
1
t＝2 t＝1 t
Figure 3 Renormalization group flow as 120583 997888rarr infin for 1205721 and 1205722
6 Summary
We have discussed the dimensional regularization approachto the renormalization group theory of the generalized sine-Gordon model There are multiple critical points for thecoupling constant 119905 given as 119905119888119899 = 81205871198992 In the case where 119905 isclose to 119905119888119899 for some 119899 the renormalization group equationsare approximated by those for the sine-Gordon model withsingle-cosine potential (conventional sine-Gordon model)A nontrivial simple generalized model is the sine-Gordonmodel with 1205721 and 1205722 When 119905 is 1199051198881 = 8120587 1205722 acts as aperturbation for 119905 and 1205721 The renormalization flow as 120583 997888rarrinfin or 120583 997888rarr 0 depends on an initial set of parameters 119905 1205721and 1205722 This can be viewed as a competition between twointeractions1205721 and1205722 Thismay lead to a generalization of theKosterlitz-Thouless transition the crossover phenomenonin the Kondo effect and other phenomena In the Kondoproblem the appearance of logarithmic singularity [22 5657] suggested the renormalizability of the model Inmaterialswith many magnetic impurities the interaction betweenmagnetic impurities called the RKKY interaction [58ndash60]should be considered In this case the renormalization groupflow is drawn on a two-dimensional plane of two parametersThere may be a relation to the generalized sine-Gordonmodel
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
The author expresses his sincere thanks to K Odagiri foruseful discussionsThis workwas supported in part byGrant-in-Aid from the Ministry of Education Culture SportsScience and Technology (MEXT) of Japan (no 17K05559)
References
[1] S Coleman ldquoQuantum sine-Gordon equation as the massiveThirring modelrdquo Physical Review vol 11 p 2088 1975
[2] R F Dashen B Hasslacher and A Neveu ldquoParticle spectrumin model field theories from semiclassical functional integraltechniquesrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 11 no 12 pp 3424ndash3450 1975
[3] J V Jose L P Kadanoff S Kirkpatrick and D R NelsonldquoRenormalization vortices and symmetry-breaking perturba-tions in the two-dimensional planar modelrdquo Physical Review BCondensed Matter andMaterials Physics vol 16 no 3 pp 1217ndash1241 1977
[4] S Samuel ldquoGrand partition function in field theory withapplications to sine-Gordon field theoryrdquo Physical Review vol18 p 1916 1978
[5] A B Zamolodchikov and A B Zamolodchikov ldquoFactorized S-matrices in two dimensions as the exact solutions of certainrelativistic quantum field theory modelsrdquoAnnals of Physics vol120 no 2 pp 253ndash291 1979
[6] D J Amit Y Y Goldschmidt and G Grinstein ldquoRenormalisa-tion group analysis of the phase transition in the 2D Coulombgas sine-Gordon theory and XY-modelrdquo Journal of Physics AMathematical and General vol 13 no 2 pp 585ndash620 1980
[7] P Weigman ldquoOne-dimensional Fermi system and plane xymodelrdquo Journal of Physics C Solid State Physics vol 11 no 8p 1583 1987
[8] J Balog and A Hegedus ldquoTwo-loop beta functions of the Sine-Gordonmodelrdquo Journal of Physics AMathematical andGeneralvol 33 p 6543 2000
[9] R Rajaraman Solitons and Instantons North-Holland Publish-ing Company Amsterdam The Netherlands 1982
[10] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2004
[11] S Coleman Aspects of Symmetry Cambridge University PressCambridge UK 1985
[12] E C Marino Quantum Field Theory Approach to CondensedMatter Physics Cambridge University Press Cambridge UK2017
[13] E Weinberg Classical Solutions in Quantum Field TheoryCambridge University Press Cambbridge UK 2015
[14] S Mandelstam ldquoSoliton operators for the quantized sine-Gordon equationrdquo Physical Review vol 11 p 3026 1975
[15] B Schroer and T Truong ldquoEquivalence of the sine-Gordon andThirring models and cumulative mass effectsrdquo Physical Reviewvol 15 p 1684 1977
[16] M Faber and A N Ivanov ldquoOn the equivalence between sine-Gordonmodel andThirring model in the chirally broken phaseof theThirringmodelrdquoEuropean Physical Journal vol 20 p 7232001
[17] V I Berezinski ldquoDestruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Con-tinuous Symmetry Group II Quantum Systemsrdquo Journal ofExperimental andTheoretical Physics vol 34 p 610 1972
[18] J M Kosterlitz and D Thouless ldquoOrdering metastabilityand phase transitions in two-dimensional systemsrdquo Journal ofPhysics vol 6 p 1181 1973
[19] J M Kosterlitz ldquoCritical Exponents of the Two-DimensionalXY Modelrdquo Journal of Physics vol 7 p 1046 1974
[20] J Jose ldquoSine-Gordon Theory and the Classical Two-Dimensional xy Modelrdquo Physical Review vol 14 p 28261976
Advances in Mathematical Physics 7
[21] J Zinn-Justin Quantum Field Theory and Critical PhenomenaOxford University Press Oxford UK 1989
[22] J Kondo ldquoResistance Minimum in Dilute Magnetic AlloysrdquoProgress of Theoretical Physics vol 32 no 1 p 34 1964
[23] J Kondo The Physics of Dilute Magnetic Alloys CambridgeUniversity Press Cambridge UK 2012
[24] P W Anderson ldquoA poor manrsquos derivation of scaling laws for theKondo problemrdquo Journal of Physics C Solid State Physics vol 3p 2436 1970
[25] P W Anderson and G Yuval Physical Review Letters vol 23 p89 1969
[26] G Yuval and P W Anderson ldquoExact Results for the KondoProblem One-Body Theory and Extension to Finite Temper-aturerdquo Physical Review vol 1 p 1522 1970
[27] P W Anderson G Yuval and D R Hamann ldquoExact Resultsin the Kondo Problem II ScalingTheory Qualitatively CorrectSolution and SomeNewResults onOne-Dimensional ClassicalStatistical Modelsrdquo Physical Review B vol 1 p 4464 1970
[28] J Solyom ldquoThe Fermi gas model of one-dimensional conduc-torsrdquo Advances in Physics vol 28 no 2 pp 201ndash303 1979
[29] F D N Haldane ldquorsquoLuttinger liquid theoryrsquo of one-dimensionalquantum fluids I Properties of the Luttinger model and theirextension to the general 1D interacting spinless Fermi gasrdquoJournal of Physics vol 14 pp 2585ndash2609 1981
[30] S-T Chui and P A Lee ldquoEquivalence of a One-DimensionalFermion Model and the Two-Dimensional Coulomb GasrdquoPhysical Review Letters vol 35 p 315 1975
[31] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979
[32] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society of London Series A Mathemati-cal and Physical vol 276 no 1365 pp 238ndash257 1963
[33] M C Gutzwiller ldquo Correlation of Electrons in a Narrow rdquoPhysical Review A Atomic Molecular and Optical Physics vol137 no 6A pp A1726ndashA1735 1965
[34] J E Hirsch ldquoMonte Carlo Study of the Two-DimensionalHubbard Modelrdquo Physical Review Letters vol 51 p 1900 1983
[35] S Sorella S Baroni R Car and M Parrinello ldquoA noveltechnique for the simulation of interacting fermion systemsrdquoEurophysics Letters vol 8 p 663 1989
[36] S R White D J Scalapino R L Sugar E Y Loh J EGubernatis and R T Scalettar ldquoNumerical study of the two-dimensional Hubbard modelrdquo Physical Review vol 40 p 5061989
[37] K Yamaji T Yanagisawa T Nakanishi and S Koike ldquoVaria-tional Monte Carlo study on the superconductivity in the two-dimensionalHubbardmodelrdquoPhysica C Superconductivity vol304 p 225 1988
[38] T Yanagisawa and Y Shimoi ldquoExact results in strongly cor-related electrons - Spin-reflection positivity and the Perron-Frobenius theoremrdquo International Journal of Modern Physicsvol 10 p 3383 1996
[39] S Koikegami ldquoVariational Monte Carlo study on the super-conductivity in the two-dimensional Hubbard modelrdquo PhysicalReview vol 67 p 134517 2003
[40] A J Leggett ldquoNumber-Phase Fluctuations in Two-Band Super-conductorsrdquo Progress of Theoretical Physics vol 36 p 901 1966
[41] Y Tanaka and T Yanagisawa ldquoChiral Ground State in Three-Band Superconductorsrdquo Journal of the Physical Society of Japanvol 79 p 114706 2010
[42] Y Tanaka and T Yanagisawa ldquoChiral state in three-gap super-conductorsrdquo Solid State Communications vol 150 no 41-42 pp1980ndash1982 2010
[43] T Yanagisawa Y Tanaka I Hase and K Yamaji ldquoChiral statein three-gap superconductors Solid State Communrdquo Journal ofthe Physical Society of Japan vol 81 p 024712 2012
[44] V Stanev and Z Tesanovic ldquoThree-band superconductivityand the order parameter that breaks time-reversal symmetryrdquoPhysical Review B Condensed Matter andMaterials Physics vol81 Article ID 134522 2010
[45] T Yanagisawa and I Hase ldquoMasslessModes and AbelianGaugeFields in Multi-Band Superconductorsrdquo Journal of the PhysicalSociety of Japan vol 82 p 124704 2013
[46] T Yanagisawa and Y Tanaka ldquoFluctuation-inducedNambundashGoldstone bosons in a HiggsndashJosephson modelrdquoNew Journal of Physics vol 16 p 123014 2014
[47] T Yanagisawa ldquoNambundashGoldstone Bosons Characterized bythe Order Parameter in Spontaneous Symmetry BreakingrdquoJournal of the Physical Society of Japan vol 86 p 104711 2017
[48] T Koyama andMTachikiPhysical ReviewB CondensedMatterand Materials Physics vol 54 no 22 pp 16183ndash16191 1996
[49] T Yanagisawa ldquoChiral sine-GordonmodelrdquoEurophysics Lettersvol 113 p 41001 2016
[50] G rsquotHooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquo Nuclear Physics vol 44 p 189 1972
[51] D Gross ldquoMethods in field theoryrdquo in Les Houches LectureNotes R Balian and J Zinn-Justin Eds North-Holland Pub-lishing Company Amsterdam Netherlands 1976
[52] T Yanagisawa Recent Studies in Perturbation Theory D IUzunov Ed InTech Open Publisher 2017
[53] I Nandori U Jentschura K Sailer and G SoffldquoRenormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for dgtsim3 dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 69 no 2 2004
[54] F J Wegner and A Houghton ldquoRenormalization GroupEquation for Critical Phenomenardquo Physical Review A AtomicMolecular and Optical Physics vol 8 no 1 pp 401ndash412 1973
[55] S Nagy I Nandori J Polonyi and K Sailer ldquoFunctionalRenormalization Group Approach to the Sine-Gordon ModelrdquoPhysical Review Letters vol 102 no 24 2009
[56] Y Nagaoka ldquoSelf-Consistent Treatment of Kondorsquos Effect inDilute AlloysrdquoPhysical ReviewA AtomicMolecular andOpticalPhysics vol 138 no 4A pp A1112ndashA1120 1965
[57] D R Hamann ldquoNewSolution for Exchange Scattering inDiluteAlloysrdquo Physical Review vol 158 p 570 1967
[58] M A Ruderman and C Kittel ldquoIndirect Exchange Coupling ofNuclearMagneticMoments by Conduction Electronsrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 96 no 1pp 99ndash102 1954
[59] T Kasuya ldquoA Theory of Metallic Ferro- and Antiferromag-netism on Zenerrsquos Modelrdquo Progress of Theoretical Physics vol16 p 45 1956
[60] K Yosida ldquoMagnetic Properties of Cu-Mn Alloysrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 106 no5 pp 893ndash898 1957
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
asymp minus 1198992321205722119899120583119889+21198864
1120598 int 119889119889119909120583119889minus2119885120601
2119905119885119905 1198992 (120597120583120601)2
+ 119874 (V119899)
(22)
The constant 119885120572119899 was absorbed for the renormalization of 120572119899Then by taking the sum from each term the kinetic partL(0)119861is renormalized to
L(2)119861 = 120583119889minus2119885120601
2119905119885119905 [1 minus sum119899
1198994321205722119899120583119889+21198864
1120598 ] (120597120583120601)2 (23)
This indicates that we choose
119885120601119885119905 = 1 + sum
119899
1198994321205722119899120583119889+21198864
1120598 (24)
119885120601 and 119885119905 appear as a ratio 119885120601119885119905 in this order and then thecoupling constant 119905 is renormalized as 1199050 = 1199051205832minus119889119885119905119885120601 or wecan choose 119885120601 = 1 The equation 120583 1205971199050120597120583 = 0 results in
The numerical coefficient is not important and this dependson the choice of the cutoff 119886(2) In the second way the divergence comes from ⟨1206012⟩where we adopt that the integral with respect to 119903 is finiteThis treatment is similar to that in [31] where the Wilson
+ +
+ ++
Figure 2The contributions to the two-point function Γ(2)(119901) up tothe order of 120572119899120572119898
renormalization group method was used The correction 119868119899119899is written as
by introducing a cutoff in the integral Then we have
119868119899119899 cong minus 1198601198991612058711989941205722119899120583119889+2119886119889+2 1120598 int 119889119889119909120583119889minus21198851206012119905119885119905 (120597120583120601)2 (29)
This results in the same beta function 120573(119905)with the numericalfactor being slightly different
120573 (119905) = (119889 minus 2) 119905 + sum119899
42MomentumSpace Formulation In themomentum spacewe evaluate the two-point function by calculating the dia-grams in Figure 2 [6] This set of diagrams gives the self-energy Σ(119901) Σ(119901) is written as a sum of Σ119899119898(119901) that comesfrom the interaction term cos(119899120601) cos(119898120601) The diagrams inFigure 2 are summed up to give
sdot int 119889119889119909 [119890119894119901sdot119909 (sinh (1198991198981198660 (119909)) minus 1198991198981198660 (119909))minus (cosh (1198991198981198660 (119909)) minus 1)]
sdot int 119889119889119909 (119890119894119901sdot119909 minus 1) exp (1198991198981198660 (119909)) (33)
Using the expansion 119890119894119901sdot119909 = 1 + 119894119901 sdot 119909 minus (12)(119901 sdot 119909)2 + sdot sdot sdot wekeep the 1199012 term By using the formula1198700(119909) asymp minus120574minus log(1199092)for small 119909 Σ119899119898(119901) is written as
The integral diverges when 119905 = 119905119888119899 = 81205871198992 and 119899 = 119898 Thenwe consider the case 119899 = 119898 which gives the correction to thetwo-point function Γ(2)119899119861 = minusΣ119899119899 when 119905 asymp 119905119888119899 as follows
where we set 119905 asymp (81205871198992) sdot (1 + V119899) This term mainly comesfrom the region where 119905 asymp 119905119888119899 The two-point function up tothis order is
Γ(2)119861 (119901)= 1
1199051205832minus119889119885119905 [1199012 minus 1199012sum
The renormalized two-point function is given as Γ(2)119877 (119901) =119885120601Γ(2)119861 (119901) The renormalization constants are determined asshown above and thus we obtain the same renormalizationgroup equation
5 Renormalization Group Flow
Let us consider the case with two parameters 1205721 and 1205722 Therenormalization group equations are
1205831205971205721120597120583 = minus21205721 (1 minus 18120587119905) (37)
1205831205971205722120597120583 = minus21205722 (1 minus 12120587119905) (38)
120583 120597119905120597120583 = (119889 minus 2) 119905 + 1
3211990512057221 +1811990512057222 (39)
We have the critical value 119905 = 1199051198881 = 8120587 for 1205721 and 119905 = 1199051198882 =2120587 for 1205722 The parameter 119905 is an increasing function of 120583 intwo dimensions 119889 = 2 The space (119905 1205721 1205722) may be dividedinto four regions which are classified by the values to whichthe pair (1205721 1205722) is renormalized as 120583 increases We call themregions I II III and IV
I (1205721 1205722) 997888rarr (infininfin) (40)
II (1205721 1205722) 997888rarr (0infin) (41)
III (1205721 1205722) 997888rarr (infin 0) (42)
IV (1205721 1205722) 997888rarr (0 0) (43)
In region 119905 asymp 8120587 we put 119905 = 8120587(1 + V1) 1199091 = 2V1 1199101 =12057214and 1199102 = 12057222 The equations read
In this region 1199101 acts as a perturbation to the scalingequation of the conventional sine-Gordon model We showthe renormalization group flow as 120583 increases in Figure 3
6 Advances in Mathematical Physics
Ｃ2
1
t＝2 t＝1 t
Figure 3 Renormalization group flow as 120583 997888rarr infin for 1205721 and 1205722
6 Summary
We have discussed the dimensional regularization approachto the renormalization group theory of the generalized sine-Gordon model There are multiple critical points for thecoupling constant 119905 given as 119905119888119899 = 81205871198992 In the case where 119905 isclose to 119905119888119899 for some 119899 the renormalization group equationsare approximated by those for the sine-Gordon model withsingle-cosine potential (conventional sine-Gordon model)A nontrivial simple generalized model is the sine-Gordonmodel with 1205721 and 1205722 When 119905 is 1199051198881 = 8120587 1205722 acts as aperturbation for 119905 and 1205721 The renormalization flow as 120583 997888rarrinfin or 120583 997888rarr 0 depends on an initial set of parameters 119905 1205721and 1205722 This can be viewed as a competition between twointeractions1205721 and1205722 Thismay lead to a generalization of theKosterlitz-Thouless transition the crossover phenomenonin the Kondo effect and other phenomena In the Kondoproblem the appearance of logarithmic singularity [22 5657] suggested the renormalizability of the model Inmaterialswith many magnetic impurities the interaction betweenmagnetic impurities called the RKKY interaction [58ndash60]should be considered In this case the renormalization groupflow is drawn on a two-dimensional plane of two parametersThere may be a relation to the generalized sine-Gordonmodel
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
The author expresses his sincere thanks to K Odagiri foruseful discussionsThis workwas supported in part byGrant-in-Aid from the Ministry of Education Culture SportsScience and Technology (MEXT) of Japan (no 17K05559)
References
[1] S Coleman ldquoQuantum sine-Gordon equation as the massiveThirring modelrdquo Physical Review vol 11 p 2088 1975
[2] R F Dashen B Hasslacher and A Neveu ldquoParticle spectrumin model field theories from semiclassical functional integraltechniquesrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 11 no 12 pp 3424ndash3450 1975
[3] J V Jose L P Kadanoff S Kirkpatrick and D R NelsonldquoRenormalization vortices and symmetry-breaking perturba-tions in the two-dimensional planar modelrdquo Physical Review BCondensed Matter andMaterials Physics vol 16 no 3 pp 1217ndash1241 1977
[4] S Samuel ldquoGrand partition function in field theory withapplications to sine-Gordon field theoryrdquo Physical Review vol18 p 1916 1978
[5] A B Zamolodchikov and A B Zamolodchikov ldquoFactorized S-matrices in two dimensions as the exact solutions of certainrelativistic quantum field theory modelsrdquoAnnals of Physics vol120 no 2 pp 253ndash291 1979
[6] D J Amit Y Y Goldschmidt and G Grinstein ldquoRenormalisa-tion group analysis of the phase transition in the 2D Coulombgas sine-Gordon theory and XY-modelrdquo Journal of Physics AMathematical and General vol 13 no 2 pp 585ndash620 1980
[7] P Weigman ldquoOne-dimensional Fermi system and plane xymodelrdquo Journal of Physics C Solid State Physics vol 11 no 8p 1583 1987
[8] J Balog and A Hegedus ldquoTwo-loop beta functions of the Sine-Gordonmodelrdquo Journal of Physics AMathematical andGeneralvol 33 p 6543 2000
[9] R Rajaraman Solitons and Instantons North-Holland Publish-ing Company Amsterdam The Netherlands 1982
[10] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2004
[11] S Coleman Aspects of Symmetry Cambridge University PressCambridge UK 1985
[12] E C Marino Quantum Field Theory Approach to CondensedMatter Physics Cambridge University Press Cambridge UK2017
[13] E Weinberg Classical Solutions in Quantum Field TheoryCambridge University Press Cambbridge UK 2015
[14] S Mandelstam ldquoSoliton operators for the quantized sine-Gordon equationrdquo Physical Review vol 11 p 3026 1975
[15] B Schroer and T Truong ldquoEquivalence of the sine-Gordon andThirring models and cumulative mass effectsrdquo Physical Reviewvol 15 p 1684 1977
[16] M Faber and A N Ivanov ldquoOn the equivalence between sine-Gordonmodel andThirring model in the chirally broken phaseof theThirringmodelrdquoEuropean Physical Journal vol 20 p 7232001
[17] V I Berezinski ldquoDestruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Con-tinuous Symmetry Group II Quantum Systemsrdquo Journal ofExperimental andTheoretical Physics vol 34 p 610 1972
[18] J M Kosterlitz and D Thouless ldquoOrdering metastabilityand phase transitions in two-dimensional systemsrdquo Journal ofPhysics vol 6 p 1181 1973
[19] J M Kosterlitz ldquoCritical Exponents of the Two-DimensionalXY Modelrdquo Journal of Physics vol 7 p 1046 1974
[20] J Jose ldquoSine-Gordon Theory and the Classical Two-Dimensional xy Modelrdquo Physical Review vol 14 p 28261976
Advances in Mathematical Physics 7
[21] J Zinn-Justin Quantum Field Theory and Critical PhenomenaOxford University Press Oxford UK 1989
[22] J Kondo ldquoResistance Minimum in Dilute Magnetic AlloysrdquoProgress of Theoretical Physics vol 32 no 1 p 34 1964
[23] J Kondo The Physics of Dilute Magnetic Alloys CambridgeUniversity Press Cambridge UK 2012
[24] P W Anderson ldquoA poor manrsquos derivation of scaling laws for theKondo problemrdquo Journal of Physics C Solid State Physics vol 3p 2436 1970
[25] P W Anderson and G Yuval Physical Review Letters vol 23 p89 1969
[26] G Yuval and P W Anderson ldquoExact Results for the KondoProblem One-Body Theory and Extension to Finite Temper-aturerdquo Physical Review vol 1 p 1522 1970
[27] P W Anderson G Yuval and D R Hamann ldquoExact Resultsin the Kondo Problem II ScalingTheory Qualitatively CorrectSolution and SomeNewResults onOne-Dimensional ClassicalStatistical Modelsrdquo Physical Review B vol 1 p 4464 1970
[28] J Solyom ldquoThe Fermi gas model of one-dimensional conduc-torsrdquo Advances in Physics vol 28 no 2 pp 201ndash303 1979
[29] F D N Haldane ldquorsquoLuttinger liquid theoryrsquo of one-dimensionalquantum fluids I Properties of the Luttinger model and theirextension to the general 1D interacting spinless Fermi gasrdquoJournal of Physics vol 14 pp 2585ndash2609 1981
[30] S-T Chui and P A Lee ldquoEquivalence of a One-DimensionalFermion Model and the Two-Dimensional Coulomb GasrdquoPhysical Review Letters vol 35 p 315 1975
[31] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979
[32] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society of London Series A Mathemati-cal and Physical vol 276 no 1365 pp 238ndash257 1963
[33] M C Gutzwiller ldquo Correlation of Electrons in a Narrow rdquoPhysical Review A Atomic Molecular and Optical Physics vol137 no 6A pp A1726ndashA1735 1965
[34] J E Hirsch ldquoMonte Carlo Study of the Two-DimensionalHubbard Modelrdquo Physical Review Letters vol 51 p 1900 1983
[35] S Sorella S Baroni R Car and M Parrinello ldquoA noveltechnique for the simulation of interacting fermion systemsrdquoEurophysics Letters vol 8 p 663 1989
[36] S R White D J Scalapino R L Sugar E Y Loh J EGubernatis and R T Scalettar ldquoNumerical study of the two-dimensional Hubbard modelrdquo Physical Review vol 40 p 5061989
[37] K Yamaji T Yanagisawa T Nakanishi and S Koike ldquoVaria-tional Monte Carlo study on the superconductivity in the two-dimensionalHubbardmodelrdquoPhysica C Superconductivity vol304 p 225 1988
[38] T Yanagisawa and Y Shimoi ldquoExact results in strongly cor-related electrons - Spin-reflection positivity and the Perron-Frobenius theoremrdquo International Journal of Modern Physicsvol 10 p 3383 1996
[39] S Koikegami ldquoVariational Monte Carlo study on the super-conductivity in the two-dimensional Hubbard modelrdquo PhysicalReview vol 67 p 134517 2003
[40] A J Leggett ldquoNumber-Phase Fluctuations in Two-Band Super-conductorsrdquo Progress of Theoretical Physics vol 36 p 901 1966
[41] Y Tanaka and T Yanagisawa ldquoChiral Ground State in Three-Band Superconductorsrdquo Journal of the Physical Society of Japanvol 79 p 114706 2010
[42] Y Tanaka and T Yanagisawa ldquoChiral state in three-gap super-conductorsrdquo Solid State Communications vol 150 no 41-42 pp1980ndash1982 2010
[43] T Yanagisawa Y Tanaka I Hase and K Yamaji ldquoChiral statein three-gap superconductors Solid State Communrdquo Journal ofthe Physical Society of Japan vol 81 p 024712 2012
[44] V Stanev and Z Tesanovic ldquoThree-band superconductivityand the order parameter that breaks time-reversal symmetryrdquoPhysical Review B Condensed Matter andMaterials Physics vol81 Article ID 134522 2010
[45] T Yanagisawa and I Hase ldquoMasslessModes and AbelianGaugeFields in Multi-Band Superconductorsrdquo Journal of the PhysicalSociety of Japan vol 82 p 124704 2013
[46] T Yanagisawa and Y Tanaka ldquoFluctuation-inducedNambundashGoldstone bosons in a HiggsndashJosephson modelrdquoNew Journal of Physics vol 16 p 123014 2014
[47] T Yanagisawa ldquoNambundashGoldstone Bosons Characterized bythe Order Parameter in Spontaneous Symmetry BreakingrdquoJournal of the Physical Society of Japan vol 86 p 104711 2017
[48] T Koyama andMTachikiPhysical ReviewB CondensedMatterand Materials Physics vol 54 no 22 pp 16183ndash16191 1996
[49] T Yanagisawa ldquoChiral sine-GordonmodelrdquoEurophysics Lettersvol 113 p 41001 2016
[50] G rsquotHooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquo Nuclear Physics vol 44 p 189 1972
[51] D Gross ldquoMethods in field theoryrdquo in Les Houches LectureNotes R Balian and J Zinn-Justin Eds North-Holland Pub-lishing Company Amsterdam Netherlands 1976
[52] T Yanagisawa Recent Studies in Perturbation Theory D IUzunov Ed InTech Open Publisher 2017
[53] I Nandori U Jentschura K Sailer and G SoffldquoRenormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for dgtsim3 dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 69 no 2 2004
[54] F J Wegner and A Houghton ldquoRenormalization GroupEquation for Critical Phenomenardquo Physical Review A AtomicMolecular and Optical Physics vol 8 no 1 pp 401ndash412 1973
[55] S Nagy I Nandori J Polonyi and K Sailer ldquoFunctionalRenormalization Group Approach to the Sine-Gordon ModelrdquoPhysical Review Letters vol 102 no 24 2009
[56] Y Nagaoka ldquoSelf-Consistent Treatment of Kondorsquos Effect inDilute AlloysrdquoPhysical ReviewA AtomicMolecular andOpticalPhysics vol 138 no 4A pp A1112ndashA1120 1965
[57] D R Hamann ldquoNewSolution for Exchange Scattering inDiluteAlloysrdquo Physical Review vol 158 p 570 1967
[58] M A Ruderman and C Kittel ldquoIndirect Exchange Coupling ofNuclearMagneticMoments by Conduction Electronsrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 96 no 1pp 99ndash102 1954
[59] T Kasuya ldquoA Theory of Metallic Ferro- and Antiferromag-netism on Zenerrsquos Modelrdquo Progress of Theoretical Physics vol16 p 45 1956
[60] K Yosida ldquoMagnetic Properties of Cu-Mn Alloysrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 106 no5 pp 893ndash898 1957
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
sdot int 119889119889119909 (119890119894119901sdot119909 minus 1) exp (1198991198981198660 (119909)) (33)
Using the expansion 119890119894119901sdot119909 = 1 + 119894119901 sdot 119909 minus (12)(119901 sdot 119909)2 + sdot sdot sdot wekeep the 1199012 term By using the formula1198700(119909) asymp minus120574minus log(1199092)for small 119909 Σ119899119898(119901) is written as
The integral diverges when 119905 = 119905119888119899 = 81205871198992 and 119899 = 119898 Thenwe consider the case 119899 = 119898 which gives the correction to thetwo-point function Γ(2)119899119861 = minusΣ119899119899 when 119905 asymp 119905119888119899 as follows
where we set 119905 asymp (81205871198992) sdot (1 + V119899) This term mainly comesfrom the region where 119905 asymp 119905119888119899 The two-point function up tothis order is
Γ(2)119861 (119901)= 1
1199051205832minus119889119885119905 [1199012 minus 1199012sum
The renormalized two-point function is given as Γ(2)119877 (119901) =119885120601Γ(2)119861 (119901) The renormalization constants are determined asshown above and thus we obtain the same renormalizationgroup equation
5 Renormalization Group Flow
Let us consider the case with two parameters 1205721 and 1205722 Therenormalization group equations are
1205831205971205721120597120583 = minus21205721 (1 minus 18120587119905) (37)
1205831205971205722120597120583 = minus21205722 (1 minus 12120587119905) (38)
120583 120597119905120597120583 = (119889 minus 2) 119905 + 1
3211990512057221 +1811990512057222 (39)
We have the critical value 119905 = 1199051198881 = 8120587 for 1205721 and 119905 = 1199051198882 =2120587 for 1205722 The parameter 119905 is an increasing function of 120583 intwo dimensions 119889 = 2 The space (119905 1205721 1205722) may be dividedinto four regions which are classified by the values to whichthe pair (1205721 1205722) is renormalized as 120583 increases We call themregions I II III and IV
I (1205721 1205722) 997888rarr (infininfin) (40)
II (1205721 1205722) 997888rarr (0infin) (41)
III (1205721 1205722) 997888rarr (infin 0) (42)
IV (1205721 1205722) 997888rarr (0 0) (43)
In region 119905 asymp 8120587 we put 119905 = 8120587(1 + V1) 1199091 = 2V1 1199101 =12057214and 1199102 = 12057222 The equations read
In this region 1199101 acts as a perturbation to the scalingequation of the conventional sine-Gordon model We showthe renormalization group flow as 120583 increases in Figure 3
6 Advances in Mathematical Physics
Ｃ2
1
t＝2 t＝1 t
Figure 3 Renormalization group flow as 120583 997888rarr infin for 1205721 and 1205722
6 Summary
We have discussed the dimensional regularization approachto the renormalization group theory of the generalized sine-Gordon model There are multiple critical points for thecoupling constant 119905 given as 119905119888119899 = 81205871198992 In the case where 119905 isclose to 119905119888119899 for some 119899 the renormalization group equationsare approximated by those for the sine-Gordon model withsingle-cosine potential (conventional sine-Gordon model)A nontrivial simple generalized model is the sine-Gordonmodel with 1205721 and 1205722 When 119905 is 1199051198881 = 8120587 1205722 acts as aperturbation for 119905 and 1205721 The renormalization flow as 120583 997888rarrinfin or 120583 997888rarr 0 depends on an initial set of parameters 119905 1205721and 1205722 This can be viewed as a competition between twointeractions1205721 and1205722 Thismay lead to a generalization of theKosterlitz-Thouless transition the crossover phenomenonin the Kondo effect and other phenomena In the Kondoproblem the appearance of logarithmic singularity [22 5657] suggested the renormalizability of the model Inmaterialswith many magnetic impurities the interaction betweenmagnetic impurities called the RKKY interaction [58ndash60]should be considered In this case the renormalization groupflow is drawn on a two-dimensional plane of two parametersThere may be a relation to the generalized sine-Gordonmodel
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
The author expresses his sincere thanks to K Odagiri foruseful discussionsThis workwas supported in part byGrant-in-Aid from the Ministry of Education Culture SportsScience and Technology (MEXT) of Japan (no 17K05559)
References
[1] S Coleman ldquoQuantum sine-Gordon equation as the massiveThirring modelrdquo Physical Review vol 11 p 2088 1975
[2] R F Dashen B Hasslacher and A Neveu ldquoParticle spectrumin model field theories from semiclassical functional integraltechniquesrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 11 no 12 pp 3424ndash3450 1975
[3] J V Jose L P Kadanoff S Kirkpatrick and D R NelsonldquoRenormalization vortices and symmetry-breaking perturba-tions in the two-dimensional planar modelrdquo Physical Review BCondensed Matter andMaterials Physics vol 16 no 3 pp 1217ndash1241 1977
[4] S Samuel ldquoGrand partition function in field theory withapplications to sine-Gordon field theoryrdquo Physical Review vol18 p 1916 1978
[5] A B Zamolodchikov and A B Zamolodchikov ldquoFactorized S-matrices in two dimensions as the exact solutions of certainrelativistic quantum field theory modelsrdquoAnnals of Physics vol120 no 2 pp 253ndash291 1979
[6] D J Amit Y Y Goldschmidt and G Grinstein ldquoRenormalisa-tion group analysis of the phase transition in the 2D Coulombgas sine-Gordon theory and XY-modelrdquo Journal of Physics AMathematical and General vol 13 no 2 pp 585ndash620 1980
[7] P Weigman ldquoOne-dimensional Fermi system and plane xymodelrdquo Journal of Physics C Solid State Physics vol 11 no 8p 1583 1987
[8] J Balog and A Hegedus ldquoTwo-loop beta functions of the Sine-Gordonmodelrdquo Journal of Physics AMathematical andGeneralvol 33 p 6543 2000
[9] R Rajaraman Solitons and Instantons North-Holland Publish-ing Company Amsterdam The Netherlands 1982
[10] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2004
[11] S Coleman Aspects of Symmetry Cambridge University PressCambridge UK 1985
[12] E C Marino Quantum Field Theory Approach to CondensedMatter Physics Cambridge University Press Cambridge UK2017
[13] E Weinberg Classical Solutions in Quantum Field TheoryCambridge University Press Cambbridge UK 2015
[14] S Mandelstam ldquoSoliton operators for the quantized sine-Gordon equationrdquo Physical Review vol 11 p 3026 1975
[15] B Schroer and T Truong ldquoEquivalence of the sine-Gordon andThirring models and cumulative mass effectsrdquo Physical Reviewvol 15 p 1684 1977
[16] M Faber and A N Ivanov ldquoOn the equivalence between sine-Gordonmodel andThirring model in the chirally broken phaseof theThirringmodelrdquoEuropean Physical Journal vol 20 p 7232001
[17] V I Berezinski ldquoDestruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Con-tinuous Symmetry Group II Quantum Systemsrdquo Journal ofExperimental andTheoretical Physics vol 34 p 610 1972
[18] J M Kosterlitz and D Thouless ldquoOrdering metastabilityand phase transitions in two-dimensional systemsrdquo Journal ofPhysics vol 6 p 1181 1973
[19] J M Kosterlitz ldquoCritical Exponents of the Two-DimensionalXY Modelrdquo Journal of Physics vol 7 p 1046 1974
[20] J Jose ldquoSine-Gordon Theory and the Classical Two-Dimensional xy Modelrdquo Physical Review vol 14 p 28261976
Advances in Mathematical Physics 7
[21] J Zinn-Justin Quantum Field Theory and Critical PhenomenaOxford University Press Oxford UK 1989
[22] J Kondo ldquoResistance Minimum in Dilute Magnetic AlloysrdquoProgress of Theoretical Physics vol 32 no 1 p 34 1964
[23] J Kondo The Physics of Dilute Magnetic Alloys CambridgeUniversity Press Cambridge UK 2012
[24] P W Anderson ldquoA poor manrsquos derivation of scaling laws for theKondo problemrdquo Journal of Physics C Solid State Physics vol 3p 2436 1970
[25] P W Anderson and G Yuval Physical Review Letters vol 23 p89 1969
[26] G Yuval and P W Anderson ldquoExact Results for the KondoProblem One-Body Theory and Extension to Finite Temper-aturerdquo Physical Review vol 1 p 1522 1970
[27] P W Anderson G Yuval and D R Hamann ldquoExact Resultsin the Kondo Problem II ScalingTheory Qualitatively CorrectSolution and SomeNewResults onOne-Dimensional ClassicalStatistical Modelsrdquo Physical Review B vol 1 p 4464 1970
[28] J Solyom ldquoThe Fermi gas model of one-dimensional conduc-torsrdquo Advances in Physics vol 28 no 2 pp 201ndash303 1979
[29] F D N Haldane ldquorsquoLuttinger liquid theoryrsquo of one-dimensionalquantum fluids I Properties of the Luttinger model and theirextension to the general 1D interacting spinless Fermi gasrdquoJournal of Physics vol 14 pp 2585ndash2609 1981
[30] S-T Chui and P A Lee ldquoEquivalence of a One-DimensionalFermion Model and the Two-Dimensional Coulomb GasrdquoPhysical Review Letters vol 35 p 315 1975
[31] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979
[32] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society of London Series A Mathemati-cal and Physical vol 276 no 1365 pp 238ndash257 1963
[33] M C Gutzwiller ldquo Correlation of Electrons in a Narrow rdquoPhysical Review A Atomic Molecular and Optical Physics vol137 no 6A pp A1726ndashA1735 1965
[34] J E Hirsch ldquoMonte Carlo Study of the Two-DimensionalHubbard Modelrdquo Physical Review Letters vol 51 p 1900 1983
[35] S Sorella S Baroni R Car and M Parrinello ldquoA noveltechnique for the simulation of interacting fermion systemsrdquoEurophysics Letters vol 8 p 663 1989
[36] S R White D J Scalapino R L Sugar E Y Loh J EGubernatis and R T Scalettar ldquoNumerical study of the two-dimensional Hubbard modelrdquo Physical Review vol 40 p 5061989
[37] K Yamaji T Yanagisawa T Nakanishi and S Koike ldquoVaria-tional Monte Carlo study on the superconductivity in the two-dimensionalHubbardmodelrdquoPhysica C Superconductivity vol304 p 225 1988
[38] T Yanagisawa and Y Shimoi ldquoExact results in strongly cor-related electrons - Spin-reflection positivity and the Perron-Frobenius theoremrdquo International Journal of Modern Physicsvol 10 p 3383 1996
[39] S Koikegami ldquoVariational Monte Carlo study on the super-conductivity in the two-dimensional Hubbard modelrdquo PhysicalReview vol 67 p 134517 2003
[40] A J Leggett ldquoNumber-Phase Fluctuations in Two-Band Super-conductorsrdquo Progress of Theoretical Physics vol 36 p 901 1966
[41] Y Tanaka and T Yanagisawa ldquoChiral Ground State in Three-Band Superconductorsrdquo Journal of the Physical Society of Japanvol 79 p 114706 2010
[42] Y Tanaka and T Yanagisawa ldquoChiral state in three-gap super-conductorsrdquo Solid State Communications vol 150 no 41-42 pp1980ndash1982 2010
[43] T Yanagisawa Y Tanaka I Hase and K Yamaji ldquoChiral statein three-gap superconductors Solid State Communrdquo Journal ofthe Physical Society of Japan vol 81 p 024712 2012
[44] V Stanev and Z Tesanovic ldquoThree-band superconductivityand the order parameter that breaks time-reversal symmetryrdquoPhysical Review B Condensed Matter andMaterials Physics vol81 Article ID 134522 2010
[45] T Yanagisawa and I Hase ldquoMasslessModes and AbelianGaugeFields in Multi-Band Superconductorsrdquo Journal of the PhysicalSociety of Japan vol 82 p 124704 2013
[46] T Yanagisawa and Y Tanaka ldquoFluctuation-inducedNambundashGoldstone bosons in a HiggsndashJosephson modelrdquoNew Journal of Physics vol 16 p 123014 2014
[47] T Yanagisawa ldquoNambundashGoldstone Bosons Characterized bythe Order Parameter in Spontaneous Symmetry BreakingrdquoJournal of the Physical Society of Japan vol 86 p 104711 2017
[48] T Koyama andMTachikiPhysical ReviewB CondensedMatterand Materials Physics vol 54 no 22 pp 16183ndash16191 1996
[49] T Yanagisawa ldquoChiral sine-GordonmodelrdquoEurophysics Lettersvol 113 p 41001 2016
[50] G rsquotHooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquo Nuclear Physics vol 44 p 189 1972
[51] D Gross ldquoMethods in field theoryrdquo in Les Houches LectureNotes R Balian and J Zinn-Justin Eds North-Holland Pub-lishing Company Amsterdam Netherlands 1976
[52] T Yanagisawa Recent Studies in Perturbation Theory D IUzunov Ed InTech Open Publisher 2017
[53] I Nandori U Jentschura K Sailer and G SoffldquoRenormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for dgtsim3 dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 69 no 2 2004
[54] F J Wegner and A Houghton ldquoRenormalization GroupEquation for Critical Phenomenardquo Physical Review A AtomicMolecular and Optical Physics vol 8 no 1 pp 401ndash412 1973
[55] S Nagy I Nandori J Polonyi and K Sailer ldquoFunctionalRenormalization Group Approach to the Sine-Gordon ModelrdquoPhysical Review Letters vol 102 no 24 2009
[56] Y Nagaoka ldquoSelf-Consistent Treatment of Kondorsquos Effect inDilute AlloysrdquoPhysical ReviewA AtomicMolecular andOpticalPhysics vol 138 no 4A pp A1112ndashA1120 1965
[57] D R Hamann ldquoNewSolution for Exchange Scattering inDiluteAlloysrdquo Physical Review vol 158 p 570 1967
[58] M A Ruderman and C Kittel ldquoIndirect Exchange Coupling ofNuclearMagneticMoments by Conduction Electronsrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 96 no 1pp 99ndash102 1954
[59] T Kasuya ldquoA Theory of Metallic Ferro- and Antiferromag-netism on Zenerrsquos Modelrdquo Progress of Theoretical Physics vol16 p 45 1956
[60] K Yosida ldquoMagnetic Properties of Cu-Mn Alloysrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 106 no5 pp 893ndash898 1957
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Advances in Mathematical Physics
Ｃ2
1
t＝2 t＝1 t
Figure 3 Renormalization group flow as 120583 997888rarr infin for 1205721 and 1205722
6 Summary
We have discussed the dimensional regularization approachto the renormalization group theory of the generalized sine-Gordon model There are multiple critical points for thecoupling constant 119905 given as 119905119888119899 = 81205871198992 In the case where 119905 isclose to 119905119888119899 for some 119899 the renormalization group equationsare approximated by those for the sine-Gordon model withsingle-cosine potential (conventional sine-Gordon model)A nontrivial simple generalized model is the sine-Gordonmodel with 1205721 and 1205722 When 119905 is 1199051198881 = 8120587 1205722 acts as aperturbation for 119905 and 1205721 The renormalization flow as 120583 997888rarrinfin or 120583 997888rarr 0 depends on an initial set of parameters 119905 1205721and 1205722 This can be viewed as a competition between twointeractions1205721 and1205722 Thismay lead to a generalization of theKosterlitz-Thouless transition the crossover phenomenonin the Kondo effect and other phenomena In the Kondoproblem the appearance of logarithmic singularity [22 5657] suggested the renormalizability of the model Inmaterialswith many magnetic impurities the interaction betweenmagnetic impurities called the RKKY interaction [58ndash60]should be considered In this case the renormalization groupflow is drawn on a two-dimensional plane of two parametersThere may be a relation to the generalized sine-Gordonmodel
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
The author expresses his sincere thanks to K Odagiri foruseful discussionsThis workwas supported in part byGrant-in-Aid from the Ministry of Education Culture SportsScience and Technology (MEXT) of Japan (no 17K05559)
References
[1] S Coleman ldquoQuantum sine-Gordon equation as the massiveThirring modelrdquo Physical Review vol 11 p 2088 1975
[2] R F Dashen B Hasslacher and A Neveu ldquoParticle spectrumin model field theories from semiclassical functional integraltechniquesrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 11 no 12 pp 3424ndash3450 1975
[3] J V Jose L P Kadanoff S Kirkpatrick and D R NelsonldquoRenormalization vortices and symmetry-breaking perturba-tions in the two-dimensional planar modelrdquo Physical Review BCondensed Matter andMaterials Physics vol 16 no 3 pp 1217ndash1241 1977
[4] S Samuel ldquoGrand partition function in field theory withapplications to sine-Gordon field theoryrdquo Physical Review vol18 p 1916 1978
[5] A B Zamolodchikov and A B Zamolodchikov ldquoFactorized S-matrices in two dimensions as the exact solutions of certainrelativistic quantum field theory modelsrdquoAnnals of Physics vol120 no 2 pp 253ndash291 1979
[6] D J Amit Y Y Goldschmidt and G Grinstein ldquoRenormalisa-tion group analysis of the phase transition in the 2D Coulombgas sine-Gordon theory and XY-modelrdquo Journal of Physics AMathematical and General vol 13 no 2 pp 585ndash620 1980
[7] P Weigman ldquoOne-dimensional Fermi system and plane xymodelrdquo Journal of Physics C Solid State Physics vol 11 no 8p 1583 1987
[8] J Balog and A Hegedus ldquoTwo-loop beta functions of the Sine-Gordonmodelrdquo Journal of Physics AMathematical andGeneralvol 33 p 6543 2000
[9] R Rajaraman Solitons and Instantons North-Holland Publish-ing Company Amsterdam The Netherlands 1982
[10] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2004
[11] S Coleman Aspects of Symmetry Cambridge University PressCambridge UK 1985
[12] E C Marino Quantum Field Theory Approach to CondensedMatter Physics Cambridge University Press Cambridge UK2017
[13] E Weinberg Classical Solutions in Quantum Field TheoryCambridge University Press Cambbridge UK 2015
[14] S Mandelstam ldquoSoliton operators for the quantized sine-Gordon equationrdquo Physical Review vol 11 p 3026 1975
[15] B Schroer and T Truong ldquoEquivalence of the sine-Gordon andThirring models and cumulative mass effectsrdquo Physical Reviewvol 15 p 1684 1977
[16] M Faber and A N Ivanov ldquoOn the equivalence between sine-Gordonmodel andThirring model in the chirally broken phaseof theThirringmodelrdquoEuropean Physical Journal vol 20 p 7232001
[17] V I Berezinski ldquoDestruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Con-tinuous Symmetry Group II Quantum Systemsrdquo Journal ofExperimental andTheoretical Physics vol 34 p 610 1972
[18] J M Kosterlitz and D Thouless ldquoOrdering metastabilityand phase transitions in two-dimensional systemsrdquo Journal ofPhysics vol 6 p 1181 1973
[19] J M Kosterlitz ldquoCritical Exponents of the Two-DimensionalXY Modelrdquo Journal of Physics vol 7 p 1046 1974
[20] J Jose ldquoSine-Gordon Theory and the Classical Two-Dimensional xy Modelrdquo Physical Review vol 14 p 28261976
Advances in Mathematical Physics 7
[21] J Zinn-Justin Quantum Field Theory and Critical PhenomenaOxford University Press Oxford UK 1989
[22] J Kondo ldquoResistance Minimum in Dilute Magnetic AlloysrdquoProgress of Theoretical Physics vol 32 no 1 p 34 1964
[23] J Kondo The Physics of Dilute Magnetic Alloys CambridgeUniversity Press Cambridge UK 2012
[24] P W Anderson ldquoA poor manrsquos derivation of scaling laws for theKondo problemrdquo Journal of Physics C Solid State Physics vol 3p 2436 1970
[25] P W Anderson and G Yuval Physical Review Letters vol 23 p89 1969
[26] G Yuval and P W Anderson ldquoExact Results for the KondoProblem One-Body Theory and Extension to Finite Temper-aturerdquo Physical Review vol 1 p 1522 1970
[27] P W Anderson G Yuval and D R Hamann ldquoExact Resultsin the Kondo Problem II ScalingTheory Qualitatively CorrectSolution and SomeNewResults onOne-Dimensional ClassicalStatistical Modelsrdquo Physical Review B vol 1 p 4464 1970
[28] J Solyom ldquoThe Fermi gas model of one-dimensional conduc-torsrdquo Advances in Physics vol 28 no 2 pp 201ndash303 1979
[29] F D N Haldane ldquorsquoLuttinger liquid theoryrsquo of one-dimensionalquantum fluids I Properties of the Luttinger model and theirextension to the general 1D interacting spinless Fermi gasrdquoJournal of Physics vol 14 pp 2585ndash2609 1981
[30] S-T Chui and P A Lee ldquoEquivalence of a One-DimensionalFermion Model and the Two-Dimensional Coulomb GasrdquoPhysical Review Letters vol 35 p 315 1975
[31] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979
[32] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society of London Series A Mathemati-cal and Physical vol 276 no 1365 pp 238ndash257 1963
[33] M C Gutzwiller ldquo Correlation of Electrons in a Narrow rdquoPhysical Review A Atomic Molecular and Optical Physics vol137 no 6A pp A1726ndashA1735 1965
[34] J E Hirsch ldquoMonte Carlo Study of the Two-DimensionalHubbard Modelrdquo Physical Review Letters vol 51 p 1900 1983
[35] S Sorella S Baroni R Car and M Parrinello ldquoA noveltechnique for the simulation of interacting fermion systemsrdquoEurophysics Letters vol 8 p 663 1989
[36] S R White D J Scalapino R L Sugar E Y Loh J EGubernatis and R T Scalettar ldquoNumerical study of the two-dimensional Hubbard modelrdquo Physical Review vol 40 p 5061989
[37] K Yamaji T Yanagisawa T Nakanishi and S Koike ldquoVaria-tional Monte Carlo study on the superconductivity in the two-dimensionalHubbardmodelrdquoPhysica C Superconductivity vol304 p 225 1988
[38] T Yanagisawa and Y Shimoi ldquoExact results in strongly cor-related electrons - Spin-reflection positivity and the Perron-Frobenius theoremrdquo International Journal of Modern Physicsvol 10 p 3383 1996
[39] S Koikegami ldquoVariational Monte Carlo study on the super-conductivity in the two-dimensional Hubbard modelrdquo PhysicalReview vol 67 p 134517 2003
[40] A J Leggett ldquoNumber-Phase Fluctuations in Two-Band Super-conductorsrdquo Progress of Theoretical Physics vol 36 p 901 1966
[41] Y Tanaka and T Yanagisawa ldquoChiral Ground State in Three-Band Superconductorsrdquo Journal of the Physical Society of Japanvol 79 p 114706 2010
[42] Y Tanaka and T Yanagisawa ldquoChiral state in three-gap super-conductorsrdquo Solid State Communications vol 150 no 41-42 pp1980ndash1982 2010
[43] T Yanagisawa Y Tanaka I Hase and K Yamaji ldquoChiral statein three-gap superconductors Solid State Communrdquo Journal ofthe Physical Society of Japan vol 81 p 024712 2012
[44] V Stanev and Z Tesanovic ldquoThree-band superconductivityand the order parameter that breaks time-reversal symmetryrdquoPhysical Review B Condensed Matter andMaterials Physics vol81 Article ID 134522 2010
[45] T Yanagisawa and I Hase ldquoMasslessModes and AbelianGaugeFields in Multi-Band Superconductorsrdquo Journal of the PhysicalSociety of Japan vol 82 p 124704 2013
[46] T Yanagisawa and Y Tanaka ldquoFluctuation-inducedNambundashGoldstone bosons in a HiggsndashJosephson modelrdquoNew Journal of Physics vol 16 p 123014 2014
[47] T Yanagisawa ldquoNambundashGoldstone Bosons Characterized bythe Order Parameter in Spontaneous Symmetry BreakingrdquoJournal of the Physical Society of Japan vol 86 p 104711 2017
[48] T Koyama andMTachikiPhysical ReviewB CondensedMatterand Materials Physics vol 54 no 22 pp 16183ndash16191 1996
[49] T Yanagisawa ldquoChiral sine-GordonmodelrdquoEurophysics Lettersvol 113 p 41001 2016
[50] G rsquotHooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquo Nuclear Physics vol 44 p 189 1972
[51] D Gross ldquoMethods in field theoryrdquo in Les Houches LectureNotes R Balian and J Zinn-Justin Eds North-Holland Pub-lishing Company Amsterdam Netherlands 1976
[52] T Yanagisawa Recent Studies in Perturbation Theory D IUzunov Ed InTech Open Publisher 2017
[53] I Nandori U Jentschura K Sailer and G SoffldquoRenormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for dgtsim3 dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 69 no 2 2004
[54] F J Wegner and A Houghton ldquoRenormalization GroupEquation for Critical Phenomenardquo Physical Review A AtomicMolecular and Optical Physics vol 8 no 1 pp 401ndash412 1973
[55] S Nagy I Nandori J Polonyi and K Sailer ldquoFunctionalRenormalization Group Approach to the Sine-Gordon ModelrdquoPhysical Review Letters vol 102 no 24 2009
[56] Y Nagaoka ldquoSelf-Consistent Treatment of Kondorsquos Effect inDilute AlloysrdquoPhysical ReviewA AtomicMolecular andOpticalPhysics vol 138 no 4A pp A1112ndashA1120 1965
[57] D R Hamann ldquoNewSolution for Exchange Scattering inDiluteAlloysrdquo Physical Review vol 158 p 570 1967
[58] M A Ruderman and C Kittel ldquoIndirect Exchange Coupling ofNuclearMagneticMoments by Conduction Electronsrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 96 no 1pp 99ndash102 1954
[59] T Kasuya ldquoA Theory of Metallic Ferro- and Antiferromag-netism on Zenerrsquos Modelrdquo Progress of Theoretical Physics vol16 p 45 1956
[60] K Yosida ldquoMagnetic Properties of Cu-Mn Alloysrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 106 no5 pp 893ndash898 1957
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Advances in Mathematical Physics 7
[21] J Zinn-Justin Quantum Field Theory and Critical PhenomenaOxford University Press Oxford UK 1989
[22] J Kondo ldquoResistance Minimum in Dilute Magnetic AlloysrdquoProgress of Theoretical Physics vol 32 no 1 p 34 1964
[23] J Kondo The Physics of Dilute Magnetic Alloys CambridgeUniversity Press Cambridge UK 2012
[24] P W Anderson ldquoA poor manrsquos derivation of scaling laws for theKondo problemrdquo Journal of Physics C Solid State Physics vol 3p 2436 1970
[25] P W Anderson and G Yuval Physical Review Letters vol 23 p89 1969
[26] G Yuval and P W Anderson ldquoExact Results for the KondoProblem One-Body Theory and Extension to Finite Temper-aturerdquo Physical Review vol 1 p 1522 1970
[27] P W Anderson G Yuval and D R Hamann ldquoExact Resultsin the Kondo Problem II ScalingTheory Qualitatively CorrectSolution and SomeNewResults onOne-Dimensional ClassicalStatistical Modelsrdquo Physical Review B vol 1 p 4464 1970
[28] J Solyom ldquoThe Fermi gas model of one-dimensional conduc-torsrdquo Advances in Physics vol 28 no 2 pp 201ndash303 1979
[29] F D N Haldane ldquorsquoLuttinger liquid theoryrsquo of one-dimensionalquantum fluids I Properties of the Luttinger model and theirextension to the general 1D interacting spinless Fermi gasrdquoJournal of Physics vol 14 pp 2585ndash2609 1981
[30] S-T Chui and P A Lee ldquoEquivalence of a One-DimensionalFermion Model and the Two-Dimensional Coulomb GasrdquoPhysical Review Letters vol 35 p 315 1975
[31] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979
[32] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society of London Series A Mathemati-cal and Physical vol 276 no 1365 pp 238ndash257 1963
[33] M C Gutzwiller ldquo Correlation of Electrons in a Narrow rdquoPhysical Review A Atomic Molecular and Optical Physics vol137 no 6A pp A1726ndashA1735 1965
[34] J E Hirsch ldquoMonte Carlo Study of the Two-DimensionalHubbard Modelrdquo Physical Review Letters vol 51 p 1900 1983
[35] S Sorella S Baroni R Car and M Parrinello ldquoA noveltechnique for the simulation of interacting fermion systemsrdquoEurophysics Letters vol 8 p 663 1989
[36] S R White D J Scalapino R L Sugar E Y Loh J EGubernatis and R T Scalettar ldquoNumerical study of the two-dimensional Hubbard modelrdquo Physical Review vol 40 p 5061989
[37] K Yamaji T Yanagisawa T Nakanishi and S Koike ldquoVaria-tional Monte Carlo study on the superconductivity in the two-dimensionalHubbardmodelrdquoPhysica C Superconductivity vol304 p 225 1988
[38] T Yanagisawa and Y Shimoi ldquoExact results in strongly cor-related electrons - Spin-reflection positivity and the Perron-Frobenius theoremrdquo International Journal of Modern Physicsvol 10 p 3383 1996
[39] S Koikegami ldquoVariational Monte Carlo study on the super-conductivity in the two-dimensional Hubbard modelrdquo PhysicalReview vol 67 p 134517 2003
[40] A J Leggett ldquoNumber-Phase Fluctuations in Two-Band Super-conductorsrdquo Progress of Theoretical Physics vol 36 p 901 1966
[41] Y Tanaka and T Yanagisawa ldquoChiral Ground State in Three-Band Superconductorsrdquo Journal of the Physical Society of Japanvol 79 p 114706 2010
[42] Y Tanaka and T Yanagisawa ldquoChiral state in three-gap super-conductorsrdquo Solid State Communications vol 150 no 41-42 pp1980ndash1982 2010
[43] T Yanagisawa Y Tanaka I Hase and K Yamaji ldquoChiral statein three-gap superconductors Solid State Communrdquo Journal ofthe Physical Society of Japan vol 81 p 024712 2012
[44] V Stanev and Z Tesanovic ldquoThree-band superconductivityand the order parameter that breaks time-reversal symmetryrdquoPhysical Review B Condensed Matter andMaterials Physics vol81 Article ID 134522 2010
[45] T Yanagisawa and I Hase ldquoMasslessModes and AbelianGaugeFields in Multi-Band Superconductorsrdquo Journal of the PhysicalSociety of Japan vol 82 p 124704 2013
[46] T Yanagisawa and Y Tanaka ldquoFluctuation-inducedNambundashGoldstone bosons in a HiggsndashJosephson modelrdquoNew Journal of Physics vol 16 p 123014 2014
[47] T Yanagisawa ldquoNambundashGoldstone Bosons Characterized bythe Order Parameter in Spontaneous Symmetry BreakingrdquoJournal of the Physical Society of Japan vol 86 p 104711 2017
[48] T Koyama andMTachikiPhysical ReviewB CondensedMatterand Materials Physics vol 54 no 22 pp 16183ndash16191 1996
[49] T Yanagisawa ldquoChiral sine-GordonmodelrdquoEurophysics Lettersvol 113 p 41001 2016
[50] G rsquotHooft and M Veltman ldquoRegularization and renormaliza-tion of gauge fieldsrdquo Nuclear Physics vol 44 p 189 1972
[51] D Gross ldquoMethods in field theoryrdquo in Les Houches LectureNotes R Balian and J Zinn-Justin Eds North-Holland Pub-lishing Company Amsterdam Netherlands 1976
[52] T Yanagisawa Recent Studies in Perturbation Theory D IUzunov Ed InTech Open Publisher 2017
[53] I Nandori U Jentschura K Sailer and G SoffldquoRenormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for dgtsim3 dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 69 no 2 2004
[54] F J Wegner and A Houghton ldquoRenormalization GroupEquation for Critical Phenomenardquo Physical Review A AtomicMolecular and Optical Physics vol 8 no 1 pp 401ndash412 1973
[55] S Nagy I Nandori J Polonyi and K Sailer ldquoFunctionalRenormalization Group Approach to the Sine-Gordon ModelrdquoPhysical Review Letters vol 102 no 24 2009
[56] Y Nagaoka ldquoSelf-Consistent Treatment of Kondorsquos Effect inDilute AlloysrdquoPhysical ReviewA AtomicMolecular andOpticalPhysics vol 138 no 4A pp A1112ndashA1120 1965
[57] D R Hamann ldquoNewSolution for Exchange Scattering inDiluteAlloysrdquo Physical Review vol 158 p 570 1967
[58] M A Ruderman and C Kittel ldquoIndirect Exchange Coupling ofNuclearMagneticMoments by Conduction Electronsrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 96 no 1pp 99ndash102 1954
[59] T Kasuya ldquoA Theory of Metallic Ferro- and Antiferromag-netism on Zenerrsquos Modelrdquo Progress of Theoretical Physics vol16 p 45 1956
[60] K Yosida ldquoMagnetic Properties of Cu-Mn Alloysrdquo PhysicalReview A Atomic Molecular and Optical Physics vol 106 no5 pp 893ndash898 1957
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
