Bond-order correlation and ground-state phase diagram of a one-dimensional V 1 –V 2 spinless fermion model This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Phys.: Condens. Matter 23 365602 (http://iopscience.iop.org/0953-8984/23/36/365602) Download details: IP Address: 132.174.255.49 The article was downloaded on 13/04/2013 at 12:27 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Bond-order correlation and ground-state phase diagram of a one-dimensional V1–V2 spinless
fermion model
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 23 (2011) 365602 (8pp) doi:10.1088/0953-8984/23/36/365602
Bond-order correlation and ground-statephase diagram of a one-dimensionalV1–V2 spinless fermion model
Cheng-Bo Duan and Wei-Zhong Wang
Department of Physics, Wuhan University, and Key Laboratory of Artificial Micro- and Nano-structuresof Ministry of Education, Wuhan 430072, People’s Republic of China
Received 17 June 2011, in final form 4 August 2011Published 25 August 2011Online at stacks.iop.org/JPhysCM/23/365602
AbstractWe establish the ground-state phase diagram of a spinless fermion model with the nearest andnext-nearest-neighbor repulsions via the density-matrix-renormalization-group method. Thebond-order (BO) correlation function and its static structure factor can be employed toaccurately determine the BO phase boundaries. The first derivative of the charge-density wave(CDW) structure factor and the second derivative of the energy are expect to diverge at the2kF-CDW–BO transition point, which indicates that a second-order transition occurs. Wedetermine the metallic-4kF-CDW boundary by analyzing the decay behavior of the chargecorrelation function near the transition point.
1. Introduction
Quantum phase transitions (QPTs) driven by electron–electron(e–e) interactions in low-dimensional systems have attractedmuch attention for several decades. The Mott transitionhas been well understood within the Hubbard model whereonly the on-site interactions are considered. However,long-range Coulomb interactions are rather prominent infrustrated systems due to their geometrical structure. Inthe transition-metal CuO and quasi-one-dimensional organicconductor [1–4], both the nearest-neighbor and the next-nearest-neighbor Coulomb repulsions play an important rolein the charge ordered insulator–metal transition and thelow-energy electronic state of the CuO double chains.
A half-filled spinless model including the nearest-neighbor as well as the next-nearest-neighbor repulsions hasbeen proposed to address the phenomena in such systems,which has the following Hamiltonian:
H = −t∑
i
(c†i ci+1 + h.c.)+ V1
∑i
nini+1
+ V2
∑i
nini+2, (1)
where c†i (ci) creates (annihilates) a spinless fermion at site i,
ni = c†i ci is the number operator; t is the hopping integral and
taken as unit energy, and V1 and V2 are the nearest-neighborand next-nearest-neighbor Coulomb interactions, respectively.
The model (1) in the case of half-filling is equivalent tothe t − U − V1 − V2 model at quarter-filling with nearestand next-nearest-neighbor interactions in the strong-couplinglimit U→∞ [5, 6]. It is worth emphasizing that the spinlessmodel is a fairly good approximation in many situations.For example, in magnetite Fe3O4 the electrons are mostlycompletely polarized because of the high Curie temperature,and it is thought that only electrons with the same spinexist [7–9].
For V2 = 0, the Hamiltonian (1) can be solved exactly bythe Bethe ansatz method. The system is in the Luttinger liquid(LL) phase (one-dimensional metal) for V1 < 2.0 and a 4kFcharge-density wave (CDW) insulator phase for V1 > 2.0. TheKosterlitz–Thouless (KT) type phase transition takes placeat V1 = 2.0 where the energy gap opens exponentially [10].For an arbitrary finite value of V2, there is no exact resultfor equation (1) but the chief features of the ground statecould be established intuitively from the atomic limit (t = 0).The ground state corresponds to the 4kF-CDW phase (1010)for V1 > 2V2 and the 2kF-CDW phase (1100) for V1 < 2V2.Here, 1 denotes an occupied site and 0 denotes a vacancy.Once the hopping energy t is switched on, the transition line
J. Phys.: Condens. Matter 23 (2011) 365602 C-B Duan and W-Z Wang
V1 = 2V2 would be broadened to a metal region. Severalworks have been concerned with the existence of the metalphase [4, 7, 11]. The gap has a different behavior on thetwo boundaries of the metal phase. For example, the gapopens exponentially [10] in the 4kF-CDW phase near thetransition point while the gap near the 2kF-CDW phase obeysthe power law [14]. The reason for these different behaviorswas attributed to the existence of the bond-order (BO) phasebetween the 2kF-CDW phase and the metal phase.
Although there are many studies on the phase diagram ofthe Hamiltonian (1) and similar models [7, 9, 11–14], little isknown about the BO phase. The model (1) can be mappedinto an XXZ spin-1/2 model with next-nearest-neighborantiferromagnetic Ising interaction. For this equivalent spinmodel, a staggered spin order or BO phase was mapped outby field-theory techniques [6] and the Lanczos method [12].However, since field-theory techniques are usually validfor weak-coupling, it is difficult to determine the exactboundary of the BO phase. On the other hand, becauseof the finite-size effect in early numerical methods up toonly 18 sites [12], the spin structure factor and the dynamiccharge transfer correlation function just give a broad transitionzone. Recently, the phase boundaries of the BO phasewere estimated by combining the ground-state curvature andlevel crossings via a density-matrix-renormalization-group(DMRG) [14]. Since the curvature is equivalent to theDrude weight of dc conductivity [15], it is convenient todetermine the metal–insulator transition point. However, boththe 2kF-CDW phase and the BO phase are insulators and theirboundaries cannot be exactly determined by the curvature. Inanother DMRG study [16], the Tomonaga–Luttinger liquidparameter Kρ was calculated to determine the boundariesbetween the metallic phase and each of the two CDW phases,while the BO phase was not reported.
In this paper, we employ the DMRG method [17] tocalculate the BO correlation function, the charge correlationfunction and their structure factors in the ground state. We cangive direct evidence for the existence of the BO phase. At the2kF-CDW–BO transition point, the BO correlation function ispositive and its structure factor has a maximum at the wavevector q = 0. At the BO–metallic transition point, the BOcorrelation function is staggered and its structure factor hasa maximum at q = π . The long-range behavior of the BOcorrelation shows a power-law decay at two critical pointsand an exponential decay far away from the critical points. Atthe 2kF-CDW–BO transition point, the first derivative of thecharge structure factor and the second derivative of the energyare expected to diverge. This feature shows a second-orderquantum phase transition, which is consistent with the resultfrom field-theory [6]. We determine the metallic-4kF-CDWboundary by analyzing the decay behavior of the chargecorrelation function near the transition point.
2. Numerical method
We study the correlations B(r) and C(r) to describe thequantum fluctuations of bond and charge, which are defined as
B(r) = 〈BiBi+r〉 − 〈Bi〉〈Bi+r〉, (2)
Figure 1. Charge and bond correlation functions in thenoninteracting case (V1 = 0,V2 = 0). The circles are the DMRGresults with system size L = 200 and crossings denote the exactresults from equations (4) and (5).
C(r) = 〈nini+r〉 − 〈ni〉〈ni+r〉. (3)
Here, Bi = c†i ci+1+ h.c. is the bond charge between site i and
i+ 1.We use the finite-size DMRG calculation with the open
boundary condition to calculate the correlations B(r) andC(r). The primary error of the DMRG method originatesfrom the truncation of the reduced-density matrix. Hence,we test the results with the system size L = 40–512 for thenumber of states retained m = 120–500. We find that thetruncated error in the reduced-density matrix is always lowerthan 10−10 for different system sizes as long as m > 200.To minimize the Friedel oscillation introduced by the openboundary condition, we cut off both ends of the chain andaverage the correlation functions over several pairs (i, j) withr = |i − j| fixed. The pairs are chosen around the center ofthe chain. Even so, the finite-size effect still exists due to thecorrelation length. Hence, we observe the convergence of thecorrelations with increasing system size L and find that it isprecise enough to evaluate the long-range behavior of thesecorrelations once L > 160. As an example, we show the CDWand BO correlation functions with the size L = 200 in thenoninteracting case (V1 = V2 = 0.0) and compare them withthe exact results which are known as [18]
C(r) =cos(rπ)− 1
2π2r2 , (4)
B(r) =1
π2r2
(2r2− 1
r2 − 1cos(rπ)+
1
r2 − 1
). (5)
In figure 1, the CDW and BO correlation functions areplotted in panels (a) and (b), respectively. We can see goodagreement between the DMRG results and the exact results.We have also performed similar processes in systems withsizes L = 240, 320 and 400. The results are almost the same
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J. Phys.: Condens. Matter 23 (2011) 365602 C-B Duan and W-Z Wang
Figure 2. Phase diagram of model (1) at half-filling on the V1–V2 plane. The dot-dashed (dotted) line indicates the phase boundary betweenthe BO (4kF-CDW) and metallic phases. The dashed line denotes the boundary between kF-CDW and the BO phase. The solid linecorresponds to V1 = 2V2.
as that shown here. For the interacting model (1), we willcarefully analyze the finite-size effect in the later sections.
To describe the CDW and BO correlation in differentphases, we also study the static structure factor
NC(q) =1L
∑ir
eiqr(〈nini+r〉 − 〈ni〉〈ni+r〉), (6)
NB(q) =1L
∑ir
eiqr(〈BiBi+r〉 − 〈Bi〉〈Bi+r〉). (7)
Furthermore, the charge gap is also utilized as additionalevidence to show the process of the phase transitions. Wedefine a charge gap with finite size as exciting a particle fromthe ground state
1c(L) = E0(L/2+ 1)− E0(L/2), (8)
where E0(N) is the ground-state energy in subspace with theparticle number N. To achieve the thermodynamic limit, weextrapolate the charge gap to L → ∞ with a second-orderpolynomial fitting [19, 20] defined as
1c(L) = 1∞c + A/L+ B/L2. (9)
where1∞c , A and B are the fitting parameters. We find that thecharge gap can be extrapolated safely with equation (9) in thewhole region of the phase diagram.
3. Results and discussion
We present the phase diagram on the V1–V2 plane in figure 2.For large V2, the ground state is the 2kF-CDW insulator.With decreasing V2, the system sequentially enters into theBO phase, a metallic phase and the 4kF-CDW phase. In thefollowing, we determine the nature of the phase transitionsand describe the characteristics of the different phases.
Figure 3. BO correlation B(r) as a function of r with r = 2n on alog–log scale for V1 = 4.0 and different V2 s. The BO correlation atV2 = 4.32 exhibits a good linear behavior on a log–log scale, whichshows a power-law decay as 1/r2 (dashed line).
3.1. 2kF-CDW–BO transition
First, we present the decay behavior of bond fluctuationcorrelation B(r) near the 2kF-CDW–BO transition point forvarious V2 s along the line V1 = 4.0 in figure 3 with a log–logscale. In a uniform system, the bond fluctuation correlationdecays rapidly along the chain. If the BO phase exists, weshould observe the slowest decay of BO correlation at thecritical point since the correlation length is divergent [21] atthis point. From figure 3, one can see that with decreasing V2,the decay of B(r) becomes slower. At V2 = 4.32, the decayis slowest. With further decreasing V2, the decay becomesfaster again. We find that the BO correlation at V2 = 4.32exhibits a good linear behavior on a log–log scale, whichshows a power-law decay as 1/r2. Away from this point, the
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J. Phys.: Condens. Matter 23 (2011) 365602 C-B Duan and W-Z Wang
Figure 4. BO correlation B(r) as a function of r for V1 = 4.0 and different V2 s. Here, r varies continuously, in contrast to r = 2n infigure 3.
BO correlation decays exponentially with a finite correlationlength. Hence, we believe that the 2kF-CDW–BO transitionpoint is at V2 = 4.32 for V1 = 4.0. Here, only the correlationswith even distance r = 2n are present due to the staggeredbehavior for smaller V2 (see figure 4). We also investigate B(r)with r = 2n+ 1 and come to the same conclusion.
In order to explore the different behaviors of the BOcorrelation B(r) for different phases, figure 4 shows B(r)as a function of the distance r; here r varies continuously,in contrast to r = 2n in figure 3. In the 2kF-CDW phase(V2 > 4.32), B(r) is always positive while for V2 < 4.32,B(r) is alternately positive and negative. This feature canbe understood by following physical pictures. For largeV2 (figures 4(e) and (f)), the system is in the 2kF-CDWphase with the configuration (1100); here, 1 denotes theelectron and 0 denotes the hole on a site. The bondcharge operator Bi in equation (2) means a charge transferbetween the nearest-neighbor sites. The bond correlationBiBi+1 means charge transfers among three sequential sites,e.g. from the configuration (1100) to (0110) or (1001). Thesecharge transfers are allowed in the 2kF-CDW phase andtherefore the expectation is that 〈BiBi+1〉 is positive. On theother hand, we expect 〈Bi〉〈Bi+1〉 to be zero because twoisolated charge transfers 〈Bi〉 and 〈Bi+1〉 cannot be realizedin any configuration of the 2kF-CDW phase. As a result,the nearest-neighbor bond fluctuation B(r = 1) defined inequation (2) is positive. Mediated by this nearest-neighborcorrelation B(r = 1), the bond fluctuation correlation B(r)for various r is always positive in the 2kF-CDW phase(see figures 4(e) and (f)). However, in the 4kF-CDW phase(small V2), the situation is quite different. For example, thenearest-neighbor bond correlation BiBi+1 is always zero inthe configuration (1010) of the 4kF-CDW phase because theelectron on the first site cannot transfer to the third occupiedsite by two sequential hoppings. On the other hand, thebond charges 〈Bi〉 and 〈Bi+1〉 are positive going from theconfiguration (1010) to (0110) or (1100). It is noticeable
that even for small V2 there is some possibility of theconfiguration (0110) or (1100), although it is much less thanthe possibility of the configuration (1010). Therefore, thenearest-neighbor bond fluctuation B(r = 1) in equation (2) isnegative. Mediated by the nearest-neighbor B(r = 1), the bondfluctuation correlation B(r) is staggered as in figures 4(a)–(d).
The above physical picture of bond fluctuation correlationB(r) can be described by the static BO structure factor NB(q)in equation (7). NB(q) in figure 5(a) shows two maxima atwave vector q = 0 and π . For large V2, the BO correlationB(r) is always positive, so that NB(q) has a higher peak atq = 0. For smaller V2, B(r) is staggered so that the peakat q = π is dominant. In order to describe the long-rangebehavior of B(r) for large V2, we investigate the evolutionof NB(q) at q = 0. In figure 5(b), we show NB(q = 0) asa function of V2 for V1 = 4.0 and different chain lengthsL. One can see that a peak develops for different systemsizes. The finite-size analysis shows that the peak is located atV2 = 4.32 for the size L > 64, although the maximum of thepeak increases with increasing L. This feature indicates thatthe 2kF-CDW–BO phase transition takes place at V2 = 4.32for V1 = 4.0. From figure 5(b), we find that the size effect ofNB(q = 0) is relatively large at the critical point V2 = 4.32 butbecomes weaker and weaker as V2 moves apart from the peak.It is known that in the region with finite correlation length allthe physical quantities are less size-dependent and convergerapidly to the thermodynamic limit. In contrast, in the criticalregion where the correlation length becomes larger or infinite,the finite-size effect is more prominent. This result coincideswith the behavior of the BO correlation B(r) in figure 3:B(r) decays as a power law at the critical point but decaysexponentially apart from this point. The evolution of NB(0)can be explained as follows: when the system approachesthe 2kF-CDW–BO transition point from the 2kF-CDW phaseside, the decay of BO correlation B(r) becomes slower andmore long-range correlations contribute to the structure factorNB(q = 0). Therefore, NB(q = 0) increases and reaches
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J. Phys.: Condens. Matter 23 (2011) 365602 C-B Duan and W-Z Wang
Figure 5. (a) Static BO structure factor NB(q) near the BO-2kF-CDW transition point for V1 = 4.0, L = 200 and different V2.(b) NB(q = 0) as a function of V2 for V1 = 4.0 and different L.
Figure 6. NB(q = 0) as a function of V2 for V1 = 0. The peak givesthe BO-2kF-CDW critical point V2 = 2.60± 0.02.
the maximum at the critical point V2 = 4.32. With furtherdecrease of V2, B(r) decays exponentially and the negativecorrelation with the odd distance r = 2n + 1 appears so thatNB(q = 0) is reduced.
It is interesting to compare our results with those obtainedwith other approaches. Figure 6 shows the BO structure factorNB(q = 0) as a function of V2 for V1 = 0. The peak gives thecritical point V2 = 2.60 ± 0.02 which is consistent with theresults V2 = 2.66 ± 0.1 given by energy curvature and levelcrossing [14].
In addition to further describing the characteristics ofdifferent phases, we also investigate the charge structure factorNC(q) shown in figure 7. The size of the system is consideredfrom 24 to 192. The parameter V2 in figures 7(a)–(d)corresponds to the 4kF-CDW, metal, BO and 2kF-CDW
phases, respectively. For a small value of V2 (0.4), a sharppeak at q = π suggests that the region is dominated by4kF-CDW correlation. From the inset in figure 7(a), onecan see that NC(q = π) increases linearly with the size L.This feature can be understood in the extreme case V1 �
V2 as follows: in the 4kF-CDW phase, the ground state canbe approximately regarded as the superposition of the twodegenerate states |G〉 = α|1010 · · ·〉 + β|0101 · · ·〉, whereα and β satisfy the normalization condition α2
+ β2= 1.
A simple calculation gives the results C(r) = (−1)r 14 and
NC(π) = L/4. For large V2, e.g. V2 = 4.8 in figure 7(d),the system is in the 2kF-CDW phase with the configuration(1100). NC(q) peaks at q = π/2 and 3π/2. For a similarreason, NC(q = π/2) also increases linearly with size L.In the region between the 2kF-CDW and 4kF-CDW phases,with increasing V2, the peak at q = π is weakened andbecomes wider, while the peak at q = π/2 develops. Thus,the CDW correlation function in such a region can beregarded as a superposition of a 2kF-CWD and 4kF-CDWcorrelations.
Based on the above results, one find that the featuresof NC(q) in both 2kF-CDW and 4kF-CDW phase are ratherobvious, but exhibit a crossover between the two regions. Itseems that there is no visible trait to reveal the 2kF-CDW–BOphase transition. However, we observe a fast change inNC(q = π/2) as a function of V2 around the 2kF-CDW–BOtransition point V2 = 4.32, as shown in figure 8(a). Wepresent the first derivative of NC(π/2) with respect to V2 infigure 8(b), and find a peak exactly located at V2 = 4.32.The height of the peak is expected to diverge with increasingsystem size L, indicating a second-order transition. We alsocalculate the second derivative of the ground-state energy withrespect to V2 shown in figure 8(c). The divergent peak in thethermodynamic limit also suggests a second-order transitionat this point. This result is consistent with that from fieldtheory [6].
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J. Phys.: Condens. Matter 23 (2011) 365602 C-B Duan and W-Z Wang
Figure 7. Charge structure factor NC(q) for V1 = 4.0 and different V2 and system sizes L. The insets in (a) and (b) show the sizedependence of NC(q = π).
Figure 8. (a) NC(q = π/2) and (b) its first derivative N ′C(π/2) withrespect to V2, and (c) the second derivative E′′0(L/2)/L of theground-state energy with respect to V2 as functions of V2 forV1 = 4.0 and different system sizes L.
3.2. BO–metal transition
From the discussion about figure 4, we have known that forsmall V2, the BO correlation B(r) is staggered so that thestructure factor NB(q) exhibits a dominant peak at q = π .This feature is shown in figure 9 for different V2. The heightof the peak is not monotonic with increasing V2 and reachesa maximum at the BO–metal transition point V2 = 2.88 asshown in figure 10(a). The large size effect at the critical
Figure 9. BO structure factor NB(q) for V1 = 4.0, L = 200 anddifferent V2. It shows a dominant peak at q = π .
point indicates a large correlation length and a slower decayof the BO correlation B(r). Figure 11 shows the decay ofbond correlation B(r) for various V2 along the line V1 = 4.0near this transition point. We find that with decreasing V2,the decay of B(r) becomes slower. At V2 = 2.88, the decayis slowest. With further decreasing V2, the decay becomesfaster again. To confirm the critical point, we present thecharge gap 1c(L) as a function of V2 for different lengthsL in figure 10(b). The line with circles corresponds to theextrapolated value in the thermodynamic limit L → ∞ byusing equation (9). The detail of the extrapolation is the sameas previous works [19, 20]. We find that the gap disappears atthe critical point V2 = 2.88.
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J. Phys.: Condens. Matter 23 (2011) 365602 C-B Duan and W-Z Wang
Figure 10. (a) NB(q = π) and (b) charge gap 1c versus V2 forV1 = 4.0 and different L. The line with circles in (b) corresponds tothe charge gap in the L→∞ limit.
Figure 11. BO correlation B(r) as a function of r with r = 2n forvarious V2 with fixed V1 = 4.0. It shows the slowest decay at theBO–metal transition point V2 = 2.88.
It is hard to determine the exact phase boundary throughthe behavior of the gap because near the phase boundarythe gap disappears slowly. In contrast, we find that NB(q =π) exhibits the BO–metal transition more clearly. Figure 12shows the BO structure factor NB(q = π) as a function of V2for V1 = 0. The peak gives the critical point V2 = 2.24± 0.1which is consistent with the results V2 = 2.16±0.1 estimatedby energy curvature and level crossing [14].
3.3. Metal–4kF-CDW transition
So far, we have figured out the boundaries of the BO phase.However, as shown in figure 10, there is no obvious featurein the BO structure factors when the system transits fromthe metal phase to the 4kF-CDW phase. From figure 10(b),we estimate the metal–4kF-CDW transition point at aboutV2 = 1.2 where the charge gap is open. This transition is
Figure 12. NB(q = π) as a function of V2 for V1 = 0. The peakgives the BO–metal transition point V2 = 2.24± 0.1.
of KT type and the energy gap opens exponentially [10]. Inprevious Lanczos numerical and analytical results [12], theCDW correlation function C(r) has the decay behavior 1/rB,where 1 < B ≤ 2 in the metallic phase, B = 1 at the KTtransition point and B = 0 well into the 4kF-CDW phase. Toinvestigate such behavior, we show C(r) for different V1 andV2 in figure 13 on a log–log scale. For V2 = 0 in figure 13(a),with increasing V1, the slope of the curve decreases. We fit thecurves with a linear function ln(C(r)) = A − B ln(r) and findthe slope B = 0.99 at the critical point V1 = 2.0, which hasbeen determined by a previous analytical method [10]. Therelative error is smaller than 0.01. Similarly, for V1 = 4.0 infigure 13(b), we obtain the critical point V2 = 1.28, which isconsistent with the result from the energy gap in figure 10(b).
We present the complete phase diagram in figure 2. Thisresult is slightly different from previous works. It is knownthat in the atomic limit (t = 0), the ground state correspondsto the 4kF-CDW phase for V1 > 2V2 and the 2kF-CDW phasefor V1 < 2V2. Once the hopping energy t is switched on,the transition line V1 = 2V2 would be broadened to a metalregion. The previous Lanczos numerical results suggest thatin this region the ground state is a Luttinger liquid (LL) forV1 < 30 and a non-LL conducting state for V1 > 30 [9].The other numerical results based on energy curvature andlevel crossing [14] indicate that the BO and LL (metallic)phases extends along the line V1 = 2V2 with the width ≈
√5t.
For strong enough interaction, the LL phase disappears andthere exists a triple point between the BO, LL and 4kF-CDWphases. The triple point estimated is around (V1 = 8.2,V2 =
3.6). Our results in figure 2 also show that the width ofthe BO and metallic phases is about
√5t. With increasing
V1, the metallic phase becomes narrow and disappears at thetriple point (V1 ≈ 16.0,V2 ≈ 7.6). We believe the differenceis due to finite-size effects in the Lanczos method [9] andthe exponential smallness of the gap in the level crossingmethod [14]. However, all these results suggest that thenon-LL behavior appears along the line V1 = 2V2 for largeinteraction V1 and V2.
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J. Phys.: Condens. Matter 23 (2011) 365602 C-B Duan and W-Z Wang
Figure 13. CDW correlation C(r) near the metal–4kF-CDW transition points (a) (V1 = 2.0,V2 = 0) and (b) (V1 = 4.0,V2 = 1.28) on alog–log scale. The decay behavior 1/rB with B ≈ 1 at the critical points is observed.
4. Summary
We have shown that the BO correlation function B(r)and its static structure factor NB(q) can be employed toaccurately determine the BO phase boundaries in a spinlessfermion model with the nearest and next-nearest-neighborrepulsive interactions. Near the 2kF-CDW–BO boundary, B(r)is positive and NB(q) peaks at q = 0. NB(q = 0) has itsmaximum at the 2kF-CDW–BO transition point. Around theBO–metal critical point, B(r) is staggered and the maximumof NB(q = π) gives the transition point. At two boundariesof the BO phase, B(r) decays most slowly as a power law,while beyond the boundaries B(r) decays exponentially. Thefirst derivative of the charge structure factor NC(q) and thesecond derivative of the energy are expect to diverge atthe 2kF-CDW–BO transition point, which indicates that asecond-order transition occurs. The boundary between themetallic and 4kF-CDW phases is determined by the decaybehavior of the charge correlation C(r).
It is noticeable that the long-range Coulomb interactionsare rather prominent in the frustrated system such asthe transition-metal CuO and quasi-one-dimensional organicconductor [1–4]. Our studies here offer intriguing insights tounderstanding the charge ordered insulator–metal transitionand the low-energy electronic state.
Acknowledgments
This work is supported by the NSFC Grant Nos 50573059and 10874132.
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