Alternating Layered Ising Models : Effects of...

Alternating Layered Ising Models :Effects of connectivity and proximity

Helen Au-Yang and Michael E Fisher

March 31, 2013

IntroductionExperimentsAlternating Layered Model

Reultss=1s=2various s

ScalingReviewnear T1c

near T2c

EnchancementDefinitionbehavior

ConclusionExact solvable model

Experiments on 4He by Gasparini an CoworkersExperiments on 4He at the superfluid transition

Critical Point Coupling and Proximity Effects in 4He at the Superfluid Transition

Justin K. Perron* and Francis M. Gasparini†

Department of Physics, University at Buffalo, The State University of New York, Buffalo, New York 14260, USA(Received 11 January 2012; published 18 July 2012)

We report measurements of the superfluid fraction �s=� and specific heat cp near the superfluid

transition of 4He when confined in an array of ð2 �mÞ3 boxes at a separation of S ¼ 2 �m and coupled

through a 32.5 nm film. We find that cp is strongly enhanced when compared with data where coupling is

not present. An analysis of this excess signal shows that it is proportional to the finite-size correlation

length in the boxes �ðt; LÞ, and it is measurable as far as S=�� 30� 50. We obtain �ð0; LÞ and the scalingfunction (within a constant) for �ðt; LÞ in an L3 box geometry. Furthermore, we find that �s=� of the film

persists a full decade closer to the bulk transition temperature T� than a film uninfluenced by proximity

effects. This excess in �s=� is measurable even when S=� > 100, which cannot be understood on the basis

of mean field theory.

DOI: 10.1103/PhysRevLett.109.035302 PACS numbers: 67.25.dr, 64.60.an, 71.45.Gm, 74.45.+c

With low temperature superconductors, coupling andproximity effects are manifest on the scale of the zero-temperature correlation length �o. This leads to the famil-iar Josephson effects in weak-link junctions and proximityeffects at a superconductor-normal metal interface [1]. Onemight suppose that in 4He near the superfluid transitiontemperature T� analogous effects would occur on the scaleof the temperature-dependent bulk correlation length �ðtÞwhere t ¼ j1� T=T�j. Indeed, Josephson effects havebeen measured between bulk superfluids separated byweak links of dimensions ��ðtÞ [2,3]. However, recentmeasurements with arrays of 4He dots have demonstratedthat proximity effects exist over a much larger scale [4].Here we report measurements which quantify both prox-imity and coupling. To see both of these effects, one mustarrange for helium to be confined in contiguous regionswith different superfluid transition temperatures. In ourcase, this is an array of L3 boxes separated by and linkedthrough a uniform thin film. We vary the coupling betweenboxes by changing their separation. At large separation, thehelium in the boxes will behave as isolated dots, while atvery small separation and, hence, large coupling, the arraywill behave like a two dimensional film of thickness L. Thethin film in equilibrium with the boxes will be influencedby the boxes both in its specific heat and its superfluiddensity. This influence will be present even in the limit oflarge separation of the boxes when the coupling amongthem is very small. Thus, even though the boxes-filmsystem should be considered together as a single thermo-dynamic system, these effects can be separated andidentified.

Surprisingly, the observed effects are manifest atdistances much larger than �ðtÞ. When one considers thatthis system is finite and the divergence of �ðtÞ is notphysically possible, this becomes even more surprising.Indeed, in this system �ðtÞ must deviate from the bulkbehavior to some finite-size correlation length �ðt; LÞ

which must round off to a value & L. Our work showsthat the observed effects, although existing over distancesmany times �ðt; LÞ, are governed by �ðt; LÞ. We note that�ðt; LÞ, just like all of the thermodynamic responses nearthe transition, can be described by a scaling function f suchthat �ðt; LÞ ¼ �ðt;1ÞfðL=�ðt;1ÞÞ [5], or equivalently

�ðt; LÞ ¼ LXðtðL=�0Þ1=�Þ [6]. The latter form is perhapsmore intuitive because at t ¼ 0, �ð0; LÞ ¼ LXð0Þ withXð0Þ � 1, since �ðt; LÞ cannot become larger than L. Incontrast to low temperature superconductors, we believethe long range effect is a reflection of the role of criticalfluctuations. Thus, these coupling-proximity effects arenew phenomena which should also be manifest in othersystems, such as magnets at the critical point.Measurements reported in this Letter are made on 4He

confined in an array of 69 million ð2 �mÞ3 boxes spaced2 �m edge-to-edge and connected through a 32:5�1:2 nm thick film (see Fig. 1). The film extends along theperimeter of the cell beyond the limits of the array of

FIG. 1 (color online). Schematic rendering, not to scale, of theconfinement cell. The cell is formed with two 50 mm diametersilicon wafers bonded at a separation determined by the 32.5 nmposts and ring. This oxide pattern is formed on the top wafer. Thebottom wafer has an array of ð2 �mÞ3 boxes at 2 �m separation.

PRL 109, 035302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending20 JULY 2012

0031-9007=12=109(3)=035302(4) 035302-1 � 2012 American Physical Society

Figure: Experiments on small boxes of helium were coupled through athin helium film.

Model

Alternating Layered Ising Model:

J1 J2

· · · · · · · · ·

J1 J2

m1 m2 m1 m2- -� -� � - -� -� �

Figure: The model consists of infinite strips of width m1 in which thecoupling energy between the nearest neighbor Ising spins is J1 separatedby other infinite strips of width m2 whose coupling J2 is “weaker”.(σ = ±1).

Relative strengths r and relative separations s

r = J2/J1 < 1, s = m2/m1.

Model

Alternating Layered Ising Model:

J1 J2

· · · · · · · · ·

J1 J2

m1 m2 m1 m2- -� -� � - -� -� �

Figure: The model consists of infinite strips of width m1 in which thecoupling energy between the nearest neighbor Ising spins is J1 separatedby other infinite strips of width m2 whose coupling J2 is “weaker”.(σ = ±1).

Relative strengths r and relative separations s

r = J2/J1 < 1, s = m2/m1.

r = 0 (J2 = 0): 1-D Ising: No discontinuities

Figure: Specific heats for r = 0 : noninteracting infinite strips of finite width m1.

I J2 = 0, the model→ 1D. Specific heatnot divergent, butrather has a fullyanalytic roundedpeak.

I The temperature ofthe maximum isblow T1c andincreases as m1

increases, andapproaches the bulkcritical point T1c asm1 →∞.

I Finite-size scalingholds.













r 6= 0 : 2-D Ising: α = 0, β = 1/8, ν = 1.

Figure: Specific heats for r = 0.3 and s = 1 form1 = m2 = 2, 4, 6, 8, 12 and 16. Dotted verticalline :Tc .

I Specific heatsdivergent at Tc

logarithmically.

I m1 increases, thedivergencebecome a barelyvisible spike.

I and two roundedpeaks appear andmove toward thelimiting valuesT1c and T2c .

r 6= 0 : 2-D Ising: α = 0, β = 1/8, ν = 1.



logarithmically.



r 6= 0 : 2-D Ising: α = 0, β = 1/8, ν = 1.



logarithmically.



r 6= 0 : 2-D Ising: α = 0, β = 1/8, ν = 1.



logarithmically.



Critical Temperature Tc(r , s)Critical temperature for random layered models [McCoy andWu, Fisher]

2〈〈Jy 〉〉 = kBTc〈〈ln coth(Jx/kBTc)〉〉,

where the brackets 〈〈·〉〉 denote an average over the distribution,

Critical temperature for alternating layered models

2J1m1+2J2m2 = kBTc [m1 ln coth(J1/kBTc)+m2 ln coth(J2/kBTc)].

The critical temperature : Tc = Tc(r , s)

2J1(1 + rs) = kBTc [ln coth(J1/kBTc) + s ln coth(rJ1/kBTc)],

The limiting values T1c and T2c(r) = rT1c

kBT1c/J1 = kBT2c/(rJ1) = 2/ ln(√

2 + 1) ' 2.269185312.










2 + 1) ' 2.269185312.










2 + 1) ' 2.269185312.










2 + 1) ' 2.269185312.

m2 = 2m1, s = 2

Figure: Specific heats for r = 0.3 and s = 2.The dashed plots: J2 = 0 ; dotted lines:J1 = 0. Go to uppersc

Go to uppersc

I Upper peaks belowT1c . Solid curveabove dash.

I Lower peaks aboveT2c , differ fromdotted lines.

m2 = 2m1, s = 2


Go to uppersc



m2 = 2m1, s = 2


Go to uppersc



m2 = 2m1, s = 2


Go to uppersc



m1 = 18, s = 1/3, 1, 5/3

Figure: Specific heats for r = 0.3 andm1 = 18. The dashed plots: J2 = 0 ; dottedlines: J1 = 0.

I Tc decreases as sincreases.

I Upper peaks belowT1c ; lower peaksabove T2c

m1 = 18, s = 1/3, 1, 5/3




m1 = 18, s = 1/3, 1, 5/3




m1 = m2 = 16, r = 0.5, 0.7, 0.9

Figure: Specific heats for r = 0.5, 0.7, 0.9 andm1 = 16; s = 1.

I T2c and Tc

increases as rincreases.

I Logarithmicdivergence isvisible forr = 0.7,

I and dominatesentirely forr = 0.9.

m1 = m2 = 16, r = 0.5, 0.7, 0.9


I T2c and Tc




m1 = m2 = 16, r = 0.5, 0.7, 0.9


I T2c and Tc




Scaling behavior near T1cGoto s2

C1(J1, J2; T )=(1 + s)[C (J1, J2; T )− C (0, J2; T )]

Figure: Plots of ∆C1(J1, J2; T ) (solid minusdotted, and subtract its value at T1c).

Data collapse:∆C1(T )=C1(T )−C1(T1c) areindependent of m2.

The solid curve is theplot of the specific heatof an infinite strip ofwidth m1 = 18 andcoupling J1 when itsvalue at T1c issubtracted.

Scaling behavior near T1cGoto s2

C1(J1, J2; T )=(1 + s)[C (J1, J2; T )− C (0, J2; T )]

Figure: Plots of ∆C1(J1, J2; T ) (solid minusdotted, and subtract its value at T1c).


The solid curve is theplot of the specific heatof an infinite strip ofwidth m1 = 18 andcoupling J1 when itsvalue at T1c issubtracted.

Scaling behavior of a single infinite strip of width m:I When m/ξ(T ) >> 1, the system behaves as the bulk 2D

Ising;

C strip(J; m; T ) = Bulk specific heat +1

mSurface contribution.

I α = 0 and ν = 1: Near Tc , it obeys the scaling hypothesis

C strip(J; m; T ) = A ln m + Q(x) + O(m−1,m−1 ln m)

where is the scaling variable of x is defined by

x = m[(T/Tc)− 1] ∝ m/ξ(T ).

∆C strip(T )=C strip(J; m; T )−C strip(J; m; Tc)=Q(x)−Q(0).

I Relationship between a single strip with a system ofnoninteracting strips.

C strip(J1; m1; T ) = (1 + s)C (J1, 0; m1,m2; T ),C strip(J2; m2; T ) = (1 + s−1)C (0, J2; m1,m2; T ).


Ising;






x = m[(T/Tc)− 1] ∝ m/ξ(T ).





Ising;






x = m[(T/Tc)− 1] ∝ m/ξ(T ).




Scaling behavior of the Upper Net Finite-size Contribution:I When T ∼ T1c , ξ1(T ) = 1/|T/T1c − 1| � 1, ξ2(T ) small, so

that when m2/ξ2(T )� 1 we have;

C (0, J2; m1,m2; T ) =m2

m1 + m2[Bulk specific heat for J2

+1

m2Surface contribution + · · · ]

C (J1, J2; m1,m2; T ) =m2


+1

m2Term similar to strips with surfaces + · · · ]

I

C1(J1, J2; T ) =m1 + m2

m1[C (J1, J2; T )−C (0, J2; T )]

is independent of m2.I In the limit: T → T1c , m1 →∞ : x1 = [(T/T1c)− 1]m1

∆C1(J1, J2; T ) = C1(J1, J2; T )− C1(J1, J2; T1c)≈ C strip(J1; m1; T )− C strip(J1; m1; T1c) ≈ Q(x1)− Q(0).



C (0, J2; m1,m2; T ) =m2


+1


C (J1, J2; m1,m2; T ) =m2


+1

m2Term similar to strips with surfaces + · · · ]I

C1(J1, J2; T ) =m1 + m2

m1[C (J1, J2; T )−C (0, J2; T )]

is independent of m2.

I In the limit: T → T1c , m1 →∞ : x1 = [(T/T1c)− 1]m1




C (0, J2; m1,m2; T ) =m2


+1


C (J1, J2; m1,m2; T ) =m2


+1

m2Term similar to strips with surfaces + · · · ]I

C1(J1, J2; T ) =m1 + m2

m1[C (J1, J2; T )−C (0, J2; T )]

is independent of m2.I In the limit: T → T1c , m1 →∞ : x1 = [(T/T1c)− 1]m1


Scaling Behavior of the Lower Net Finite-size Contribution

I Similarly T ∼ T2c , ξ2(T ) = 1/|T/T2c − 1| � 1, ξ1(T ) small,so that when m1/ξ1(T )� 1

C2(J1, J2; T ) = (1 + s−1)[C (J1, J2; T )−C (J1, 0; T )],


I Scaling behavior, in the limit T → T2c , m2 →∞ with fixedx2 = [(T/T2c)− 1]m2, of the lower net finite size contributionis

∆C2(J1, J2; T ) = C2(J1, J2; T )− C2(J1, J2; T2c)≈ Q(−x2)− Q(0).

Scaling Behavior of the Lower Net Finite-size Contribution

I Similarly T ∼ T2c , ξ2(T ) = 1/|T/T2c − 1| � 1, ξ1(T ) small,so that when m1/ξ1(T )� 1

C2(J1, J2; T ) = (1 + s−1)[C (J1, J2; T )−C (J1, 0; T )],


I Scaling behavior, in the limit T → T2c , m2 →∞ with fixedx2 = [(T/T2c)− 1]m2, of the lower net finite size contributionis

∆C2(J1, J2; T ) = C2(J1, J2; T )− C2(J1, J2; T2c)≈ Q(−x2)− Q(0).

Scaling behavior near T2c

Figure: Plots of ∆C2(J1, J2; T ) for m2 = 16,and m1 = 4, 8, · · · , 32.


The solid curve isscaling functionQ(−x2)− Q(0), anddashed line form2 = 60.

Scaling behavior near T2c

Figure: Plots of ∆C2(J1, J2; T ) for m2 = 16,and m1 = 4, 8, · · · , 32.


The solid curve isscaling functionQ(−x2)− Q(0), anddashed line form2 = 60.

Enhancement E(J1, J2;T )

E(J1, J2; T )=C (J1, J2; T )− C (J1, 0; T )− C (0, J2; T )

Figure: Plots of E(T ) for r = 0.3 andm1 = 8 and various s.

T (T ) = T2c−(T−T2c).

Enhancement (m1 + m2)E(J1, J2;T )

The enhancement E(t): (a) for m1 = 8 and (b) for m1 = 16, butmultiplied by m1 + m2. The short vertical lines locate thecorresponding upper limiting critical points, T1c .





Enchancement (m1 + m2)E(J1, J2;T )

Figure: Plots of the rescaled enhancement(m1 + m2)E(T ) for m1 = 8, 16 and 32 andseparations s = 1/2, 1, 2, 4, 8 versus thereduced temperature variable, t†(t), definedin the text.

t†(T ) = t/c1 + (1 + s)−1

for t ≥ 0;

t†(T ) = t/c2 + (1 + s)−1

for t ≤ 0.

c1 = (1+s−1)[(T1c/Tc)−1],c2 = (1 + s)[(T2c/Tc)− 1].

E1c(m1) = E(m1,m2; T1c)

(m1 + m2)E1c(m1) '0.0800 ln(m1/m0)

m0 = 0.36.



t†(T ) = t/c1 + (1 + s)−1

for t ≥ 0;

t†(T ) = t/c2 + (1 + s)−1

for t ≤ 0.

c1 = (1+s−1)[(T1c/Tc)−1],c2 = (1 + s)[(T2c/Tc)− 1].

E1c(m1) = E(m1,m2; T1c)

(m1 + m2)E1c(m1) '0.0800 ln(m1/m0)

m0 = 0.36.



t†(T ) = t/c1 + (1 + s)−1

for t ≥ 0;

t†(T ) = t/c2 + (1 + s)−1

for t ≤ 0.

c1 = (1+s−1)[(T1c/Tc)−1],c2 = (1 + s)[(T2c/Tc)− 1].

E1c(m1) = E(m1,m2; T1c)

(m1 + m2)E1c(m1) '0.0800 ln(m1/m0)

m0 = 0.36.

Summary

The enhancements of the specific heats of the strong layers and ofthe overall critical temperature, Tc(J1, J2; m1,m2), arising fromthe collective effects reflect the observations of Gasparini andcoworkers in experiments on confined superfluid helium.

Explicitly, we demonstrate that finite-size scaling holds in thevicinity of the upper limiting critical point T1c and close to thecorresponding lower critical limit T2c when m1 and m2 increase.

The residual enhancement, defined via appropriate subtractions ofleading contributions from the total specific heat, is dominated(away from T1c) by a decay factor 1/(m1 + m2) arising from theseams (or boundaries) separating the strips; close to T1c the decayis slower by a factor ln(m1/m0).

Summary




Summary




Exact Solvable Model

Exact solvable Models

I Ising model: a simplification. Less variables.

I By solving it exactly, it may shade light on real experiments.

I Conjectures on 3D Ising is useless and worse if it does notagree with series expansions.

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Gasparini’s Results

Critical Point Coupling and Proximity Effects in 4He at the Superfluid Transition

Justin K. Perron* and Francis M. Gasparini†

Department of Physics, University at Buffalo, The State University of New York, Buffalo, New York 14260, USA(Received 11 January 2012; published 18 July 2012)

We report measurements of the superfluid fraction �s=� and specific heat cp near the superfluid

transition of 4He when confined in an array of ð2 �mÞ3 boxes at a separation of S ¼ 2 �m and coupled

through a 32.5 nm film. We find that cp is strongly enhanced when compared with data where coupling is

not present. An analysis of this excess signal shows that it is proportional to the finite-size correlation

length in the boxes �ðt; LÞ, and it is measurable as far as S=�� 30� 50. We obtain �ð0; LÞ and the scalingfunction (within a constant) for �ðt; LÞ in an L3 box geometry. Furthermore, we find that �s=� of the film

persists a full decade closer to the bulk transition temperature T� than a film uninfluenced by proximity

effects. This excess in �s=� is measurable even when S=� > 100, which cannot be understood on the basis

of mean field theory.

DOI: 10.1103/PhysRevLett.109.035302 PACS numbers: 67.25.dr, 64.60.an, 71.45.Gm, 74.45.+c

With low temperature superconductors, coupling andproximity effects are manifest on the scale of the zero-temperature correlation length �o. This leads to the famil-iar Josephson effects in weak-link junctions and proximityeffects at a superconductor-normal metal interface [1]. Onemight suppose that in 4He near the superfluid transitiontemperature T� analogous effects would occur on the scaleof the temperature-dependent bulk correlation length �ðtÞwhere t ¼ j1� T=T�j. Indeed, Josephson effects havebeen measured between bulk superfluids separated byweak links of dimensions ��ðtÞ [2,3]. However, recentmeasurements with arrays of 4He dots have demonstratedthat proximity effects exist over a much larger scale [4].Here we report measurements which quantify both prox-imity and coupling. To see both of these effects, one mustarrange for helium to be confined in contiguous regionswith different superfluid transition temperatures. In ourcase, this is an array of L3 boxes separated by and linkedthrough a uniform thin film. We vary the coupling betweenboxes by changing their separation. At large separation, thehelium in the boxes will behave as isolated dots, while atvery small separation and, hence, large coupling, the arraywill behave like a two dimensional film of thickness L. Thethin film in equilibrium with the boxes will be influencedby the boxes both in its specific heat and its superfluiddensity. This influence will be present even in the limit oflarge separation of the boxes when the coupling amongthem is very small. Thus, even though the boxes-filmsystem should be considered together as a single thermo-dynamic system, these effects can be separated andidentified.

Surprisingly, the observed effects are manifest atdistances much larger than �ðtÞ. When one considers thatthis system is finite and the divergence of �ðtÞ is notphysically possible, this becomes even more surprising.Indeed, in this system �ðtÞ must deviate from the bulkbehavior to some finite-size correlation length �ðt; LÞ

which must round off to a value & L. Our work showsthat the observed effects, although existing over distancesmany times �ðt; LÞ, are governed by �ðt; LÞ. We note that�ðt; LÞ, just like all of the thermodynamic responses nearthe transition, can be described by a scaling function f suchthat �ðt; LÞ ¼ �ðt;1ÞfðL=�ðt;1ÞÞ [5], or equivalently

�ðt; LÞ ¼ LXðtðL=�0Þ1=�Þ [6]. The latter form is perhapsmore intuitive because at t ¼ 0, �ð0; LÞ ¼ LXð0Þ withXð0Þ � 1, since �ðt; LÞ cannot become larger than L. Incontrast to low temperature superconductors, we believethe long range effect is a reflection of the role of criticalfluctuations. Thus, these coupling-proximity effects arenew phenomena which should also be manifest in othersystems, such as magnets at the critical point.Measurements reported in this Letter are made on 4He

confined in an array of 69 million ð2 �mÞ3 boxes spaced2 �m edge-to-edge and connected through a 32:5�1:2 nm thick film (see Fig. 1). The film extends along theperimeter of the cell beyond the limits of the array of

FIG. 1 (color online). Schematic rendering, not to scale, of theconfinement cell. The cell is formed with two 50 mm diametersilicon wafers bonded at a separation determined by the 32.5 nmposts and ring. This oxide pattern is formed on the top wafer. Thebottom wafer has an array of ð2 �mÞ3 boxes at 2 �m separation.

PRL 109, 035302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending20 JULY 2012

0031-9007=12=109(3)=035302(4) 035302-1 � 2012 American Physical Society

Figure: Experiments on small boxes of helium were coupled through athin helium film.

References

J. K. Perron, and F. M. Gasparini, M. O. Kimball, K. P.Mooney, M Diaz-Avila, Finite-size Scaling of 4He at theSuperfuid Transition, Rev. Mod. Phys. 80, 1009–1059 (2008).

J. K. Perron, M. O. Kimball, K. P. Mooney, and F. M.Gasparini, Lack of Correlation-length Scaling for an Array ofBoxes, J. Phys.: Conf. Ser. 50, 032082–032084 (2009).

J. K. Perron, M. O. Kimball, K. P. Mooney, and F. M.Gasparini, Coupling and Proximity Effects in the SuperfluidTransition in 4He Dots, Nature Physics 6, 499–502 (2010).

J. K. Perron, and F. M. Gasparini Critical Point Coupling andProximity Effects in 4He at the Superfuid Transition, Phys.Rev. Lett. 109, 035302 (2012).

J. K. Perron, M. O. Kimball, K. P. Mooney and F. M.Gasparini Critical Behavior of Couplec 4He regions near theSuperfuid Transition, Phys. Rev. B. 87, 094507 (2013).

Spontaneous Magnetization β = 1/8

T < T2c < T1c

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ + + +

T2c < T < Tc

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ ± + ±

Tc < T < T1c

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ ± – ±


T < T2c < T1c

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ + + +

T2c < T < Tc

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ ± + ±

Tc < T < T1c

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ ± – ±


T < T2c < T1c

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ + + +

T2c < T < Tc

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ ± + ±

Tc < T < T1c

· · · · · · · · ·m1 m2 m1 m2- -� -� � - -� -� �

+ ± – ±

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