The Derivation of the Exact Internal Energies for Spin Glass Models by Applying the Gauge Theory to the Fortuin-Kasteleyn Representation

# The Derivation of the Exact Internal Energies for Spin Glass Models by Applying the Gauge Theory to the Fortuin-Kasteleyn Representation

Chiaki Yamaguchi

## 1 Introduction

The theoretical studies of spin glasses have been widely done [1]. There are special lines on the phase diagrams for several spin glass models, where the lines are called the Nishimori line [2, 3, 4]. It is known that several physical quantities and several bounds for physical quantities are exactly calculated on the Nishimori line by using gauge transformations. The exact internal energies and the rigorous upper bounds of specific heats for several spin glass models have already been derived on the Nishimori line [2, 3, 4]. The aim of this article is to derive the exact internal energies and the rigorous upper bounds of specific heats for several spin glass models by applying the gauge theory to the Fortuin-Kasteleyn representation. As the spin glass models, the Ising model [2, 3] and a Potts gauge glass model [4] are studied. The Potts gauge glass model is a more complex version of the Ising model.

The Fortuin-Kasteleyn representation is a representation based on a percolation picture for spin-spin correlation [5, 6, 7]. By using the Fortuin-Kasteleyn representation, spin-spin correlations are directly treated. In the previous methods, the solutions have been directly calculated from the Boltzmann factor [2, 3, 4]. Instead, in the present method, the solutions are directly calculated from the Fortuin-Kasteleyn representation.

In this article, gauge transformations are used. The gauge transformations are treated in Refs. 2, 3, 4, 9, 8, 10 for example. It is known that a gauge transformation has no effect on thermodynamic quantities [9].

## 2 The Fortuin-Kasteleyn representation

We briefly explain the Fortuin-Kasteleyn representation [5, 6]. The Fortuin-Kasteleyn representation introduces auxiliary variables called graph G. The graph is a state by the weights between spins which are directly connected by the interaction. The partition function is expressed in the double summation over state and graph as [7]

 Z=∑S,GV(G)\mathchar28929\relax(S,G), (2.1)

where is a function that takes the value one when is compatible to and takes the value zero otherwise. A bond that probabilistically connects two spins by the weight of graph is especially called the active bond. A graph consists of a set of active bonds. The active bond is fictitious, and is used in order to generate a cluster composed of spins, which is often referred to as the Fortuin-Kasteleyn cluster. is the weight for the graph . The partition function is expressed as , where . This partition function is expressed in the summation over graph instead of state . Since the weight for the graph is used, the study of the percolation problem of the graph is expected to have a physical significance. This representation for graph is called the Fortuin-Kasteleyn representation.

We define the number of active bonds as . The number of states for the active bond number, , is given by

 \mathchar28938\relax(nb)=∑{G|nb(G)=nb}∑S\mathchar28929\relax(S,G). (2.2)

By using the , the partition function is expressed as

 Z=NB∑nb=0\mathchar28938\relax(nb)V(nb), (2.3)

where is the weight for the active bond number, and is the number of nearest-neighbor pairs in the whole system.

If two spins are on the same cluster, the two spins are correlated. If two spins are not on the same cluster, the two spins are not correlated. In the ferromagnetic Ising model, the percolation transition point of the Fortuin-Kasteleyn cluster agrees with the phase transition point [12]. On the other hand, in the Ising model, the percolation transition point of the Fortuin-Kasteleyn cluster disagrees with the phase transition point [13]. Instead, it is pointed out that, in the Ising model, there is a possibility that the percolation transition point of the Fortuin-Kasteleyn cluster agrees with a dynamical transition point [14]. For the applications of the Fortuin-Kasteleyn representation, the Swendsen-Wang algorithm [11] is probably the prime example. This algorithm is a Markov chain Monte Carlo method. By performing this algorithm, the Fortuin-Kasteleyn clusters are generated, and the states on each cluster are simultaneously updated. This algorithm produces a faster thermal equilibration when this algorithm is applied to the ferromagnetic Ising model [11]. In this article, we concentrate ourselves on the number of the active bonds, which generate the Fortuin-Kasteleyn clusters, and the fluctuation of the number of active bonds.

## 3 The ±J Ising model and the present results

The Hamiltonian for the Ising model, , is given by [1, 2, 3]

 \@fontswitchH=−∑⟨i,j⟩Ji,jSiSj, (3.1)

where denotes nearest-neighbor pairs, is a state of the spin at the site , and . is a strength of the exchange interaction between the spins at the sites and . The value of is given with a distribution . The distribution is given by

 P(Ji,j)=pδJi,j,J+(1−p)δJi,j,−J, (3.2)

where , and is the Kronecker delta. is the probability that the interaction is ferromagnetic, and is the probability that the interaction is antiferromagnetic. By using Eq. (3), the distribution is written as [2, 3, 8]

 P(Ji,j)=eβPJi,j2cosh(βPJ),Ji,j=±J, (3.3)

where is given by [2, 3, 8]

 βP=12Jlnp1−p. (3.4)

When the value of is consistent with the value of the inverse temperature , the line on the phase diagram for the temperature and , where Eq. (3) is satisfied, is called the Nishimori line.

A gauge transformation [2, 3, 9, 8] given by

 Ji,j→Ji,jσiσj,Si→Siσi (3.5)

is used where . By using the gauge transformation, the Hamiltonian part becomes , and the distribution part becomes

 ∏⟨i,j⟩P(Ji,j) = eβP∑⟨i,j⟩Ji,j[2cosh(βPJ)]NB (3.6) → ∑{σi}eβP∑⟨i,j⟩Ji,jσiσj2N[2cosh(βPJ)]NB,

where is the number of sites.

For the Ising model, is given by

 V(nb)=(e2βJ−1)nbe−NBβJ, (3.7)

where , is the temperature, and is the Boltzmann constant. The way of deriving Eq. (3.7) is described in Appendix A. We define the probability for putting the active bond as . The value of depends on the exchange interaction and the states of spins [5, 6, 7, 14, 8, 10]. For the Ising model, is given by [14, 8]

 PFK(Si,Sj,Ji,j)=1−e−βJi,jSiSj−β|Ji,j|. (3.8)

The way of deriving Eq. (3.8) is also described in Appendix A. By using the gauge transformation, the part becomes .

The internal energy is given by

 E=−∂∂β[lnZ]R, (3.9)

where denotes the random configuration average. By using Eqs. (2), (3.7) and (3.9), we obtain

 E=NBJ−2Je2βJe2βJ−1[⟨nb⟩T]R, (3.10)

where denotes the thermal average. is given by

 [⟨nb⟩T]R=∑⟨i,j⟩[⟨PFK(Si,Sj,Ji,j)⟩T]R. (3.11)

When , is obtained by using the gauge transformation as [8]

 [⟨PFK(Si,Sj,Ji,j)⟩T]R (3.12) = ∑{Jl,m}∏⟨l,m⟩P(Jl,m)∑{Sl}PFK(Si,Sj,Ji,j)e−βP\@fontswitchH({Sl},{Jl,m})∑{Sl}e−βP\@fontswitchH({Sl},{Jl,m}) = 12N[2cosh(βPJ)]NB∑{Jl,m}∑{Sl}PFK(Si,Sj,Ji,j)e−βP\@fontswitchH({Sl},{Jl,m}) = tanh(βPJ),

where , is the temperature on the Nishimori line. By using Eqs. (3.11) and (3.12), we obtain

 [⟨nb⟩T]R=NBtanh(βPJ). (3.13)

By using Eqs. (3.10) and (3.13), the internal energy is obtained as

 E=−NBJtanh(βPJ). (3.14)

This solution is exact, and is equivalent to the solution in Ref. 2.

The specific heat is given by

 C=kBβ2∂2∂β2[lnZ]R. (3.15)

By using Eqs. (2), (3.7) and (3.15), we obtain

 C = kB(βJ)2cosech2(βJ){−[⟨nb⟩T]R (3.16) + e2βJ([⟨n2b⟩T]R−[⟨nb⟩2T]R)}.

is given by

 [⟨n2b⟩T]R (3.17) = ∑⟨i,j⟩∑⟨k,l⟩[⟨PFK(Si,Sj,Ji,j)PFK(Sk,Sl,Jk,l)(1−δi,kδj,l−δi,lδj,k) + PFK(Si,Sj,Ji,j)(δi,kδj,l+δi,lδj,k)⟩T]R.

By performing a similar calculation with the calculation in Eq. (3.12), we obtain

 [⟨n2b⟩T]R = NB(NB−1)tanh2(βPJ) (3.18) + NBtanh(βPJ).

By applying the Cauchy-Schwarz inequality, we obtain

 [⟨nb⟩2]R≥[⟨nb⟩]2R=N2Btanh2(βPJ). (3.19)

Therefore, by using Eqs. (3.13), (3.16), (3.18) and (3.19), we obtain the upper bound of the specific heat as

 C≤kBNB(βPJ)2sech2(βPJ). (3.20)

This solution is rigorous, and is equivalent to the solution in Refs. 2, 3.

## 4 A Potts gauge glass model and the present results

The Hamiltonian for a Potts gauge glass model, , is given by [4]

 \@fontswitchH=−Jq∑⟨i,j⟩q−1∑ri,j=1e2πiq(νi,j+qi−qj)ri,j, (4.1)

where is a state of the spin at the site , and . is a variable related to the strength of the exchange interaction between the spins at the sites and , and . is the total number of states that a spin takes. The value of is given with a distribution . The distribution is given by

 P(νi,j)=pδνi,j,0+1−pq−1(1−δνi,j,0). (4.2)

The normalization of is given by

 q−1∑νi,j=0P(νi,j)=1. (4.3)

When for all pairs, the model becomes the ferromagnetic Potts model. When , the model becomes the Ising model. By using Eqs. (4.2) and (4.3), the distribution is written as [4, 10]

 P(νi,j)=AeβPq∑q−1ri,j=1J(ri,j)i,j(νi,j), (4.4)

where and are given by [4, 10]

 A = 1eβPJq(q−1)+(q−1)e−βPJq, (4.5) βP = 1Jln[p(q−11−p)] (4.6)

respectively. When the value of is consistent with the value of the inverse temperature , the line on the phase diagram for the temperature and , where Eq. (4.6) is satisfied, is called the Nishimori line.

We use representations: and . A gauge transformation [4, 10] given by

 J(ri,j)i,j→J(ri,j)i,jμq−ri,jiμri,jj,λi→λiμi (4.7)

is used where , is an arbitrary value for the spin state at the site , and . By using the gauge transformation, the Hamiltonian part becomes , and the distribution part becomes

 ∏⟨i,j⟩P(νi,j) = ANBeβPq∑⟨i,j⟩∑q−1ri,j=1J(ri,j)i,j(νi,j) (4.8) → ANBqN∑{μi}eβPq∑⟨i,j⟩∑q−1ri,j=1J(ri,j)i,j(νi,j)μq−ri,jiμri,jj.

For the Potts gauge glass model, is given by

 V(nb)=(eβJ−1)nbe−NBβJq. (4.9)

The way of deriving Eq. (4.9) is described in Appendix B. For the Potts gauge glass model, is given by [10]

 PFK(qi,qj,νi,j)= 1−exp{−βJq[q−1∑ri,j=1e2πiq(νi,j+qi−qj)ri,j+1]}. (4.10)

The way of deriving Eq. (4.10) is also described in Appendix B. By using the gauge transformation, the part becomes .

By using Eqs. (2), (3.9) and (4.9), the internal energy is given by

 E=NBJq−JeβJeβJ−1[⟨nb⟩T]R. (4.11)

is given by

 [⟨nb⟩T]R=∑⟨i,j⟩[⟨PFK(qi,qj,νi,j)⟩T]R. (4.12)

When , is obtained by using the gauge transformation as [10]

 [⟨PFK(qi,qj,νi,j)⟩T]R (4.13) = ∑{νl,m}∏⟨l,m⟩P(νl,m)∑{ql}PFK(qi,qj,νi,j)e−βP\@fontswitchH({ql},{νl,m})∑{ql}e−βP\@fontswitchH({ql},{νl,m}) = ANBqN∑{νl,m}∑{ql}PFK(qi,qj,νi,j)e−βP\@fontswitchH({ql},{νl,m}) = eβPJ−1eβPJ+q−1,

where is the inverse temperature on the Nishimori line. By using Eqs. (4.12) and (4.13), we obtain

 [⟨nb⟩T]R=NB(eβPJ−1)eβPJ+q−1. (4.14)

By using Eqs. (4.11) and (4.14), the internal energy is obtained as

 E=NBJq−NBJeβPJeβPJ+q−1. (4.15)

This solution is exact, and is equivalent to the solution in Ref. 4.

By using Eqs. (2), (3.15) and (4.9), the specific heat is given by

 C = kB(βJ2)2cosech2(βJ2){−[⟨nb⟩T]R (4.16) + eβJ([⟨n2b⟩T]R−[⟨nb⟩2T]R)}.

is given by

 [⟨n2b⟩T]R (4.17) = ∑⟨i,j⟩∑⟨k,l⟩[⟨PFK(qi,qj,νi,j)PFK(qk,ql,νk,l)(1−δi,kδj,l−δi,lδj,k) + PFK(qi,qj,νi,j)(δi,kδj,l+δi,lδj,k)⟩T]R.

By performing a similar calculation with the calculation in Eq. (4.13), we obtain

 [⟨n2b⟩T]R = NB(NB−1)(eβPJ−1)2(eβPJ+q−1)2 (4.18) + NB(eβPJ−1)eβPJ+q−1.

By applying the Cauchy-Schwarz inequality, we obtain

 [⟨nb⟩2]R≥[⟨nb⟩]2R=N2B(eβPJ−1)2(eβPJ+q−1)2. (4.19)

Therefore, by using Eqs. (4.14), (4.16), (4.18) and (4.19), we obtain the upper bound of the specific heat as

 C≤kBNB(βPJ)2eβPJ(q−1)(eβPJ+q−1)2. (4.20)

This solution is rigorous, and is equivalent to the solution in Ref. 4.

## 5 Summary

We derived the exact internal energies and the rigorous upper bounds of specific heats for the Ising model and a Potts gauge glass model by applying the gauge theory to the Fortuin-Kasteleyn representation. The results were derived on the Nishimori lines. The present solutions agreed with the previous solutions in Refs. [2, 3, 4]. The Fortuin-Kasteleyn representation is a representation based on a percolation picture for spin-spin correlation. The derivation of the solutions by the present method must be useful for understanding the relationship between the percolation picture for spin-spin correlation and the physical quantities on the Nishimori line.

## A The weight and the probability for active bond in the ±J Ising model

We will derive Eqs. (3.7) and (3.8). The framework for the way to derive Eqs. (3.7) and (3.8) is described in Ref. 7. We define the weight of two spins as . is given by

 w(Si,Sj,Ji,j)=exp(βJi,jSiSj). (A.1)

We define the weight for as . We obtain

 wpara(Si,Sj,Ji,j)=exp(βJ). (A.2)

We define the weight for as . We obtain

 wanti(Si,Sj,Ji,j)=exp(−βJ). (A.3)

We define the weight of graph for connecting two spins as . We define the weight of graph for disconnecting two spins as . We are able to write

 wpara(Si,Sj,Ji,j) = w(gconn)+w(gdisc), (A.4) wanti(Si,Sj,Ji,j) = w(gdisc). (A.5)

By using Eqs. (A.2), (A.3), (A.4) and (A.5), we obtain

 w(gconn) = exp(βJ)−exp(−βJ), (A.6) w(gdisc) = exp(−βJ). (A.7)

By using Eqs. (A.6) and (A.7), we obtain the weight for the active bond number as

 V(nb)=(eβJ−e−βJ)nb(e−βJ)NB−nb (A.8)

The above equation is equal to Eq. (3.7). We define the probability of connecting two spins for as . We define the probability of connecting two spins for as . We are able to write

 Ppara(gconn) = w(gconn)w(gconn)+w(gdisc), (A.9) Panti(gconn) = 0. (A.10)

By using Eqs. (A.6), (A.7), (A.9), (A.10), we derive Eq. (3.8).

## B The weight and the probability for active bond in a Potts gauge glass model

We will derive Eqs. (4.9) and (4.10). The framework for the way to derive Eqs. (4.9) and (4.10) is described in Ref. 7. We define the weight of two spins as . is given by

 w(qi,qj,νi,j) (B.1) =

We define the weight for as . We obtain

 wpara(qi,qj,νi,j)=exp[βJ(q−1)q]. (B.2)

We define the weight for as . We obtain

 (B.3)

We define the weight of graph for connecting two spins as . We define the weight of graph for disconnecting two spins as . We are able to write

 wpara(qi,qj,νi,j) = w(gconn)+w(gdisc), (B.4) wanti(qi,qj,νi,j) = w(gdisc). (B.5)

By using Eqs. (B.2), (B.3), (B.4) and (B.5), we obtain

 w(gconn) = exp[βJ(q−1)q]−exp(−βJq), (B.6) w(gdisc) = exp(−βJq). (B.7)

By using Eqs. (B.6) and (B.7), we obtain the weight for the active bond number as

 V(nb)=[eβJ(q−1)q−e−βJq]nb(e−βJq)NB−nb. (B.8)

The above equation is equal to Eq. (4.9). We define the probability of connecting two spins for as . We define the probability of connecting two spins for as . We are able to write

 Ppara(gconn) = w(gconn)w(gconn)+w(gdisc), (B.9) Panti(gconn) = 0. (B.10)

By using Eqs. (B.6), (B.7), (B.9) and (B.10), we derive Eq. (4.10).

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