Analyticity of the energy in an Ising spin glass with correlated disorder

We first show that the average energy can be expressed in a simple formula in terms of the $Z_{2}$ gauge Higgs model in a subspace of the phase diagram.

The average energy $E$ of the model defined in the preceding section is written as

where $H$ is the Hamiltonian of Eq. (6) and the angular brackets denote the thermal average,

with the partition function of the Ising model,

Notice that the constant multiplicative factor $K$ is dropped for simplicity in Eq. (8) of the energy.

Following the standard prescription [6, 1], we apply the gauge transformation

to Eqs. (7) and (9). The thermal average $\langle H\rangle_{K}$ is gauge invariant, and only the term in the exponent of Eq. (7) with coefficient $K_{1}$ is affected. We sum the resulting expression over all possible values of $\{\sigma_{i}\}$, and divide the expression by $2^{N}$, and find

Since $\tau_{\Box}$ is gauge invariant and $\{\sigma_{i}\}$ appears only in the term with coefficient $K_{1}$, the factor in the numerator $\sum_{\{\sigma_{i}\}}e^{K_{1}\sum\tau_{ij}\sigma_{i}\sigma_{j}}$ cancels with the denominator $Z_{s}=\sum_{\{S_{i}\}}e^{K\sum\tau_{ij}S_{i}S_{j}}$ when $K=K_{1}$ and we are left with a simplified expression

where we have applied the gauge transformation $\tau_{ij}\to\tau_{ij}S_{i}S_{j}$ to the second line to derive the third line.

In the case of uncorrelated distribution of disorder ($K_{2}=0$), the condition $K=K_{1}$ defines a line in the $K$-$K_{1}$ phase diagram, sometimes called the Nishimori line [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 9]. It is known that the low-temperature side of this line $(K=K_{1}\gg 1)$ lies in the ferromagnetic phase in two and higher dimensions [8, 39] and that the spin glass phase exists away from the line [6, 1]. The line generally goes across a multicritical point where the paramagnetic, ferromagnetic and spin glass phases meet in three and higher dimensions [1, 40]. Our present model with correlation in disorder has an additional parameter $K_{2}$ to control the degree of correlation, and the condition $K=K_{1}$ defines a subspace in the three-dimensional phase diagram drawn in terms of $K_{1},K_{2}$, and $K$.

The expression of the average energy in Eq. (12) in the subspace $K=K_{1}$ can be interpreted as the expectation value of the local bond variable $\tau_{ij}$ summed over $\langle ij\rangle$ for the $Z_{2}$ gauge Higgs model [16] ²²2 Notice that the general expression of the energy of Eq. (7) is also regarded as the expectation value of a function of $\{\tau_{ij}\}$, $\langle H\rangle_{K}(\{\tau_{ij}\})$, with respect to the gauge Higgs probability weight $P_{\tau}$. However, this $\langle H\rangle_{K}(\{\tau_{ij}\})$ is a very complicated function of the bond variables $\{\tau_{ij}\}$ involving all types of products and summations of $\{\tau_{ij}\}$, which makes it hard to analyze its properties.. The condition $K=K_{1}$ has greatly simplified the expression of the average energy, leaving only the expectation value of a single $\tau_{ij}$ summed up over $\langle ij\rangle$.

The average energy of Eq. (12) is an analytic function of $K(=K_{1})$ for sufficiently small $K_{2}(>0)$:

This result is a known property of the $Z_{2}$ lattice gauge theory, see e.g. Ref. [42] for a theoretical analysis and Refs. [43, 44, 45, 46, 47] for numerical studies. Since the argument presented in Ref. [42] is fairly sketchy, we provide a more formal proof in Appendix A.

When $K_{2}=0$ (uncorrelated disorder), the analyticity is trivial. Equation (12) immediately gives a simple formula

where $N_{B}$ is the number of bonds. Nevertheless, the line defined by $K=K_{1}$ in the $K$-$K_{1}$ phase diagram has a transition point on it since the low-temperature side ($K\gg 1)$ is in the ferromagnetic phase in two and higher dimensions [8] and the high-temperature side is trivially in the paramagnetic phase. Therefore the absence of singularity in $E(K,K,0)$ is highly non-trivial, a consequence of delicate balance of the probability weight $P(\tau,K_{1},0)$ and the partition function $Z_{s}(K)$ in Eq. (11). Theorem 1 claims that this non-trivial analyticity remains valid even after the introduction of correlation in disorder. This may look rather trivial mathematically, but we believe it to be highly non-trivial physically.

Although it is difficult to prove that the subspace $K=K_{1}$ has a transition point (critical point) on it when $K_{2}>0$, in contrast to the case $K_{2}=0$ where it is proven [8, 9], we expect that the physical properties of the system are unlikely to change dramatically by the introduction of small but finite $K_{2}$ as suggested by the continuation of analyticity of the average energy as in Theorem 1. An indirect indication of the stability of the low-temperature ($K=K_{1}\gg 1$) ferromagnetic phase after the introduction of $K_{2}$ is provided by small-$K_{2}$ perturbation of the distribution function of a single-bond variable defined by

where $\delta$ is Kronecker’s delta. We define the probability for a bond to be positive (ferromagnetic) $p_{+}$ by

Then, as shown in Appendix B, $p_{+}$ is found by first-order perturbation in $\tanh K_{2}$ as

where $c$ is a positive constant and $p$ is the probability of $\tau_{ij}=1$ when $K_{2}=0$ (i.e., $p_{+}$ for $K_{2}=0$), defined through the ratio of probabilities for negative $(1-p)$ and positive $(p)$ values of $\tau_{ij}$: $e^{-K_{1}}/e^{K_{1}}=(1-p)/p$. Equation (16) suggests that the probability of a single bond to be positive (ferromagnetic) increases as we introduce small $K_{2}$. This implies (but does not prove) that the low-temperature ferromagnetic phase becomes more stable by the introduction of $K_{2}>0$. This is intuitively reasonable because $K_{2}$ suppresses frustration, increasing stability of the ferromagnetic phase. Since the existence of a paramagnetic phase for $K=K_{1}\ll 1$ is trivial, we may reasonably expect that there is a phase transition as a function of $K=K_{1}$ for $K_{2}>0$ in spite of the absence of singularity in the average energy.

The range of $K_{2}$ where the analyticity of average energy holds depends on the spatial dimension and the lattice structure. In the case of the two-dimensional square lattice, the $Z_{2}$ gauge Higgs model of Eq. (12) is equivalent to the conventional ferromagnetic Ising model in a finite field on the square lattice according to the duality relation [48]. Since the ferromagnetic Ising model in a field has no singularity, the energy of Eq. (12) is analytic for any positive value of $K_{2}$ ³³3 This duality equivalence to the two-dimensional ferromagnetic Ising model in a field also implies the impossibility of the exact closed-form formula of the average energy. . In three dimensions, Monte Carlo simulation shows the range of analyticity to be $0\leq K_{2}<0.6893$ on the cubic lattice [47].

It is generally difficult to derive an explicit compact expression of the average energy for $K=K_{1}$ except for the case of $K_{2}=0$ as written in Eq. (13). Nevertheless, identities involving the average energy can be obtained in the case of the three-dimensional cubic lattice using the self duality of the model [48].

Let us define the average energy per spin as

and the average plaquette energy per spin by

For these quantities, we can derive the following relation for a sufficiently large system on the cubic lattice where the boundary effects are sufficiently small,

where the dual couplings $K_{1}^{*}$ and $K_{2}^{*}$ are defined as

The proof is based on the duality relation of Wegner [48] for the $Z_{2}$ gauge Higgs model in three dimensions,

Self duality manifests itself as the same function $Z_{\tau}$ appearing on both sides. Taking the logarithmic derivative of both sides with respect to $K_{1}$ and dividing both sides by $N$, we obtain Eq. (19).

On the self-dual line $K_{1}=K_{1}^{*}$ (equivalently $K_{2}=K_{2}^{*})$ in the $K_{1}$-$K_{2}$ plane, the above relation (19) becomes

This is a weighted sum of the averages of $\tau_{ij}$ and $\tau_{\Box}$. A further simplification is realized when we impose another condition $K_{1}=K_{2}$ in addition to the self duality, which, together with $K_{1}=K_{1}^{*}$ and $K_{2}=K_{2}^{*}$, yields $\sinh 2K_{1}=\sinh 2K_{2}=1$, resulting in

where $K_{c}=\frac{1}{2}\ln(\sqrt{2}+1)$ is the solution to $\sinh 2K_{c}=1$. This is the thermal average of the effective Hamiltonian $K_{c}\sum\tau_{ij}+K_{c}\sum\tau_{\Box}$ of the $Z_{2}$ gauge Higgs model under the probability weight $P_{\tau}$ at a special point $(K_{c},K_{c})$ on the $K_{1}$-$K_{2}$ phase diagram.

Identities and inequalities can be proven for the specific heat and correlation functions in the subspace $K=K_{1}$, generalizing the results known in the case of uncorrelated disorder [6, 1].

The specific heat in the subspace $K=K_{1}$, $C(K,K,K_{2})$, is bounded from above as follows.

In deriving the third line from the second, we have replaced the average of the squared quantity $(\cdots)^{2}$ by the square of the average and have applied the condition $K_{1}=K$. The last inequality is based on the analyticity of the average energy for $K(=K_{1})$. This inequality means that the specific heat does not diverge in the subspace $K=K_{1}$ for small $K_{2}>0$, generalizing the result known for uncorrelated disorder [6, 1].

Similarly, we can verify the following relations on correlation functions using the method of gauge transformation as above, generalizing known relations [6, 1],

Here the square brackets $[\cdots]_{K_{1},K_{2}}$ denote the average over $\{\tau_{ij}\}$ by the probability $P(\tau,K_{1},K_{2})$. Notice that Eqs. (25) and (26) hold for any values of $K_{1},K_{2}$, and $K$, not just in the subspace $K=K_{1}$. The first identity means that there is no spin glass phase, which has $[\langle S_{i}\rangle_{K}]_{K_{1},K_{2}}=0$ and $[\langle S_{i}\rangle_{K}^{2}]_{K_{1},K_{2}}>0$, when $K=K_{1}$. It is assumed that boundary spins are fixed to $+1$ such that the thermal average $\langle S_{i}\rangle_{K}$ is finite in the ferromagnetic phase.

Analyticity of the average energy holds for a more general class of correlation distributions as long as the invariance property under gauge transformation is satisfied. A simple example is an additional term of the product of two neighboring plaquette variables,

where $\Box_{1}$ and $\Box_{2}$ are two neighboring plaquettes sharing a bond. It is straightforward to confirm that the proof of Theorem 1 in Appendix A remains valid with minor changes after this modification with sufficiently small $K_{2}$ and $K_{3}$.

It is unclear whether or not Theorem 1 holds in the case of infinitely many (in the thermodynamic limit) terms in the exponent of the probability distribution of bond variables, not just a finite number of terms as in Eq. (28). Nevertheless, the following example suggests that similar analyticity may be valid for a large class of probability distributions. Let us consider the probability distribution of correlated disorder,

where $Z_{\rm I}(K_{0},\{\tau_{ij}\})$ is the partition function of the Ising model on the same lattice with coupling $K_{0}$ and $F(K_{0},\{\tau_{ij}\})$ is the corresponding free energy,

The denominator $Z_{\tau}(K_{1},K_{0})$ is for normalization. The free energy can be expressed in terms of a cluster expansion (high-temperature expansion) for small $K_{0}$, generalizing Eq. (28) to infinitely many (in the thermodynamic limit) gauge invariant terms,

where the summations run over connected clusters with positive coefficients $K_{1},K_{2},K_{3},\cdots$ [50].

The following result holds for this distribution of correlated disorder.