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

Hidetoshi Nishimori  Institute of Innovative Research, Tokyo Institute of
Technology, Yokohama 226-8503, Japan
Graduate School of Information Sciences, Tohoku University, Sendai 980-8579,
Japan
RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program
(iTHEMS), Wako, Saitama 351-0198, Japan

###### Abstract

The average energy of the Ising spin glass is known to have no singularity
along a special line in the phase diagram although there exists a critical
point on the line. This result on the model with uncorrelated disorder is
generalized to the case with correlated disorder. For a class of correlations
in disorder that suppress frustration, we show that the average energy in a
subspace of the phase diagram is expressed as the expectation value of a local
gauge variable of the $Z_{2}$ gauge Higgs model, from which we prove that the
average energy has no singularity although the subspace is likely to have a
phase transition on it. This is a generalization of the result for
uncorrelated disorder. Though it is difficult to obtain an explicit expression
of the energy in contrast to the case of uncorrelated disorder, an exact
closed-form expression of a physical quantity related to the energy is derived
in three dimensions using a duality relation. Identities and inequalities are
proved for the specific heat and correlation functions.

##  I Introduction

The problem of spin glass is one of the most challenging topics in statistical
physics [1, 2]. Only a very limited number of exact solutions are known so
far, among which the Parisi solution of the mean-field-type Sherrington-
Kirkpartick model [3] stands out as a distinguished achievement [4, 5].
Another rare example is the exact solution for the average energy of the Ising
spin glass in finite dimensions on a special line in the phase diagram [6, 7,
1]. The resulting expression of the energy is written as a simple hyperbolic
tangent of the inverse temperature without any singularity. Since a
multicritical point (transition point) lies on this special line in the phase
diagram, it is highly non-trivial and counter-intuitive that the exact average
energy is an analytic function. Other physical quantities such as the specific
heat and magnetic susceptibility are expected to have a singularity at the
phase transition, but nothing is known so far exactly or rigorously on those
quantities in finite-dimensional models except for the existence of a phase
transition on the line as concluded from the existence of ferromagnetic order
in the low-temperature side of the line [8] (see also [9] for the continuous-
variable case).

Those results concern the Edwards-Anderson model [10], in which disorder of a
given interaction bond distributes independently of disorder of other bonds.
In real spin glass materials, disorder exists mostly in sites, not in bonds
[11]. Edwards and Anderson assumed that properties of spin glasses will be
independent of the type of disorder and proposed their model with uncorrelated
disorder. One of the important differences between site and bond disorder is
the existence or absence of correlation of disorder among nearby bond
variables. For example, if the position of a magnetic atom in a metal is
affected by disorder, all interactions between this and other atoms are
changed in a correlated manner. It therefore makes sense to study how
correlations in disorder would affect the system properties.

There have been several attempts to study the effects of correlation in
disorder [12, 13, 14]. It is generally concluded that correlations, if weak,
do not significantly modify the system properties qualitatively. These studies
exploit numerical methods or mean-field-type models, and it is desirable to
establish analytical results in finite-dimensional systems.

We introduce a model with correlations of a specific type in disorder, which
makes it possible to reduce the expression of the average energy to the
expectation value of a local variable in the $Z_{2}$ gauge Higgs model [15].
Based on this reduction, we prove that the average energy is an analytic
function in a subspace of the phase diagram, a generalization of a special
line in the case of uncorrelated disorder. It is difficult to derive a closed-
form exact solution for the average energy when correlations are introduced.
Nevertheless, using self duality, we show in the case of three spatial
dimensions that a simple formula holds for a physical quantity related to the
energy. We also derive several non-trivial identities and inequalities for the
specific heat and correlation functions, generalizing the results for
uncorrelated disorder. Also discussed are more complex types of correlations
in disorder.

In Sec. II, we define the model and analyze its properties. Conclusion is
given in Sec. III, and some technical details are described in Appendixes.

##  II Ising spin glass with correlated disorder

We first define the model of correlated disorder and then study the properties
of the corresponding Ising spin glass, in particular the behavior of the
average energy in a subspace of the phase diagram.

###  II.1 Model of correlated disorder

As mentioned in the Introduction, it is reasonable to study a spin-glass
system with short-range correlations in disorder variables. It is likely that
site disorder has a different degree of frustration than in the case of
uncorrelated bond disorder, a prominent example of which is the Mattis model
[16] without frustration but with site-dependent quenched disorder
$\xi_{i}=\pm 1$,

$\displaystyle H_{\rm Mattis}=-J\sum_{\langle
ij\rangle}\xi_{i}\xi_{j}S_{i}S_{j},$ (1)

where $S_{i}(\pm 1)$ is the Ising variable at site $i$, and $\langle
ij\rangle$ runs over interacting spin pairs on the lattice with interaction
strength $J$. An example with site disorder but with frustration is the
Hopfield model [17, 18, 19, 20],

$\displaystyle H_{\rm Hopfield}=-\frac{1}{N}\sum_{i,j}J_{ij}S_{i}S_{j},$ (2)

where $N$ is the number of sites, and the interaction is composed of site-
dependent quenched variables $\\{\xi_{i}^{\mu}=\pm 1\\}$,

$\displaystyle J_{ij}=\sum_{\mu=1}^{r}\xi_{i}^{\mu}\xi_{j}^{\mu}.$ (3)

This is a model with site-dependent disorder, in which correlation exists
between disorder of different bonds $J_{ij}$ and $J_{jk}$ sharing a site index
$j$. When $r=1$, the model reduces to the Mattis model of Eq. (1) without
frustration, and in the limit of $r\to\infty$, the model becomes equivalent to
the Sherrington-Kirkpatrick model with uncorrelated bond disorder with strong
frustration according to the central limit theorem. The Hopfield model thus
interpolates the unfrustrated and fully frustrated cases by the parameter $r$.

These examples motivate us to introduce a system with correlated disorder in
bond variables with a parameter to control the degree of frustration. We
therefore propose and analyze the following probability distribution of bond-
assigned disorder variables $\\{\tau_{ij}\\}$ with control parameters $K_{1}$
and $K_{2}$,

$\displaystyle
P_{\tau}(K_{1},K_{2})=\frac{1}{Z_{\tau}(K_{1},K_{2})}\exp\Big{(}K_{1}\sum_{\langle
ij\rangle}\tau_{ij}+K_{2}\sum_{\Box}\tau_{\Box}\Big{)},$ (4)

where $Z_{\tau}(K_{1},K_{2})$ is the normalization factor (or the partition
function of the disorder variables)

$\displaystyle Z_{\tau}(K_{1},K_{2})=\sum_{\\{\tau_{ij}=\pm
1\\}}\exp\Big{(}K_{1}\sum_{\langle
ij\rangle}\tau_{ij}+K_{2}\sum_{\Box}\tau_{\Box}\Big{)},$ (5)

and $\tau_{\Box}$ is the product of four disorder variables around a unit
plaquette denoted by $\Box$ on the $d$-dimensional hypercubic lattice,
$\tau_{\Box}=\tau_{ij}\tau_{jk}\tau_{kl}\tau_{li}$ 111 We can consider other
possibilities. For example, on the two-dimensional triangular lattice, the
product will run over three bond variables around a unit triangle,
$\tau_{ij}\tau_{ji}\tau_{ki}$. We work on the hypercubic lattice in the
present paper for simplicity.. This is a minimal model with a parameter to
control the degree of frustration and correlations in bond disorder. Equation
(4) reduces to the case of uncorrelated disorder when $K_{2}=0$. For positive
$K_{2}$, the last term in the exponent of Eq. (4) suppresses the probability
for a plaquette to be frustrated, $\tau_{\Box}=-1$. The first term with
coefficient $K_{1}$ controls the relative probability of positive and negative
values of the disorder variable $\tau_{ij}$ at each bond $\langle ij\rangle$.
Notice that the exponent in the probability distribution Eq. (4) is of the
same form as the action of the $Z_{2}$ gauge Higgs model [15], and this
analogy facilitates our analysis below.

The Hamiltonian of the Ising spin glass is

$\displaystyle H=-K\sum_{\langle ij\rangle}\tau_{ij}S_{i}S_{j},$ (6)

where $K$ is the dimensionless coupling constant (the inverse temperature
$K=1/T$). We study the properties of this Hamiltonian with the quenched
disorder variables $\\{\tau_{ij}\\}$ chosen from the probability distribution
of Eq. (4). Possible generalization of the probability distribution will be
discussed at the end of this section.

###  II.2 Average energy in a subspace of the phase diagram

####  II.2.1 Expression of the average energy

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

$\displaystyle
E(K,K_{1},K_{2})=\frac{1}{Z_{\tau}(K_{1},K_{2})}\sum_{\\{\tau_{ij}=\pm
1\\}}e^{K_{1}\sum\tau_{ij}+K_{2}\sum\tau_{\Box}}\langle H\rangle_{K},$ (7)

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

$\displaystyle\langle H\rangle_{K}=-\frac{1}{Z_{s}(K)}\frac{\partial}{\partial
K}Z_{s}(K)$ (8)

with the partition function of the Ising model,

$\displaystyle Z_{s}(K)=\sum_{\\{S_{i}=\pm 1\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}.$
(9)

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

$\displaystyle S_{i}\to
S_{i}\sigma_{i},\,\tau_{ij}\to\tau_{ij}\sigma_{i}\sigma_{j}~{}(\sigma_{i}=\pm
1)~{}(\forall i,j)$ (10)

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

$\displaystyle-E(K,K_{1},K_{2})$
$\displaystyle=\frac{1}{2^{N}Z_{\tau}(K_{1},K_{2})}\sum_{\\{\tau_{ij}=\pm
1\\}}\sum_{\\{\sigma_{i}=\pm
1\\}}e^{K_{1}\sum\tau_{ij}\sigma_{i}\sigma_{j}+K_{2}\sum\tau_{\Box}}\,\frac{\partial_{K}\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}{\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}.$
(11)

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

$\displaystyle-E(K,K,K_{2})$
$\displaystyle=\frac{1}{2^{N}Z_{\tau}(K,K_{2})}\sum_{\\{\tau_{ij}\\}}e^{K_{2}\sum\tau_{\Box}}\frac{\partial}{\partial
K}\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}$
$\displaystyle=\frac{1}{2^{N}Z_{\tau}(K,K_{2})}\sum_{\\{S_{i}\\}}\sum_{\\{\tau_{ij}\\}}\Big{(}\sum_{\langle
ij\rangle}\tau_{ij}S_{i}S_{j}\Big{)}e^{K\sum\tau_{ij}S_{i}S_{j}+K_{2}\sum\tau_{\Box}}$
$\displaystyle=\frac{1}{Z_{\tau}(K,K_{2})}\sum_{\langle
ij\rangle}\Big{(}\sum_{\\{\tau_{ij}\\}}\tau_{ij}\Big{)}\,e^{K\sum\tau_{ij}+K_{2}\sum\tau_{\Box}},$
(12)

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 [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 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, 38] 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,
39]. 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 [15] 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 $\langle ij\rangle$.

####  II.2.2 Analyticity of the average energy

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

######  Theorem 1.

The average energy $E(K,K,K_{2})$ of Eq. (12) is analytic in $K$ for
sufficiently small $K_{2}$ in the thermodynamic limit.

This result is a known property of the $Z_{2}$ lattice gauge theory, see e.g.
Ref. [41] for a theoretical analysis and Refs. [42, 43, 44, 45, 46] for
numerical studies. Since the argument presented in Ref. [41] is fairly
sketchy, we provide a formal proof in Appendix A.

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

$\displaystyle E(K,K,0)=-N_{B}\tanh K,$ (13)

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.

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

$\displaystyle P_{l}(K_{1},K_{2})=\sum_{\\{\tau_{ij}=\pm
1\\}}\delta_{l,\tau_{ij}}\,P_{\tau}(K_{1},K_{2})\quad(l=\pm 1),$ (14)

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

$\displaystyle
P_{l}(K_{1},K_{2})=p_{+}\,\delta_{l,1}+(1-p_{+})\,\delta_{l,-1}.$ (15)

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

$\displaystyle p_{+}=p\,\Big{(}1+2c(1-p)(2p-1)^{3}\tanh
K_{2}+\mathcal{O}(\tanh K_{2})^{2}\Big{)},$ (16)

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 not proves) 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 the
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 [47]. 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 [46].

####  II.2.3 Identities for the average energy in three dimensions

It is generally difficult to derive an explicit compact expression of the
average energy for $K=K_{1}$ except for the trivial 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 [47].

Let us define the average energy per spin as

$\displaystyle
e(K_{1},K_{2})=\frac{1}{N}\,E(K_{1},K_{1},K_{2})=-\frac{1}{NZ_{\tau}(K_{1},K_{2})}\sum_{\\{\tau_{ij}=\pm
1\\}}\Big{(}\sum_{\langle
ij\rangle}\tau_{ij}\Big{)}\,e^{K_{1}\sum\tau_{ij}+K_{2}\sum\tau_{\Box}}.$ (17)

and the average plaquette energy per spin by

$\displaystyle
e_{p}(K_{1},K_{2})=-\frac{1}{NZ_{\tau}(K_{1},K_{2})}\sum_{\\{\tau_{ij}=\pm
1\\}}\Big{(}\sum_{\Box}\tau_{\Box}\Big{)}\,e^{K_{1}\sum\tau_{ij}+K_{2}\sum\tau_{\Box}}.$
(18)

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,

$\displaystyle e(K_{1},K_{2})+3\tanh K_{2}=-\frac{1}{\sinh
2K_{2}}\,\big{(}e_{p}(K_{1}^{*},K_{2}^{*})+3\big{)},$ (19)

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

$\displaystyle\tanh K_{1}^{*}=e^{-2K_{2}},~{}\tanh K_{2}^{*}=e^{-2K_{1}}.$
(20)

The proof is based on the duality relation of Wegner [47] for the $Z_{2}$
gauge Higgs model in three dimensions,

$\displaystyle\frac{Z_{\tau}(K_{1},K_{2})}{2^{3N}(\cosh K_{1}\cosh
K_{2})^{3N}}=\frac{Z_{\tau}(K_{1}^{*},K_{2}^{*})}{e^{3N(K_{1}^{*}+K_{2}^{*})}}.$
(21)

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

$\displaystyle\sinh 2K_{2}\,\cdot e(K_{1},K_{2})+e_{p}(K_{1},K_{2})=-3\tanh
K_{2}\,\sinh K_{2}-3.$ (22)

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

$\displaystyle e(K_{c},K_{c})+e_{p}(K_{c},K_{c})=-3\sqrt{2},$ (23)

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.

####  II.2.4 Specific heat and correlation functions

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.

$\displaystyle T^{2}\,C(K,K,K_{2})=-\left.\frac{\partial
E(K,K_{1},K_{2})}{\partial K}\right|_{K_{1}=K}$
$\displaystyle=\frac{1}{2^{N}Z_{\tau}(K_{1},K_{2})}\sum_{\\{\tau_{ij}=\pm
1\\}}\sum_{\\{\sigma_{i}=\pm
1\\}}e^{K_{1}\sum\tau_{ij}\sigma_{i}\sigma_{j}+K_{2}\sum\tau_{\Box}}$
$\displaystyle\left.\hskip
56.9055pt\times\left\\{\frac{\partial^{2}_{K}\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}{\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}-\left(\frac{\partial_{K}\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}{\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}\right)^{2}\right\\}\right|_{K_{1}=K}$
$\displaystyle\leq\frac{1}{2^{N}Z_{\tau}(K_{1},K_{2})}\sum_{\\{\tau_{ij}=\pm
1\\}}e^{K_{2}\sum\tau_{\Box}}\frac{\partial^{2}}{\partial
K^{2}}\sum_{\\{S_{i}=\pm 1\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}$
$\displaystyle\hskip
56.9055pt-\left(\frac{1}{Z_{\tau}(K_{1},K_{2})}\frac{\partial}{\partial
K}\sum_{\\{\tau_{ij}=\pm
1\\}}e^{K\sum\tau_{ij}+K_{2}\sum\tau_{\Box}}\right)^{2}$
$\displaystyle=-\frac{\partial}{\partial K}E(K,K,K_{2})<\infty.$ (24)

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 $K_{2}>0$,
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],

$\displaystyle\left[\langle
S_{i}\rangle_{K}\right]_{K_{1},K_{2}}=\left[\langle
S_{i}\rangle_{K_{1}}\langle S_{i}\rangle_{K}\right]_{K_{1},K_{2}}$ (25)
$\displaystyle\left[\langle
S_{i}\rangle_{K}\right]_{K_{1},K_{2}}\leq\left[|\langle\sigma_{i}\rangle_{K_{1}}||\langle
S_{i}\rangle_{K}|\right]_{K_{1},K_{2}}\leq\left[|\langle\sigma_{i}\rangle_{K_{1}}|\right]_{K_{1},K_{2}}$
(26) $\displaystyle\left[\frac{1}{\langle
S_{i}\rangle_{K_{1}}}\right]_{K_{1},K_{2}}=1.$ (27)

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.

####  II.2.5 More general correlations

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,

$\displaystyle
P_{\tau}(K_{1},K_{2},K_{3})=\frac{1}{Z_{\tau}(K_{1},K_{2},K_{3})}\exp\Big{(}K_{1}\sum_{\langle
ij\rangle}\tau_{ij}+K_{2}\sum_{\Box}\tau_{\Box}+K_{3}\sum_{\Box_{1}\Box_{2}}\tau_{\Box_{1}}\tau_{\Box_{2}}\Big{)},$
(28)

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 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,

$\displaystyle P_{\tau}(K_{1},K_{\rm I})=\frac{1}{Z_{\tau}(K_{1},K_{\rm
I})}\,e^{K_{1}\sum\tau_{ij}}\,Z_{\rm I}(K_{\rm
I},\\{\tau_{ij}\\})=\frac{1}{Z_{\tau}(K_{1},K_{\rm
I})}\,e^{K_{1}\sum\tau_{ij}-F(K_{\rm I},\\{\tau_{ij}\\})},$ (29)

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

$\displaystyle Z_{\rm I}(K_{\rm I},\\{\tau_{ij}\\})=\sum_{\\{\xi_{i}=\pm
1\\}}e^{K_{\rm I}\sum_{\langle ij\rangle}\tau_{ij}\xi_{i}\xi_{j}}=e^{-F(K_{\rm
I},\\{\tau_{ij}\\})}\,.$ (30)

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

$\displaystyle-F(K_{\rm I},\\{\tau_{ij}\\})=K_{1}(K_{\rm
I})\sum\tau_{\Box}+K_{2}(K_{\rm I})\sum_{\rm
connected}\tau_{\Box}\tau_{\Box}+K_{3}(K_{\rm I})\sum_{\rm
connected}\tau_{\Box}\tau_{\Box}\tau_{\Box}+\cdots,$ (31)

where the summations run over connected clusters with positive coefficients
$K_{1},K_{2},K_{3},\cdots$ [49].

The following result holds for this distribution of correlated disorder.

######  Theorem 2.

For sufficiently small $K_{\rm I}$, the average energy is analytic in $K$
under the distribution function of Eq. (29) of correlated disorder.

###### Proof.

Similarly to Eq. (11), the average energy after gauge transformation is

$\displaystyle-E(K,K_{1},K_{\rm I})$
$\displaystyle=\frac{1}{2^{N}Z_{\tau}(K_{1},K_{\rm
I})}\sum_{\\{\tau_{ij}\\}}\sum_{\\{\sigma_{i}\\}}e^{K_{1}\sum\tau_{ij}\sigma_{i}\sigma_{j}}\sum_{\\{\xi_{i}\\}}e^{K_{\rm
I}\sum\tau_{ij}\xi_{i}\xi_{j}}\,\frac{\partial_{K}\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}{\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}}.$
(32)

When $K=K_{1}$, the summation over $\\{\sigma_{i}\\}$ cancels out with the
partition function (summation over $\\{S_{i}\\}$) in the denominator,

$\displaystyle-E(K,K,K_{\rm I})=\frac{1}{2^{N}Z_{\tau}(K,K_{\rm
I})}\sum_{\\{\tau_{ij}\\}}\sum_{\\{\xi_{i}\\}}e^{K_{\rm
I}\sum\tau_{ij}\xi_{i}\xi_{j}}\,\frac{\partial}{\partial
K}\sum_{\\{S_{i}\\}}e^{K\sum\tau_{ij}S_{i}S_{j}}\,.$ (33)

We can carry out the summation over $\\{\tau_{ij}\\}$ at each $\langle
ij\rangle$ independently,

$\displaystyle\sum_{\\{\tau_{ij}\\}}\prod_{\langle
ij\rangle}e^{\tau_{ij}(K_{\rm I}\xi_{i}\xi_{j}+KS_{i}S_{i})}\propto
e^{\tilde{K}\sum_{\langle ij\rangle}\xi_{i}\xi_{j}S_{i}S_{j}},$ (34)

where $\tanh\tilde{K}=\tanh K_{\rm I}\tanh K$. This relation can be verified
by inserting all possible values of $\xi_{i}\xi_{j}=\pm 1$ and $S_{i}S_{j}=\pm
1$. By the gauge transformation $\xi_{i}S_{i}\to S_{i}$ for fixed $\xi_{i}$,
we find that $E(K,K,K_{\rm I})$ is the energy of the ferromagnetic Ising model
with uniform coupling $\tilde{K}$ without disorder. Since the effective
coupling $\tilde{K}$ runs from 0 to $K_{\rm I}$ as $K$ changes from 0 to
$\infty$, the effective coupling $\tilde{K}$ stays below the critical point of
the ferromagnetic Ising model for sufficiently small $K_{\rm I}$. Therefore
the energy is analytic in $K$. ∎

##  III Conclusion

We have introduced correlations in disorder for the Ising spin glass problem
in such a way that the degree of frustration is controlled. We have analyzed
the properties of the system in a subspace of the phase diagram in which the
average energy is known to be analytic even across a transition point in the
case of uncorrelated disorder. We have proved that the analyticity of the
average energy is preserved under the introduction of correlations in
disorder. This result remains valid for a few more complex types of
correlations as long as correlations are gauge invariant. Bounds on the
specific heat and identities and inequalities have been derived, generalizing
the results for uncorrelated disorder.

Our derivation of these results relies on gauge invariance of correlations. It
is difficult to analyze the case without gauge invariance such as a simple
product of two neighboring bond variables $\tau_{ij}\tau_{jk}$. It would be
safe not to expect similar analyticity to hold in such a case because our
result has been derived through a delicate balance of the term in the
probability distribution and the partition function, which breaks down for
gauge non-invariant probabilities.

For uncorrelated disorder, outstanding properties of the system in the special
subspace have been found useful in applications in many fields including
statistical inference [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 36],
quantum error correction [38, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92,
93, 94, 95, 96, 97], quantum Hall effect [98], and localization [99, 100]. The
present result may stimulate further developments in these fields in addition
to the spin glass theory itself.

##  Appendix A Proof of Theorem 1

We prove by a cluster expansion that the following expression in Eq. (12)

$\displaystyle
C_{01}(K_{1},K_{2})=\frac{\sum_{\\{\tau_{ij}\\}}\tau_{01}\,e^{K_{1}\sum\tau_{ij}+K_{2}\sum(\tau_{\Box}+1)}}{\sum_{\\{\tau_{ij}\\}}e^{K_{1}\sum\tau_{ij}+K_{2}\sum(\tau_{\Box}+1)}}$
(35)

is analytic in $K_{1}$ for sufficiently small $K_{2}(>0)$.

Notice that the exponent above has an additional term $K_{2}\sum 1$ both in
the numerator and the denominator compared to Eq. (12) but they cancel out to
give the same quantity as in Eq. (12). This term is useful in the proof. The
proof below is basically an adaptation of the theory in Refs. [101] to the
present context of the $Z_{2}$ lattice gauge model. See also Ref. [41], where
an abridged description of the theory is provided.

If we define $P_{\tau}(K_{1})$ as $P_{\tau}(K_{1},0)$, the above Eq. (35) is
expressed as

$\displaystyle
C_{01}(K_{1},K_{2})=\frac{\sum_{\\{\tau_{ij}\\}}\tau_{01}P_{\tau}(K_{1})\,e^{K_{2}\sum(\tau_{\Box}+1)}}{\sum_{\\{\tau_{ij}\\}}P_{\tau}(K_{1})\,e^{K_{2}\sum(\tau_{\Box}+1)}}.$
(36)

To show that the small-$K_{2}$ expansion of $C_{01}(K_{1},K_{2})$ converges
absolutely and uniformly and thus $C_{01}(K_{1},K_{2})$ is analytic in
$K_{1}$, we introduce $\rho_{\Box}$ as

$\displaystyle e^{K_{2}(\tau_{\Box}+1)}=1+\rho_{\Box}.$ (37)

This $\rho_{\Box}$ is positive semi-definite and small for small $K_{2}$. The
exponential factor in Eq. (36) is expanded as

$\displaystyle
e^{K_{2}\sum_{\Box}(\tau_{\Box}+1)}=\prod_{\Box}(1+\rho_{\Box})=\sum_{Q}\prod_{\Box\in
Q}\rho_{\Box},$ (38)

where $Q$ is the set of products of plaquettes. Let us divide $Q$ into $Q_{1}$
and $Q_{2}$, where $Q_{1}$ is the set of connected products of plaquettes
involving the bond $(01)$ and $Q_{2}$ is $Q\setminus Q_{1}$. Then

$\displaystyle
C_{01}(K_{1},K_{2})=\frac{1}{Z_{\tau}^{+}(K_{1},K_{2})}\sum_{Q_{1}}\sum_{Q_{2}}\sum_{\\{\tau_{ij}\\}}\tau_{01}P_{\tau}(K_{1})\prod_{\Box\in
Q_{1}}\rho_{\Box}\prod_{\Box\in Q_{2}}\rho_{\Box},$ (39)

where

$\displaystyle
Z_{\tau}^{+}(K_{1},K_{2})=\sum_{\\{\tau_{ij}\\}}P_{\tau}(K_{1})e^{K_{2}\sum(\tau_{\Box}+1)}=Z_{\tau}(K_{1},K_{2})e^{K_{2}\sum
1}.$ (40)

Since $Q_{1}$ and $Q_{2}$ are disjoint,

$\displaystyle
C_{01}(K_{1},K_{2})=\frac{1}{Z_{\tau}^{+}(K_{1},K_{2})}\sum_{Q_{1}}\sum_{\tau\in
Q_{1}}\tau_{01}P_{\tau\in Q_{1}}(K_{1})\Big{(}\prod_{\Box\in
Q_{1}}\rho_{\Box}\Big{)}\cdot Z_{Q_{2}}(K_{1},K_{2}),$ (41)

where $Z_{Q_{2}}(K_{1},K_{2})$ is the partial partition function,

$\displaystyle Z_{Q_{2}}(K_{1},K_{2})=\sum_{Q_{2}}\sum_{\tau\in
Q_{2}}P_{\tau\in Q_{2}}(K_{1})\prod_{\Box\in Q_{2}}\rho_{\Box}.$ (42)

This partial partition function is bounded from above by
$Z_{\tau}^{+}(K_{1},K_{2})$ because the coupling $K_{2}$ for $Q_{1}$ is set to
0 in $Z_{Q_{2}}$ in comparison with the full $Z_{\tau}$ and thus the summand
is smaller. See Remark 1 of Lemma 3.2 of Ref. [101]. We therefore replace
$Z_{Q_{2}}(K_{1},K_{2})/Z_{\tau}^{+}(K_{1},K_{2})$ by 1 to have a bound

$\displaystyle C_{01}(K_{1},K_{2})$
$\displaystyle\leq\sum_{Q_{1}}\sum_{\tau\in Q_{1}}\tau_{01}P_{\tau\in
Q_{1}}\prod_{\Box\in Q_{1}}\rho_{\Box}$
$\displaystyle\leq\sum_{k}\sum_{|Q_{1}|=k}\,\sum_{\tau\in
Q_{1}:|Q_{1}|=k}P_{\tau\in Q_{1}}\prod_{\Box\in Q_{1}}\rho_{\Box}.$ (43)

We have classified $Q_{1}$ by the number of elements in it. Using the trivial
inequality

$\displaystyle\rho_{\Box}\leq e^{2K_{2}}-1,$ (44)

we can replace the average of $\prod_{\Box\in Q_{1}}\rho_{\Box}$ with respect
to the weight $P_{\tau}$ by the upper bound,

$\displaystyle\sum_{\tau\in Q_{1}:|Q_{1}|=k}P_{\tau\in Q_{1}}\prod_{\Box\in
Q_{1}}\rho_{\Box}\leq(e^{2K_{2}}-1)^{k}.$ (45)

Equation (43) is then simplified as

$\displaystyle
C_{01}(K_{1},K_{2})\leq\sum_{k}\sum_{|Q_{1}|=k}(e^{2K_{2}}-1)^{k}.$ (46)

As stated in Lemma 1 below, the number of possible connected graphs involving
the bond $(01)$ is bounded as

$\displaystyle N(k)\leq c_{1}\cdot c_{2}{}^{k},$ (47)

where $c_{1}$ and $c_{2}$ are positive constants. We therefore obtain

$\displaystyle C_{01}(K_{1},K_{2})\leq
c_{1}\sum_{k}c_{2}{}^{k}\,(e^{2K_{2}}-1)^{k},$ (48)

which converges for sufficiently small $K_{2}$. Uniform convergence follows
this absolute convergence of an upper bound, proving analyticity of the left-
hand side $C_{01}(K_{1},K_{2})$ as a function of $K_{1}$ if we regard $K_{1}$
as a complex number in the neighborhood of the origin. $\square$
Remark. The proof remains valid when $\tau_{01}$ is replaced by the product of
a finite number of bond variables.

######  Lemma 1.

The number of graphs in $Q_{1}$ is bounded as in Eq. (47).

This is Lemma 3.3 of Ref. [101] and is a plaquette version of the well-
established bound for the number of connected graphs with variables on the
bonds (edges), not on plaquettes, as described, for example, in Exercise 5.3
of Ref. [49]. It is straightforward to replace bonds with plaquettes to derive
the above Lemma.

##  Appendix B Evaluating $p_{+}$ to first order in $\tanh K_{2}$

According to the definition of Eq. (15), $p_{+}$ is written as

$\displaystyle
p_{+}=\frac{\sum_{\\{\tau_{ij}\\}}\delta_{\tau_{ij},1}e^{K_{1}\sum\tau_{ij}}\prod_{\Box}(1+\tau_{\Box}\tanh
K_{2})}{\sum_{\\{\tau_{ij}\\}}e^{K_{1}\sum\tau_{ij}}\prod_{\Box}(1+\tau_{\Box}\tanh
K_{2})}.$ (49)

The denominator is expanded to first order in $\tanh K_{2}$ as

$\displaystyle(\cosh K_{1})^{N_{B}}+N_{p}\tanh K_{2}\,(\cosh
K_{1})^{N_{B}-4}(\sinh K_{1})^{4},$ (50)

where $N_{B}$ is the number of bonds and $N_{p}$ is for the number of
plaquettes. Similarly, the numerator is

$\displaystyle(\cosh
K_{1})^{N_{B}-1}\sum_{\tau_{ij}}\delta_{\tau_{ij},1}e^{K_{1}\tau_{ij}}$
$\displaystyle+(\cosh K_{1})^{N_{B}-5}\tanh K_{2}\,(\sinh
K_{1})^{4}\sum_{\tau_{ij}}\delta_{\tau_{ij},1}e^{K_{1}\tau_{ij}}(N_{B}-a)$
$\displaystyle+(\cosh K_{1})^{N_{B}-4}\tanh K_{2}\,(\sinh
K_{1})^{3}\sum_{\tau_{ij}}\delta_{\tau_{ij},1}\tau_{ij}\,e^{K_{1}\tau_{ij}}\cdot
a,$ (51)

where $a$ the number of plaquettes to which a given single bond belongs.
Inserting these expansions to Eq. (49) and leaving the leading order, we have

$\displaystyle p_{+}=p\Big{(}1-a(2p-1)^{4}\tanh K_{2}+a(2p-1)^{3}\tanh
K_{2}\Big{)},$ (52)

where we used $(1-p)/p=e^{-2K_{1}}$. This is Eq. (16).

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