Revisiting the Replica Trick: Competition Between Spin Glass and Conventional Order

Abstract

There is an ambiguity in how to apply the replica trick to spin glass models which have additional order parameters unrelated to spin glass order—with respect to which quantities does one minimize vs maximize the action, and in what sequence? Here we show that the correct procedure is to first maximize with respect to “replica” order parameters, and then minimize with respect to “conventional” order parameters. With this result, we further elucidate the relationship between quenched free energies, annealed free energies, and replica order—it is possible for the quenched and annealed free energies to differ even while all replica order parameters remain zero.

Appendices

A Extremizing with Respect to Order Parameters vs Lagrange Multipliers

Entirely unrelated to the replica trick and spin glass physics, there are subtleties in how one extremizes over (conventional) order parameters and their associated Lagrange multipliers. This is an old topic and we are certainly not the first to consider it (to the point where it often passes without comment in the literature). Yet since the present work is specifically concerned with the order in which one extremizes an action with respect to various quantities, we feel that it is appropriate to give a clear discussion of the issue here.

As a concrete example, consider a classical spin-1/2 Ising model with “mean-field” interactions:

$$\begin{aligned} H(\sigma ) = N \epsilon \bigg ( N^{-1} \sum _i \sigma _i \bigg ), \end{aligned}$$

(96)

for some function $\epsilon (m)$. In other words, the energy can be written as a function solely of the magnetization density $N^{-1} \sum _i \sigma _i$. To evaluate the partition function, we can separate the trace into an outer sum over values of the magnetization m and an inner sum over $\sigma$ such that $N^{-1} \sum _i \sigma _i = m$:

$$\begin{aligned} Z \equiv \sum _{\sigma } \exp {\big [ -\beta H(\sigma ) \big ]} = \int _{-1}^1 \text {d}m \exp {\big [ -N \beta \epsilon (m) \big ]} \sum _{\sigma } \delta \bigg ( m - N^{-1} \sum _i \sigma _i \bigg ). \end{aligned}$$

(97)

Defining $\sum _{\sigma } \delta (m - N^{-1} \sum _i \sigma _i) \equiv \exp {[Ns(m)]}$ and evaluating the integral over m by saddle point, we have that

$$\begin{aligned} -\lim _{N \rightarrow \infty } (N \beta )^{-1} \log {Z} = \min _{m \in [-1, 1]} \Big [ \epsilon (m) - \beta ^{-1} s(m) \Big ]. \end{aligned}$$

(98)

Let us pretend that we do not have an explicit expression for $\exp {[Ns(m)]}$—while it is simply a binomial coefficient in the present example, it may not have a closed form more generally. There are then two ways to proceed. One often sees the $\delta$-function expressed in integral form as $(2\pi )^{-1} \int _{-i \infty }^{i \infty } N \text {d}h \exp {\big [ -Nhm + h \sum _i \sigma _i \big ]}$ (note that h runs along the imaginary axis). We will discuss this approach momentarily. Alternatively, one can use a method more along the lines of large deviation theory [50] and consider the auxiliary quantity

$$\begin{aligned} Z_0(h) \equiv \sum _{\sigma } \exp {\Big [ \beta h \sum _i \sigma _i \Big ]} = \exp {\big [ -N \beta g(h) \big ]}, \end{aligned}$$

(99)

where $g(h) \equiv -\beta ^{-1} \log {2 \cosh {\beta h}}$. Since one could again separate the sum over $\sigma$ into an outer and inner sum just as in Eq. (97), we have that¹⁷

$$\begin{aligned} g(h) = \min _{m \in [-1, 1]} \Big [ -hm - \beta ^{-1} s(m) \Big ]. \end{aligned}$$

(100)

Denote the location of the minimum, which will be a function of h, by $m^*(h)$. There is an explicit expression for $m^*(h)$: just as Eq. (99) is dominated by $\sigma$ with magnetizations close to $m^*(h)$, so is

$$\begin{aligned} \frac{\partial \log {Z_0(h)}}{\partial h} = \sum _{\sigma } \bigg ( \beta \sum _i \sigma _i \bigg ) \frac{\exp {\big [ \beta h \sum _i \sigma _i \big ]}}{\sum _{\sigma '} \exp {\big [ \beta h \sum _i \sigma '_i \big ]}} \sim N \beta m^*(h), \end{aligned}$$

(101)

i.e., $m^*(h) = -\partial g(h) / \partial h$. At this point, note that if one chooses h so that¹⁸$m^*(h) = m$, then from Eq. (100), one has $\beta ^{-1} s(m) = -hm - g(h)$ and

$$\begin{aligned} -\lim _{N \rightarrow \infty } (N \beta )^{-1} \log {Z} = \min _{m \in [-1, 1]} \Big [ \epsilon (m) + hm + g(h) \Big ]. \end{aligned}$$

(102)

Since we have an explicit expression¹⁹ for g(h), Eq. (102) can readily be evaluated (keeping in mind that h is a function of m defined by $m^*(h) = m$).

In fact, since h solves the equation $m = m^*(h) = -\partial g(h) / \partial h$, we can view h as being determined by extremizing the “action” $\epsilon (m) + hm + g(h)$ at fixed m. Thus the free energy is determined by extremizing with respect to both m and h. However, note that the second derivative with respect to h is

$$\begin{aligned} \frac{\partial ^2 g(h)}{\partial h^2} = -N \beta \left[ \Big< \bigg ( N^{-1} \sum _i \sigma _i \bigg )^2 \Big> - \Big < N^{-1} \sum _i \sigma _i \Big >^2 \right] , \end{aligned}$$

(103)

where $\langle \, \cdot \, \rangle$ denotes a thermal expectation value with respect to $h \sum _i \sigma _i$. Thus the second derivative is automatically negative, and the free energy is maximized with respect to h. Since h is really a function of m in Eq. (102), the maximization occurs inside the minimization, meaning we can write

$$\begin{aligned} -\lim _{N \rightarrow \infty } (N \beta )^{-1} \log {Z} = \min _{m \in [-1, 1]} \max _h \Big [ \epsilon (m) + hm + g(h) \Big ]. \end{aligned}$$

(104)

Interestingly, this is another “min-max” prescription, albeit one unrelated to that of the main text (although see Ref. [51] for an alternate derivation using the interpolation techniques of Sect. 4).

The same min-max prescription is hidden within the approach to calculating s(m) based on the integral representation of $\delta (m - N^{-1} \sum _i \sigma _i)$. In this approach, starting from Eq. (97), we have that

$$\begin{aligned} Z \sim \int _{-1}^1 \text {d}m \int _{-i \infty }^{i \infty } \text {d}h \exp {\Big [ -N \beta \big [ \epsilon (m) + hm + g(h) \big ] \Big ]}, \end{aligned}$$

(105)

with the same g(h) as defined in Eq. (99). The right-hand side can be evaluated by saddle point, but since h initially runs along the imaginary axis, its contour must be deformed to pass through the (real) solution to $m = -\partial g(h) / \partial h$. We do need that the action be minimized with respect to h along the trajectory of the contour, but this is fully consistent with the fact that $\partial ^2 g(h) / \partial h^2 < 0$ for real h since the contour passes through the solution vertically. The second derivative being negative in the real direction implies that it is positive in the imaginary direction, as required. Thus we are in fact maximizing the action with respect to real h after all.

Regardless of the approach, it is clear that m and h play different roles. m is undeniably an order parameter—from the beginning, we use it to decompose the original partition function (Eq. (97)). h is instead a Lagrange multiplier—we use it to enforce the constraint that $N^{-1} \sum _i \sigma _i = m$. The conclusion here is that one should first maximize the effective action with respect to Lagrange multipliers, and then minimize with respect to order parameters.

With the proper ordering in mind, let us lastly consider the solution to the saddle point equations. We have the pair

$$\begin{aligned} \frac{\partial g(h)}{\partial h} = -m, \qquad \frac{\partial \epsilon (m)}{\partial m} = -h, \end{aligned}$$

(106)

where the former is to be solved for h, and then the latter is to be solved for m. Nonetheless, it is tempting to interpret the latter equation as determining h and then use that expression in the former to obtain an equation for m. In fact, this is what we do in Sect. 2 of the main text—we use Eq. (22) to solve for the Lagrange multipliers K and $\Lambda$. Although decidedly not the procedure we have derived thus far, the substitution $h = -\partial \epsilon (m) / \partial m$ turns out to be justified, as we now show.

To be precise, let $h^*(m)$ be the solution to $\partial g(h) / \partial h = -m$, and let $h^{\times }(m)$ denote the function $-\partial \epsilon (m) / \partial m$. We have already established that the correct free energy is obtained by minimizing $S^*(m) \equiv \epsilon (m) + h^* m + g(h^*)$ with respect to m—taking a derivative (assuming the minimum lies in the interior²⁰ of $[-1, 1]$) leads to the equation $\partial \epsilon (m) / \partial m = -h^*(m)$. Now instead consider minimizing $S^{\times }(m) \equiv \epsilon (m) + h^{\times } m + g(h^{\times })$—taking a derivative gives $[m + \partial g(h^{\times }) / \partial h^{\times }] \partial h^{\times } / \partial m = 0$. Thus unless $\partial h^{\times } / \partial m = 0$ (a case that can often be treated separately²¹), the extrema of $S^{\times }$ occur where $m = -\partial g(h^{\times }) / \partial h^{\times }$. Either way — whether minimizing $S^*$ or $S^{\times }$—the same equations are being solved (namely Eq. (106)) and the same two-parameter action is being evaluated (namely $\epsilon (m) + hm + g(h)$). Thus the correct global minimum is identified (except for points at which $\partial h^{\times } / \partial m = 0$). This is true even though $S^*(m) \ne S^{\times }(m)$ for general values of m.

B Consequences of Sub-additivity

We demonstrated in Sect. 4.1 of the main text that the disorder-averaged free energy of the classical p-spin model is sub-additive, $F^{(N+L)} \le F^{(N)} + F^{(L)}$ (where the superscript indicates the system size). This implies both that $f \equiv \lim _{N \rightarrow \infty } F^{(N)}/N$ exists and that it can be written as in Eq. (47), reproduced here:

$$\begin{aligned} f = \lim _{N \rightarrow \infty } \limsup _{L \rightarrow \infty } \frac{F^{(N+L)} - F^{(L)}}{N}. \end{aligned}$$

(107)

For completeness, we prove this statement here (following Ref. [23]).

The fact that sub-additivity implies the existence of $\lim _{N \rightarrow \infty } F^{(N)}/N$ goes by the name of Fekete’s lemma. To prove it, pick integers M and P, and note that we inductively have

$$\begin{aligned} \frac{F^{(KM+P)}}{KM+P} \le \frac{K F^{(M)}}{KM+P} + \frac{F^{(P)}}{KM+P}. \end{aligned}$$

(108)

Taking $K \rightarrow \infty$ gives

$$\begin{aligned} \limsup _{K \rightarrow \infty } \frac{F^{(KM+P)}}{KM+P} \le \frac{F^{(M)}}{M}. \end{aligned}$$

(109)

This holds for all $P \in \{0, 1, \cdots , M-1\}$, thus $\limsup _{N \rightarrow \infty } F^{(N)}/N \le F^{(M)}/M$. Taking the liminf as $M \rightarrow \infty$ then gives

$$\begin{aligned} \limsup _{N \rightarrow \infty } \frac{F^{(N)}}{N} \le \liminf _{M \rightarrow \infty } \frac{F^{(M)}}{M}, \end{aligned}$$

(110)

i.e., the two must be equal and the limit exists.

Having established that $f \equiv \lim _{N \rightarrow \infty } F^{(N)}/N$ exists, now turn to Eq. (107). Since $F^{(N)} \ge F^{(N+L)} - F^{(L)}$ for all L, we certainly have that $F^{(N)} \ge \limsup _{L \rightarrow \infty } [F^{(N+L)} - F^{(L)}]$. Dividing by N and taking $N \rightarrow \infty$ then gives

$$\begin{aligned} f \ge \limsup _{N \rightarrow \infty } \limsup _{L \rightarrow \infty } \frac{F^{(N+L)} - F^{(L)}}{N}. \end{aligned}$$

(111)

At the same time, we have that for all N,

$$\begin{aligned} \begin{aligned} f = \lim _{j \rightarrow \infty } \frac{F^{(jN)}}{jN}&= \lim _{j \rightarrow \infty } \frac{1}{j} \sum _{i=0}^{j-1} \frac{F^{((i+1)N)} - F^{(iN)}}{N} \\&\quad \le \limsup _{j \rightarrow \infty } \frac{F^{((j+1)N)} - F^{(jN)}}{N} \le \limsup _{L \rightarrow \infty } \frac{F^{(N+L)} - F^{(L)}}{N}. \end{aligned} \end{aligned}$$

(112)

Taking $N \rightarrow \infty$ then gives

$$\begin{aligned} f \le \liminf _{N \rightarrow \infty } \limsup _{L \rightarrow \infty } \frac{F^{(N+L)} - F^{(L)}}{N}, \end{aligned}$$

(113)

and Eq. (107) follows.

Lastly, we needed analogues of these results in Sect. 4.4, where we considered the partition function $Z^{(N)}(R)$ and corresponding free energy $F^{(N)}(R)$ of states restricted to have a certain value R of the self-overlap (technically a matrix $R_{\tau \tau '}$). We proved in Sect. 4.4 the following analogue of sub-additivity: $F^{(N,L)}(R, R) \le F^{(N)}(R) + F^{(L)}(R)$, where $F^{(N,L)}(R, R)$ is the free energy of states in a size-$(N+L)$ system restricted to separately have $\sigma \cdot \sigma = R$ and $\alpha \cdot \alpha = R$ — recall that we divided the spins into $\{ \vec {\sigma }_i \}_{i=1}^N$ and $\{ \vec {\alpha }_j \}_{j=1}^L$, and defined

$$\begin{aligned} \sigma (\tau ) \cdot \sigma (\tau ') \equiv \frac{1}{N} \sum _i \sigma _i(\tau ) \sigma _i(\tau '), \qquad \alpha (\tau ) \cdot \alpha (\tau ') \equiv \frac{1}{L} \sum _j \alpha _j(\tau ) \alpha _j(\tau '). \end{aligned}$$

(114)

In fact, sub-additivity of the sequence $F^{(N)}(R)$ follows from this result simply by observing that the set of states with $\sigma \cdot \sigma = \alpha \cdot \alpha = R$ is a subset of the states with total self-overlap R (note that the total self-overlap can be written $(N \sigma \cdot \sigma + L \alpha \cdot \alpha )/(N+L)$). Thus $Z^{(N+L)}(R) \ge Z^{(N,L)}(R, R)$ since the sum that is $Z^{(N+L)}(R)$ includes every term of $Z^{(N,L)}(R, R)$, and $F^{(N+L)}(R) \le F^{(N,L)}(R, R)$. Fekete’s lemma then proves that $f(R) \equiv \lim _{N \rightarrow \infty } F^{(N)}(R)/N$ exists, and a straightforward generalization of Eqs. (111) through (113) gives

$$\begin{aligned} \begin{aligned}&\limsup _{N \rightarrow \infty } \limsup _{L \rightarrow \infty } \frac{F^{(N,L)}(R, R) - F^{(L)}(R)}{N} \le f(R) \le \liminf _{N \rightarrow \infty } \limsup _{L \rightarrow \infty } \frac{F^{(N+L)}(R) - F^{(L)}(R)}{N} \\&\quad \le \liminf _{N \rightarrow \infty } \limsup _{L \rightarrow \infty } \frac{F^{(N,L)}(R, R) - F^{(L)}(R)}{N}. \end{aligned} \end{aligned}$$

(115)

Thus

$$\begin{aligned} f(R) = \lim _{N \rightarrow \infty } \limsup _{L \rightarrow \infty } \frac{F^{(N,L)}(R, R) - F^{(L)}(R)}{N}, \end{aligned}$$

(116)

which is Eq. (84) from the main text.