1. Introduction and main results
The Ising model is one of the oldest, most famous and most studied models in statistical physics, see [FV18] for a thorough introduction, description, results and references. In this paper, we look at the computational complexity and complex zeros of the partition function in the Ising model. This is a classical and also currently very active area of research, see [G+19], [J+19], [L+12], [L+19a], [L+19b], [PR18], [S+14] and [Z+11] for some recent results.
Formally, the partition function we work with is described as follows. Let be the -dimensional Boolean cube of all, where for . We consider a quadratic polynomial ,
and define the partition function as
Often, there is an underlying with vertices numbered and the property that if and only if and span an edge of . In this case, the variable is interpreted as the spin of a vertex and as the energy of the configuration . The coefficients describe the interactions of vertices and : the interaction is ferromagnetic if and antiferromagnetic if . The coefficients describe the external field, see [FV18] for a thorough discussion.
In this paper, we discuss the computational complexity of computing (approximating) . Via the method of polynomial interpolation [Ba16], we relate the computational complexity to the absence of complex zeros of in the vicinity of real coefficients and . We prove the following main result.
Suppose that for some , we have
for . Then
Some remarks are in order.
The first remark is regarding notation. We treat indices in as an unordered pair, so is the coefficient of the monomial in the quadratic polynomial and the sum accounts for all coefficients of the monomials containing . For a complex number , we denote by and the real and imaginary parts of respectively.
The second remark concerns algorithmic consequences of Theorem 1.1. From Theorem 1.1, the by now standard polynomial interpolation argument (see [Ba16], [G+19], [L+19a], [L+19b], [PR17]), produces an algorithm for approximating the partition function when the coefficients and are real and satisfy the condition
where is fixed in advance. As there is no restriction on , the sum can be exponentially large in . To avoid dealing with exponentially large numbers, we assume that we are provided with numbers for . Then the complexity of the algorithm is quasi-polynomial: we approximate within relative error in time.
We describe the algorithm in Section 3 and prove Theorem 1.1 in Section 2. In the remainder of this section we relate Theorem 1.1 to what is known about the partition function in the Ising model.
(1.2) The bounds for the zero-free region are asymptotically optimal
Let be a graph with vertices . For a real number , let us choose
and let us choose for some for all . Let us fix some positive integer and choose either
(all interactions are ferromagnetic) or
(all interactions are antiferromagnetic). We consider the partition function as a function of a complex parameter . It is known that for a fixed , as grows and ranges over all graphs with the largest degree of a vertex, the zeros of the univariate function with either choice of can get arbitrarily close to , see [BG01], [PR18]. We have
The right hand side approaches as , which shows that “1” in the “” bound of Theorem 1.1 cannot be replaced by a larger number.
(1.3) The bounds for approximation are asymptotically optimal in the antiferromagnetic case
As in Section 1.2, let be a graph of the largest degree , let us choose all for all and let us define by (1.2.1). Suppose that
It is shown in [SS14] and also in [G+16] that the problem of approximating is NP-hard under randomized reduction. Hence unless the computational complexity hierarchy collapses, we cannot approximate in quasi-polynomial time in the class of problems where all and
for an arbitrarily small , fixed in advance.
(1.4) The ferromagnetic case is special
Suppose that for all and that for and some complex parameter . Lee and Yang [LY52] showed that the zeros of the univariate function lie on the line . If are allowed to vary, then as long as for . The ferromagnetic case is also special from the complexity point of view: Jerrum and Sinclair [JS93] constructed a randomized polynomial time algorithm approximating when for some real and . Deterministic approximation algorithms of quasi-polynomial (genuinely polynomial, if the largest degree of the underlying graph is fixed in advance) complexity, are constructed in [L+19b], assuming that for some constant , fixed in advance, and , see also Section 7.4 of [Ba16]. The complexity status of the approximation problem in the ferromagnetic case of and by a deterministic algorithm appears to be not known.
(1.5) The bounds for phase transition are asymptotically optimal
Theorem 1.1 can be interpreted as saying that there is no phase transition in the Lee - Yang sense [YL52] provided the parameters and are real and satisfy (1.1.1) for some , fixed in advance. There is a related, though not identical, concept of phase transition, based on the disappearance of correlation decay, see [FV18]. If we choose for all and define as in (1.2.1), then the correlation decay occurs precisely in the interval
where is the largest degree of a vertex of , see [Z+11], [S+14], and [L+19a]. In view of (1.2.2), the bound (1.1.1) is asymptotically optimal as .
(1.6) What’s new
The main novelty of our approach to approximation compared to those of [G+19], [L+12], [L+19a], [L+19b], [S+14] and [Z+11] is that we state our condition in terms of the mixed norm on the interactions , see (1.1.1), as opposed to the uniform bound
on the strength of individual interactions [Z+11], [S+14], where is the largest degree of the underlying graph. As we remarked above, our results are asymptotically optimal, when . Generally, the condition (1.1.1) appears to be more robust than (1.6.1), as (1.1.1) is independent on the degree and allows individual coefficients to be relatively large, as long as the sum for all interactions of any given vertex remains appropriately bounded. Of course, for any particular , the conditions (1.1.1) and (1.6.1) are in general position, as it is easy to construct examples where one holds and the other is violated.
Another novelty of our approach with respect to locating zero-free regions of , compared to those of [G+19], [L+19a], and [PR18], is that we allow all parameters and to vary: this concerns both Lee - Yang zeros [LY52], [PR18] of as a function of with fixed and the Fisher zeros [L+19a] of as a function of with fixed. It appears that Theorem 1.1 is the first result establishing an asymptotically optimal zero-free region when the interactions are allowed to differ for different pairs and even to be of different signs, so that we have a mixture of ferromagnetic and antiferromagnetic interactions.
Finally we note that when the energy is described by a higher degree polynomial
on the Boolean cube, some (apparently non-optimal) estimates can be found in[Ba17].
2. Proof of Theorem 1.1
(2.1) Definitions and notation
We consider the Boolean cube of vectors , where for . Let us choose a set and numbers for . The set
is called a face of . Any index is called a free index of the face and the number of free indices is called the dimension of and denoted . Indexes are called fixed. For example, if then the face is the whole cube and if then the face consists of a single point. Generally, a face of dimension consists of points.
Let be a function and let be a face. We define the partial sum of on by
We will use the following straightforward identity. Let be a face of of dimension at least . Let be a free index of and let and be the faces obtained by fixing the -th coordinate of vectors to and respectively. Then
Let us fix a real number and a vector such that for . We denote by the set of all polynomials ,
We can view as a convex subset of with a non-empty interior. In particular, for any and any , we have for .
In what follows, we view non-zero complex numbers as vectors in the plane and measure angles between them.
We start with a simple geometric lemma.
Let be non-zero numbers such that the angle between and does not exceed some and let .
Let , and be the orthogonal projections of , and respectively onto the bisector of the angle between and . Then
and the proof of Part 1 follows.
To prove Part 2, let and . Then
is real and the angle between and does not exceed . Let
Without loss of generality, we may assume that and hence . Let
by Part 1. Since , we have
Since the function is convex on the interval , the minimum value of
subject to the constraints and is attained at . Consequently,
which proves Part 2. ∎
Part 1 can be extended to the sum of more than two vectors, for which one should require , see Lemma 3.6.3 in [Ba16]. Part 2 with a stronger condition is extended to more than two vectors in [BD20], see Lemma 2.1 there.
Let be a face of . Suppose that for all we have and, moreover, the following condition is satisfied: if is a free index of and and are the faces obtained by setting the -th coordinate and respectively, then the angle between the numbers and does not exceed for some .
Since for all ,
and the set is simply connected, we can choose a branch of the function for .
Let us fix two indices , at most one of which is free for . Then
for all .
Differentiating, we get
Suppose first that neither of the indices and is free. Then the value of is constant for all and hence
from which the proof follows.
Suppose now that only one of the indices and , say , is free. Let and be the faces obtained by setting the -th coordinate and respectively. Applying (2.1.1), we get
We apply Lemma 2.2 with
By (2.1.1), we have
and the angle between and does not exceed by the assumption of the lemma. Applying Part 1 of Lemma 2.2, we get
Applying Part 2 of Lemma 2.2, we conclude that
which completes the proof. ∎
Let be a face as in Lemma 2.3 and let be a polynomial. Suppose that is a fixed index of and let be a polynomial obtained by replacing the coefficient in for some by . Then the angle between and does not exceed
For , let be the polynomial obtained by replacing with in , so and and for . Then
Applying Lemma 2.3, we complete the proof. ∎
(2.5) Proof of Theorem 1.1
First, we show that there is such that
Indeed, we can just choose . Using that
We prove by induction for the following statement.
Let be a face of dimension of . Then . Moreover, if and is a free index of then the following holds. Let and be the faces of obtained by fixing the -th variable to and respectively. Then the angle between and does not exceed .
The statement clearly holds for . Suppose that , let be a free index of and let be the corresponding faces, so . By the induction hypothesis, we have and . Moreover, , where is obtained from
by replacing the coefficients for by and the coefficient by . Since is a fixed index for , replacing by leads to multiplying by , which results in the rotation of by an angle of . By Corollary 2.4, replacing all with leads to a rotation of by at most an angle of
Hence the angle between and does not exceed
by (2.5.1). Since
applying Part 1 of Lemma 2.2 with
we conclude that
This concludes the proof of the induction step and hence of Theorem 1.1. ∎
We suppose that the coefficients and are real and satisfy
and some , fixed in advance (there are no restrictions on ). Here we sketch an algorithm for approximating
by the method of polynomial interpolation. Without loss of generality, we assume that and that is large enough,
since for smaller the sum can be computed by brute force.
(3.1) Writing as for a univariate polynomial
First, we rewrite as a polynomial in some new variables. We have
For given , we consider a polynomial
of degree in the variables . Thus we have
and we want to approximate .
Next, for given and , we consider a univariate polynomial
of a complex variable . Our goal is to approximate .
(3.2) Showing that in a neighborhood of
Our next goal is to show that
Since and , from (3.1.1) we deduce that
From (3.1.1), we obtain
Let us choose an arbitrary such that and let us define
where we choose the principal branch of the logarithm, so that . Let
From Section 3.1, we have
Combining (3.2.2)–(3.2.4), we obtain
In addition, if , then
Hence by Theorem 1.1 we have that and therefore by (3.2.5) we conclude that (3.2.1) holds.
As discussed in Section 2.2 of [Ba16], see also [PR17] and [Ba19] for some enhancements, as soon as in some neighborhood of the interval , to approximate within relative error , it suffices to compute the derivatives for where . Since in our case , to approximate within relative error , it suffices to compute for , where the implied constant in the “” notation depends on alone.
From (3.1.2), we obtain