Functional representation of the partition function and phase transitions in the Hubbard model

The functional representation of the partition function for the Hubbard model with strong Coulomb repulsion is obtained in the form of an integral with respect to auxiliary Grassmann variables. The approximate study of the phase transitions shows:

1. for a small doping constant δ, the system is an antiferromagnetic dielectric;
2. for δ≥t/V, the system is a ferromagnetic metal if U/t>10 and superconducting metal if U/t? 10. Bibliography:13 titles.

     

Integrable equations for the partition function of the six-vertex model

The partition function of the six-vertex model with the domain-wall boundary condition is considered in the homogeneous and inhomogeneous cases. The determinant representation allows us to show that the partition function is a solution of the Toda equation in the homogeneous case and a solution of the Hirota equation in the inhomogeneous case. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 207–215. Translated by N. A. Kitanin.     

On the classification of partition quantum groups

This is a survey on some results obtained recently in the classification of compact quantum groups associated to partitions, with a focus on the non-crossing case. We take a global look at the main results in the subject and highlight some key features of the methods used. We conclude by several suggestions for pushing further the classification.     

Parity of the partition function

Although much is known about the partition function, little is known about its parity. For the polynomials D(x):=(Dx²+1)/24, where , we show that there are infinitely many m (resp. n) for which p(D(m)) is even (resp. p(D(n)) is odd) if there is at least one such m (resp. n). We bound the first m and n (if any) in terms of the class number h(−D). For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products, and we employ Galois representations and locally nilpotent Hecke algebras.     

The disordered phase of the inhomogeneous potts model is extremal on the Cayley tree

We find the sufficient condition on the parameters of the inhomogeneous Potts model on the Cayley tree under which the disordered phase becomes extremal. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 410–418, September, 2000.     

On the restricted partition function

For a vector \(\mathbf a = (a_1,\ldots ,a_r)\) of positive integers, we prove formulas for the restricted partition function \(p_{\mathbf a}(n): = \) the number of integer solutions \((x_1,\dots ,x_r)\) to \(\sum _{j=1}^r a_jx_j=n\) with \(x_1\ge 0, \ldots , x_r\ge 0\) and its polynomial part.     

Computing the integer partition function

In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of for for primes up to and small powers of and .

     

Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p_i(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π^′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p₀(n)-p₂(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that

(i)

and that for sufficiently large n

(ii)

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:

Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.     

Multiplicative properties of the partition function

A lower bound for the number of multiplicatively independent values ofp(n) forN ≤ n <N + R is given. The proof depends on the Hardy-Ramanujan formula and is of an elementary nature.     

On the parity of the partition function

Let N be the set of all positive integers and D a subset of N. Let p(D,n) be the number of partitions of n with parts in D and let |D(x)| denote the number of elements of D not exceeding x. It is proved that if D is an infinite subset of N such that p(D,n) is even for all n?n₀, then |D(x)|?logx/log2−logn₀/log2. Moreover, if D is an infinite subset of N such that p(D,n) is odd for all n?n₀ and , then |D(x)|?logx/log2−logn₀/log2. These lower bounds are essentially the best possible.     

Convolution congruences for the partition function

Ahlgren and Boylan recently proved the uniqueness of the Ramanujan congruences for the primes , , and by using the modularity of a certain partition function. Here we use their result to find universal congruences, of a different type, which hold for the partition function modulo all primes .

     

The solution to the BCS gap equation and the second-order phase transition in superconductivity

The existence and the uniqueness of the solution to the BCS gap equation of superconductivity are established in previous papers, but the temperature dependence of the solution is not discussed. In this paper, in order to show how the solution varies with the temperature, we first give another proof of the existence and the uniqueness of the solution and point out that the unique solution belongs to a certain set. Here this set depends on the temperature T. We define another certain subset of a Banach space consisting of continuous functions of both T and x. Here, x stands for the kinetic energy of an electron minus the chemical potential. Let the solution be approximated by an element of the subset of the Banach space above. Second, we show, under this approximation, that the transition to a superconducting state is a second-order phase transition.     

Complex-temperatures zeros of the partition function in spin-glass models

We propose a method for approximating the density of partition zeros for real-dimensional spin-glass models. The method uses the complex-temperature data of the ferromagnetic model on the same lattice. We obtain the universality line in the complex-temperature space. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 520–528, September, 2000.     

A model Hamiltonian of the BCS type for a Bose gas

Theoretical and Mathematical Physics -