Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks

Abstract

The XX0 Heisenberg model on a cyclic chain is considered. The representation of the Bethe wave functions via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The determinantal expressions of the form-factors and of the thermal correlation functions are obtained. The q-binomial determinants enable the connection of the form-factors with the generating functions both of boxed plane partitions and of self-avoiding lattice paths. The asymptotical behavior of the thermal correlation functions is studied in the limit of low temperature provided that the characteristic parameters of the system are large enough.

Correlation Functions as Nests of Self-Avoiding Paths

“…In this paper we shall be restricted with the case of finite size of the system in order to consider the dynamical correlation function called the persistence of ferromagnetic string and related to the projection operatorΠ n that forbids spin "down" states on n consecutive sites of the chain [18]:…”

“…In the above formulas G(µ L ; µ R |K) is given either by (28) or (34), u N and v N stand for an arbitrary parametrization, and two independent summations over λ L,R = µ L,R − δ N are analogous to those in (36). Substituting (31) into (53), we obtain the transition amplitude in the form, [18]:…”

“…Some sections of enumerative combinatorics [1] and the theory of symmetric functions [15] has come to play an important role in the theory of integrable models [16,17] and especially in the studies of correlation functions [18,19]. The aim of this paper is to represent correlation functions of XX0 spin chain as sums over nests of self-avoiding lattice paths.…”

The Ground-State Vector of the XY Heisenberg Chain and the Gauss Decomposition

“…The correlation functions of certain quantum integrable models demonstrate connection with enumerative combinatorics and with the theory of symmetric functions [1][2][3][4][5]. For instance, random lattice walks and boxed plane partitions, as subjects of enumerative combinatorics [6], are related to the correlation functions of the XX model [7][8][9][10][11]. Various spin lattice models [12], including the XY Heisenberg chain model, as well as its isotropic limit, the XX model, provide a base for such actively developing subjects in the theory of quantum information [13] as random lattice walks [14] and entanglement entropy [15].…”

Combinatorics of a strongly coupled boson system

“…Отметим, что сумма функций Шура в левой части (20) может быть выражена через q-биномиальный определитель [28].…”