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Journal of Statistical Mechanics Theory and Experiment

The phase model and the norm-trace generating function of plane partitions

N. M. Bogoliubov, Cyril Leonidovich Malyshev

Published Aug 1, 2018• 11 citations
Publisher

Abstract

Abstract A quantum phase model is considered and the temporal evolution of the exponential of the first moment of particles distribution is studied. The calculation of the temporal evolution is based on the properties of Schur functions. The form-factor of the considered distribution function is expressed in the determinant form and it is proved that in the q -parametrization it is equal to the generating function of the norm-trace generating function of the boxed plane partitions. The asymptotics of the temporal evolution in the double scaling limit and decreasing temperature is obtained. It is shown that the amplitudes of the leading asymptotics depend on the generating function of plane partitions with the fixed sum of diagonal elements.

DOI
10.1088/1742-5468/aad3f8

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Abstract A quantum phase model is considered and the temporal evolution of the exponential of the first moment of particles distribution is studied. The calculation of the temporal evolution is based on the properties of Schur functions. The form-factor of the considered distribution function is expressed in the determinant form and it is proved that in the q -parametrization it is equal to the generating function of the norm-trace generating function of the boxed plane partitions. The asymptotics of the temporal evolution in the double scaling limit and decreasing temperature is obtained. It is shown that the amplitudes of the leading asymptotics depend on the generating function of plane partitions with the fixed sum of diagonal elements.
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.
Abstract Relations between the mean values of distributions of flipped spins on periodic Heisenberg XX chain and some aspects of enumerative combinatorics are discussed. The Bethe vectors, which are the state-vectors of the model, are considered both as on- and off-shell. It is this approach that makes it possible to represent and to study the correlation functions in the form of non-intersecting nests of lattice walks and related plane partitions. We distinguish between two types of walkers, namely lock step models and random turns. Of particular interest is the connection of random turns walks and a circulant matrix. The determinantal representation for the norm-trace generating function of plane partitions with fixed height of diagonal parts is obtained as the expectation of the generating exponential over off-shell N -particle Bethe states. The asymptotics of the dynamical mean value of the generating exponential is calculated in the double scaling limit provided that the evolution parameter is large. It is shown that the amplitudes of the leading asymptotics depend on the number of diagonally constrained plane partitions.
We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of N particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large N, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable β (β is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as N → ∞
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We consider quantum dynamics of the order parameter in the discrete pairing model (Richardson model) in thermodynamic equilibrium. The integrable Richardson Hamiltonian is represented as a direct sum of Hamiltonians acting in different Hilbert spaces of single-particle and paired/empty states. This allows us to factorize the full thermodynamic partition function into a combination of simple terms associated with real spins on singly occupied states and the partition function of the quantum XYXYXY model for Anderson pseudospins associated with the paired/empty states. Using coherent-state path integral, we calculate the effects of superconducting phase fluctuations exactly. The contribution of superconducting amplitude fluctuations to the partition function in the broken-symmetry phase is shown to follow from the Bogoliubov-de Gennes equations in imaginary time. These equations in turn allow several interesting mappings, e.g., they are shown to be in a one-to-one correspondence with the one-dimensional Schr\"odinger equation in supersymmetric quantum mechanics. However, the most practically useful approach to calculate functional determinants is found to be via an analytical continuation of the quantum order parameter to real time, $\ensuremath{\Delta}(\ensuremath{\tau}\ensuremath{\rightarrow}it)$, such that the problem maps onto that of a driven two-level system. The contribution of a particular dynamic order parameter, $\ensuremath{\Delta}(\ensuremath{\tau})$, to the partition function is shown to correspond to the sum of the Berry phase and dynamic phase accumulated by the pseudospin. We also examine a family of exact solutions for two-level-system dynamics on a class of elliptic functions and suggest a compact expression to estimate the functional determinants on such trajectories. The possibility of having quantum soliton solutions coexisting with classical BCS mean field is discussed.