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arXiv:2011.05148

Heisenberg XX chain, non-homogeneously parameterised generating exponential, and diagonally restricted plane partitions

Cyril Leonidovich Malyshev, N. M. Bogoliubov

Published Nov 10, 2020• 2 citations
arXivHTMLPDF

Abstract

The mean values of non-homogeneously parameterized generating exponential are obtained and investigated for the periodic Heisenberg XX model. The norm-trace generating function of boxed plane partitions with fixed volume of their diagonal parts is obtained as N-particles average of the generating exponential. The generating function of self-avoiding walks of random turns vicious walkers is obtained in terms of the circulant matrices that leads to generalizations of the Ramus's identity. Under various specifications of the generating exponential, the N-particles averages arise for a set of inconsecutive flipped spins and for powers of the first moment of flipped spins distribution at large length of the chain. These averages are expressed through the numbers of closed trajectories with constrained initial/final positions. The estimates at large temporal parameter are expressed through the numbers of diagonally restricted plane partitions characterized by fixed values of the main diagonal trace or by fixed heights of the diagonal columns in one-to-one correspondence with the flipped spins positions.

DOI
10.48550/arxiv.2011.05148

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The mean values of non-homogeneously parameterized generating exponential are obtained and investigated for the periodic Heisenberg XX model. The norm-trace generating function of boxed plane partitions with fixed volume of their diagonal parts is obtained as N-particles average of the generating exponential. The generating function of self-avoiding walks of random turns vicious walkers is obtained in terms of the circulant matrices that leads to generalizations of the Ramus's identity. Under various specifications of the generating exponential, the N-particles averages arise for a set of inconsecutive flipped spins and for powers of the first moment of flipped spins distribution at large length of the chain. These averages are expressed through the numbers of closed trajectories with constrained initial/final positions. The estimates at large temporal parameter are expressed through the numbers of diagonally restricted plane partitions characterized by fixed values of the main diagonal trace or by fixed heights of the diagonal columns in one-to-one correspondence with the flipped spins positions.
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.
The correlation functions of the XXZXXZXXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers
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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.
Using contour-integral representations of the permanent and of the Hafnian, we turn the computation of their moments for random matrices into a replica calculation, analogous to that of the moments of the partition function of a mean-field spin glass. For the 2k2k2k-th moment of N×NN\times NN×N matrices with independent entries, the dominant saddle point is replica symmetric for k>2k>2k>2, and we obtain in closed form the exponential growth rate, the prefactor and the first 1/N1/N1/N correction, in agreement with exact enumerations. The growth rate depends only on whether the entries are real or complex; higher cumulants affect only the prefactor and the 1/N1/N1/N term. At k=2k=2k=2 the replica-symmetric solution becomes unstable, separating a power-law bulk of the distribution from a large-deviation tail, which we compute. The results apply to boson sampling with single photons and with squeezed light.
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.