A group theoretical approach of thermodynamic limits in spin systems

Nick G. Duffield

Zhihan Zheng, Pei Sun, Xiaotian Xu +3 more

The thermodynamic limit and boundary energy of the isotropic spin-1 Heisenberg chain with non-diagonal boundary fields are studied. The finite size scaling properties of the inhomogeneous term in the T-Q relation at the ground state are calculated by the density matrix renormalization group. Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields. These results can be generalized to the SU(2) symmetric high spin Heisenberg model directly.

С. М. Сергеев

Yukun Huang

We give a thorough biorthonormal matrix-product-state (BMPS) analysis of the transfer-matrix renormalization group (TMRG) for non-Hermitian matrices in the thermodynamic limit. The BMPS is built on dual series of reduced biorthonormal bases for the left and right Perron states of a non-Hermitian matrix. We propose two alternative infinite-size biorthonormal TMRG (iBTMRG) algorithms and compare their numerical performances for both finite and infinite systems. We show that both iBTMRGs produce dual infinite-BMPS (iBMPS) which are translationally invariant in the thermodynamic limit. We also develop an efficient wavefunction transformation of the iBTMRG, in analogy with that of McCulloch for the infinite-DMRG (McCulloch, 2008 arXiv: 0804.2509 ), for predicting the wavefunction as the lattice size is increased. The resulting iBMPS allows for probing bulk properties of the system in the thermodynamic limit without boundary effects and allows for reducing the computational cost to be independent of the lattice size; these findings are illustrated by calculating the magnetization as a function of the temperature and the critical spin–spin correlation in the thermodynamic limit for a 2D classical Ising model.

Yan Gu

Fourier transforms of the von Neumann density operators could be regarded as functions on a Lie group whose infinitesimal generators correspond to dynamical variables characterizing a given quantum system. Called ``characteristic functions'' in this paper, they are discussed with a view to reformulating quantum statistical mechanics in a group-theoretical formalism. It is shown that these characteristic functions allow a universal differentiating procedure for calculating averages of arbitrary ordered noncommuting observables and satisfy the same criterion of non-negative definiteness as the Fourier transforms of the everywhere non-negative probability distribution (the characteristic functions in classical probability theory). The group-theoretical characteristic-function formalism is especially useful in providing easy reductions of the dynamics of complicated quantum systems to the evolutions of a set of quantum variables that are of particular interest. As examples of applications, the following results are derived: (a) a hierarchy of equations of motion for subsystems of a given N-body quantum system (a quantum analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy), (b) an equation of motion for centers of mass of two interacting quantum systems, and (c) the behavior of spin fluctuations of an ideal spin gas in the thermodynamic limit.

Masanori Ohya, Dėnes Petz