Transfer matrices and matrix product operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic overview of applications, exact results, and computational ...aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.
The simplest Tree-Tensor-States (TTS) respecting the Parity and the Time-Reversal symmetries are studied in order to describe the ground states of long-ranged quantum spin chains with or without ...disorder. Explicit formulas are given for the one-point and two-point reduced density matrices that allow to compute any one-spin and two-spin observable. For Hamiltonians containing only one-body and two-body contributions, the energy of the TTS can be then evaluated and minimized in order to obtain the optimal parameters of the TTS. This variational optimization of the TTS parameters is compared with the traditional block-spin renormalization procedure based on the diagonalization of some intra-block renormalized Hamiltonian.
•The entanglement properties are essential to understand quantum many-body systems.•Tree-Tensors describe the entanglement on various scales.•Explicit calculations are possible for simple elementary building-block tensors.•Optimized Tree-Tensors are inhomogeneous for disordered systems.•Scale-invariant Tree-Tensors describe the critical points of pure systems.
One of the fundamental aspects of quantum information theory is the study of distance measures between quantum states. Hilbert-Schmidt distance serves as a convenient choice for exploring a wide ...range of quantum information-related phenomena. This study contributes by deriving precise closed-form expressions for the variance of the squared Hilbert-Schmidt distance, both between one fixed and one random density matrix and between two independent random density matrices. Notably, our findings go beyond recently acquired results concerning the mean-square Hilbert-Schmidt distance. By incorporating both mean and variance, we extend our analysis to provide a gamma-distribution-based approximation for the probability density function of the squared Hilbert-Schmidt distance. The validity of these analytical results is confirmed through Monte Carlo simulations. Additionally, we perform a comparative analysis, aligning our analytical predictions with the mean and variance of the squared Hilbert-Schmidt distance calculated in correlated spin chain simulations, revealing a satisfactory agreement.
•We have obtained exact closed-form expressions for the variance of the squared Hilbert-Schmidt distance between a pair of density matrices, where either one or both matrices can be random and taken from the Hilbert-Schmidt ensemble.•In derivation, we have utilized the canonical relationship between the Hilbert-Schmidt and the Wishart-Laguerre ensemble of random matrices.•Using the mean and variance, we have presented a gamma-distribution-based approximation for the probability density function of squared Hilbert-Schmidt distance.•We have validated our analytical results by performing Monte-Carlo simulations of the underlying random matrix models and found consistent agreement.•Additionally, we have compared our RMT analytical results and results obtained from correlated spin chain simulations.
This work supports the existence of extended nonergodic states in the intermediate region between the chaotic (thermal) and the many‐body localized phases. These states are identified through an ...extensive analysis of static and dynamical properties of a finite one‐dimensional system with onsite random disorder. The long‐time dynamics is particularly sensitive to changes in the spectrum and in the structures of the eigenstates. The study of the evolution of the survival probability, Shannon information entropy, and von Neumann entanglement entropy enables the distinction between the chaotic and the intermediate region.
Despite the consensus that the transition from a metal to an insulator can still take place in quantum systems with many interacting particles, the details are not entirely understood. It has been debated, for instance, whether there is an intermediate phase between the chaotic and the many‐body localized phase. Our results for the long‐time evolution of the survival probability makes clear the existence of the intermediate region.
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