Efficient evaluation of high-order moments and cumulants in tensor network states

West, Colin G.; García-Sáez, Artur; Wei, Tzu-Chieh

doi:10.1103/physrevb.92.115103

Cited by 11 publications

(9 citation statements)

References 73 publications

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“…Hence, the critical point obtained from it is much more precise than using solely the order parameter. The occurrence of a crossing point in the Binder cumulant is also observed in infinite translationally-invariant systems represented by an iMPS, where the crossing occurs between U 4 of different bond dimension m instead of L [6].…”

Section: Introductionmentioning

confidence: 75%

“…The advantage this recursive method has over other tensor network methods of calculating higher-order moments and cumulants such as that in Ref. [6] is that it unifies the method of computing the expectation values of local and string operators whereby the latter ends up being treated as a second order operator (i.e. a two-point correlator).…”

Section: Higher-order Moments and Cumulants In Tensor Networkmentioning

confidence: 99%

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Cumulants and scaling functions of infinite matrix product states

Pillay

McCulloch

2019

Phys. Rev. B

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The order parameter cumulants of infinite matrix product ground states are evaluated across a quantum phase transition. A scheme using the Binder cumulant, finite-entanglement scaling and scaling functions to obtain the critical point and exponents of the correlation length and cumulants is presented. Analogous to the scaling relations that relate the exponents of various thermodynamic quantities, a cumulant exponent relation is derived and used to check the consistency and relationship between the cumulant exponents. This scheme gives a numerically economical way of accurately obtaining the critical exponents. Examples of this scheme are shown for four one dimensional models -the transverse field Ising model, the topological Kondo insulator, the S = 1 Heisenberg chain with single-ion anisotropy and the Bose-Hubbard model. A two dimensional model is also exemplified in the square lattice transverse field Ising model on an infinite cylinder. These exemplary systems portray a variety of local and string order parameters as well as phase transition classes that can be studied with the scaling functions and infinite matrix product states.

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Section: Introductionmentioning

confidence: 75%

Section: Higher-order Moments and Cumulants In Tensor Networkmentioning

confidence: 99%

Cumulants and scaling functions of infinite matrix product states

Pillay

McCulloch

2019

Phys. Rev. B

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“…Indeed, generating functions for the moments and cumulants have been introduced in Ref. [40], which took exponential forms with possible Trotter error and a finite difference method was then employed. In contrast, our approach does not have Trotter error, nor finite difference error, and the generating functions for the state and operator can be unified in a simple way.…”

mentioning

confidence: 99%

Generating Function for Tensor Network Diagrammatic Summation

Tu,

Wu,

Schuch

et al. 2021

Preprint

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The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among other things, excitation spectrum and long-range interacting systems can be studied using TNs, where one however confronts the intricate summation over an extensive number of tensor diagrams. Here, we introduce a set of generating functions, which encode the diagrammatic summations as leading order series expansion coefficients. Combined with automatic differentiation, the generating function allows us to solve the problem of TN diagrammatic summation. We illustrate this scheme by computing variational excited states and dynamical structure factor of a quantum spin chain, and further investigating entanglement properties of excited states. Extensions to infinite size systems and higher dimension are outlined.

show abstract

Section: Introductionmentioning

confidence: 75%

“…The advantage this recursive method has over other tensor network methods of calculating higher-order moments and cumulants such as that in Ref. [145] is that it unifies the method of computing the expectation values of local and string operators whereby the latter ends up being treated as a second order operator (i.e. a two-point correlator).…”

Section: Higher-order Moments and Cumulants In Tensor Networkmentioning

confidence: 99%

Matrix product state study of strongly-interacting systems and quantum phase transitions

Pillay¹

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The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks -an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) -a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and nonlocal order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system's correlation length diverges at the critical point.However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length...

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Efficient evaluation of high-order moments and cumulants in tensor network states

Cited by 11 publications

References 73 publications

Cumulants and scaling functions of infinite matrix product states

Cumulants and scaling functions of infinite matrix product states

Generating Function for Tensor Network Diagrammatic Summation

Matrix product state study of strongly-interacting systems and quantum phase transitions

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