The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among others, 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 the 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.
The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among others, 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 the 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.
Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin- 1 / 2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin- 1 / 2 J 1 - J 2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations. Published by the American Physical Society 2024
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Efficient methods to access the entanglement of a quantum many-body state, where the complexity generally scales exponentially with the system size N, have long been a concern. Here we propose the Schmidt tensor network state (Schmidt TNS) that efficiently represents the Schmidt decomposition of finite- and even infinite-size quantum states with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients (i.e., entanglement spectrum) and transformations in the decomposition to tensor networks (TNs) with linearly scaled complexity versus N. Specifically, the transformations are written as the TNs formed by local unitary tensors, and the Schmidt coefficients are encoded in a positive-definite matrix product state (MPS). Translational invariance can be imposed on the TNs and MPS for the infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating the ground state of the quasi-one-dimensional spin model with geometrical frustration. Our results show that the MPS encoding the Schmidt coefficients is weakly entangled even when the entanglement entropy of the decomposed state is strong. This justifies the efficiency of using MPS to encode the Schmidt coefficients, and promises an exponential speedup on the full-state sampling tasks.
Feynman's diagrammatic series is a common language for a formally exact theoretical description of systems of infinitely-many interacting quantum particles, as well as a foundation for precision computational techniques. Here we introduce a universal framework for efficient summation of connected or skeleton Feynman diagrams for generic quantum many-body systems. It is based on an explicit combinatorial construction of the sum of the integrands by dynamic programming, at a computational cost that can be made only exponential in the diagram order on a classical computer and potentially polynomial on a quantum computer. We illustrate the technique by an unbiased diagrammatic Monte Carlo calculation of the equation of state of the 2D SU(N) Hubbard model in an experimentally relevant regime, which has remained challenging for state-of-the-art numerical methods.
We introduce Neural Tensor Network States (νTNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the νTNS framework, a neural network serves as a disentangler of the wave-function, transforming the physical degrees of freedom into renormalized variables with much less entanglement. The renormalized state is then efficiently encoded by a back-flow tensor network. This construction yields a compact yet highly expressive representation of strongly correlated quantum states. Using convolutional neural networks combined with matrix product states as a concrete implementation, we obtain state-of-the-art variational energies for the spin-1/2J1-J2 Heisenberg model on the square lattice at the highly frustrated point J2/J1=0.5, for systems up to 20×20 with periodic boundary conditions. Finite-size scaling of spin, dimer, and plaquette correlations exhibits power-law decay without magnetic or valence-bond long-range order, consistent with a gapless quantum spin-liquid ground state at that point.This νTNS framework is flexible and naturally extensible to other neural and tensor-network structures, offering a general platform for investigating strongly correlated quantum many-body systems.
The classical simulation of many-body quantum systems is an essential tool in understanding many fundamental aspects of condensed matter physics. But a major obstacle arises from the number of degrees of freedom involved in describing such systems, which is exponential in the system size. Recently, however, a class of numerical techniques based on structures called "tensor networks" has emerged, which allows many "typical" quantum states (such as the ground states of gapped, local Hamiltonians) to be represented much more efficiently. In this work we extend and apply these techniques to consider several central topics in quantum many-body physics. After reviewing the relevant background material from the field of tensor networks and tensor network states, we demonstrate a method for computing high order moments and cumulants of operators with respect to such states, including the so-called "Binder cumulant," a powerful tool for detecting phase transitions. Next, we employ tensor network algorithms to characterize the ground state phase diagram of a quantum spin model, including both symmetry-breaking phases and symmetry protected topological order, and find a signicant variety of phases and phase transitions. Finally, we consider the entanglement properties of quantum states exhibiting many-body localization, using a combination of exact diagonalization and tensor network techniques. | 231 pages