The tensor network representation of high order cumulant and algorithm for their calculation.

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 pagesThe 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

Jan 1, 20160 citations

Introduction to Tensor Network Methods

Timo Felser, Simone Montangero

Graduate texts in physicsJan 1, 20260 citations

Vectorized Symmetric and Fermionic Tensor Network Implementations for GPU-Accelerated Variational Monte Carlo

Si-Jing Du, Johnnie Gray, Ao Chen +1 more

Variational Monte Carlo (VMC) calculations based on tensor networks (TN) have recently achieved competitive accuracy in ground-state calculations of strongly correlated spin and fermionic systems. However, existing tensor network VMC (TN-VMC) algorithms have not been formulated in a manner that can fully utilize GPU acceleration. We tackle the key missing ingredient, namely, the vectorized evaluation of tensor network amplitudes and tensor network operations. This ensures high GPU utilization by batching over computations with identical structure. In particular, we show how to achieve vectorization for the practically relevant case of abelian symmetric tensor networks (which includes fermionic tensor networks) by developing a ``flat'' tensor network formalism for block-sparse tensor representation and contraction. Using this, we construct a GPU-adapted symmetric TN-VMC workflow with batched tensor network computation. In the two-dimensional Fermi--Hubbard model, we demonstrate a GPU speedup of up to $300 \times$ over single core CPU implementations, for fermionic TN variational wavefunctions.Variational Monte Carlo (VMC) calculations based on tensor networks (TN) have recently achieved competitive accuracy in ground-state calculations of strongly correlated spin and fermionic systems. However, existing tensor network VMC (TN-VMC) algorithms have not been formulated in a manner that can fully utilize GPU acceleration. We tackle the key missing ingredient, namely, the vectorized evaluation of tensor network amplitudes and tensor network operations. This ensures high GPU utilization by batching over computations with identical structure. In particular, we show how to achieve vectorization for the practically relevant case of abelian symmetric tensor networks (which includes fermionic tensor networks) by developing a ``flat'' tensor network formalism for block-sparse tensor representation and contraction. Using this, we construct a GPU-adapted symmetric TN-VMC workflow with batched tensor network computation. In the two-dimensional Fermi--Hubbard model, we demonstrate a GPU speedup of up to $300 \times$ over single core CPU implementations, for fermionic TN variational wavefunctions.

physics.comp-phCross-listedarXiv:2608.27861Aug 31, 20260 citations

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

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

We present a numerical scheme for efficiently extracting the higher-order moments and cumulants of various operators on spin systems represented as tensor product states, for both finite and infinite systems, and present several applications for such quantities. For example, the second cumulant of the energy of a state, $\ensuremath{\langle}\mathrm{\ensuremath{\Delta}}{H}^{2}\ensuremath{\rangle}$, gives a straightforward method to check the convergence of numerical ground-state approximation algorithms. Additionally, we discuss the use of moments and cumulants in the study of phase transitions. Of particular interest is the application of our method to calculate the so-called Binder cumulant, which we use to detect critical points and study the critical exponent of the correlation length with only small finite numerical calculations. We apply these methods to study the behavior of a family of one-dimensional models (the transverse Ising model, the spin-1 Ising model, and the spin-1 Ising model in a crystal field), as well as the two-dimensional Ising model on a square lattice. Our results show that in one dimension, cumulant-based methods can produce precise estimates of the critical points at a low computational cost. The method shows promise for two-dimensional systems as well.We present a numerical scheme for efficiently extracting the higher-order moments and cumulants of various operators on spin systems represented as tensor product states, for both finite and infinite systems, and present several applications for such quantities. For example, the second cumulant of the energy of a state, $\ensuremath{\langle}\mathrm{\ensuremath{\Delta}}{H}^{2}\ensuremath{\rangle}$, gives a straightforward method to check the convergence of numerical ground-state approximation algorithms. Additionally, we discuss the use of moments and cumulants in the study of phase transitions. Of particular interest is the application of our method to calculate the so-called Binder cumulant, which we use to detect critical points and study the critical exponent of the correlation length with only small finite numerical calculations. We apply these methods to study the behavior of a family of one-dimensional models (the transverse Ising model, the spin-1 Ising model, and the spin-1 Ising model in a crystal field), as well as the two-dimensional Ising model on a square lattice. Our results show that in one dimension, cumulant-based methods can produce precise estimates of the critical points at a low computational cost. The method shows promise for two-dimensional systems as well.

arXiv:1503.07045Physical Review BSep 1, 201517 citations

300 \times

over single core CPU implementations, for fermionic TN variational wavefunctions.