The Tensor Networks Anthology: Simulation techniques for many-body  quantum lattice systems

Silvi, Pietro; Tschirsich, Ferdinand; Gerster, Matthias; Jünemann, Johannes; Jaschke, Daniel; Rizzi, Matteo; Montangero, Simone

doi:10.21468/scipostphyslectnotes.8

Cited by 119 publications

(120 citation statements)

References 199 publications

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“…The study of one-dimensional quantum many-body systems has motivated the emergence of a number of techniques, based on tensor network states (TNS). More concretely, they use matrix product states (MPS) and matrix product density operators (MPDO) [1][2][3][4][5] to approximate the ground states, low-lying excitations, thermal states, as well as time evolution. These methods have enabled the in-depth study of a multitude of models and the analysis of relevant physical phenomena.…”

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confidence: 99%

Probing Thermalization through Spectral Analysis with Matrix Product Operators

et al. 2020

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We combine matrix product operator techniques with Chebyshev polynomial expansions and present a method that is able to explore spectral properties of quantum many-body Hamiltonians. In particular, we show how this method can be used to probe thermalization of large spin chains without explicitly simulating their time evolution, as well as to compute full and local densities of states. The performance is illustrated with the examples of the Ising and PXP spin chains. For the non-integrable Ising chain, our findings corroborate the presence of thermalization for several initial states, well beyond what direct time-dependent simulations have been able to achieve so far.The study of one-dimensional quantum many-body systems has motivated the emergence of a number of techniques, based on tensor network states (TNS). More concretely, they use matrix product states (MPS) and matrix product density operators (MPDO) [1-5] to approximate the ground states, low-lying excitations, thermal states, as well as time evolution. These methods have enabled the in-depth study of a multitude of models and the analysis of relevant physical phenomena.The success of such techniques is rooted in the ability of MPS and related ansatzes to accurately describe states that fulfill an area law of entanglement [6,7], satisfied (or only slightly violated) by many of the problems mentioned above [8]. There are, however, important open questions that such techniques cannot easily solve. In particular, excited states at finite energy density are difficult to approximate, except in very particular cases [9, 10], as they generically display volume law entanglement and, additionally, are embedded in highly dense spectral regions, which severely hinders the convergence of the algorithms. Out-of-equilibrium dynamics is also problematic: under time evolution a volume law often emerges, that makes an MPS approximation inadequate, except for short times. As a consequence, it is virtually impossible for standard MPS techniques to address the fundamental questions of equilibration and thermalization of relatively large closed quantum systems.A few alternative tensor network algorithms have tried to overcome these problems by avoiding the explicit representation of the states [11][12][13][14]. Although they extend the applicability of the toolbox and allow access to additional dynamical quantities in some scenarios, the fundamental goal of accessing the long time behavior in a general case, and thus deciding the appearance of equilibration or thermalization, has not been achieved.Here we introduce a new powerful tool to fill in these gaps. Our method is based on the use of MPO to approximate a family of generalized densities of states, and provides a means to directly address thermalization. More concretely, we combine TNS and the kernel polynomial method (KPM) [15] in a general scheme that provides access not only to the full density of states (DOS) of a given many-body Hamiltonian [16], but also to energy functions that are intimately related to the out-o...

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Probing Thermalization through Spectral Analysis with Matrix Product Operators

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“…The matrix product states (MPS) results presented in the main text were performed by our own implementation of a U(1) symmetric code preserving the total number of particles, based on the anthology of tensor networks build on a symmetry-preserving library in collaboration with the group of S. Montangero at the University of Ulm [75].…”

Section: Appendix E: Details On the Mps Simulationsmentioning

confidence: 99%

Drude weight increase by orbital and repulsive interactions in fermionic ladders

Haller

Rizzi

Filippone

2020

Phys. Rev. Research

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In strictly one-dimensional systems, repulsive interactions tend to reduce particle mobility on a lattice. Therefore, the Drude weight, controlling the divergence at zero-frequency of optical conductivities in perfect conductors, is lower than in non-interacting cases. We show that this is not the case when extending to quasi one-dimensional ladder systems. Relying on bosonization, perturbative and matrix product states (MPS) calculations, we show that nearest-neighbor interactions and magnetic fluxes provide a bias between back-and forward-scattering processes, leading to linear corrections to the Drude weight in the interaction strength. As a consequence, Drude weights counter-intuitively increase (decrease) with repulsive (attractive) interactions. Our findings are relevant for the efficient tuning of Drude weights in the framework of ultracold atoms trapped in optical lattices and equally affect topological edge states in condensed matter systems. arXiv:1911.11160v1 [cond-mat.quant-gas]

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“…Here we focus on its introduction and applications in the field of quantum many body systems. In this realm, and generally speaking, the name tensor network states (or tensor product states) is used to designate families of ansatzes which can efficiently parametrize the state of such quantum systems and fundamentally encode particular patterns of entanglement [202][203][204][205][206]. Alternatively, a TN (without open indices) may be used to represent the partition function of a certain (classical or quantum) model.…”

Section: A Tensor Network: a New Tool For Classical Computationsmentioning

confidence: 99%

“…For the sake of completeness, we review here the fundamental aspects of the main techniques employed in the context of lattice gauge theories, and we refer the reader to the literature on the subject for detailed technical descriptions of the algorithms (see, e.g., Refs. [203,204,206,[244][245][246][247][248]). Most of the algorithms we describe here are easy to implement, and there exist several libraries available to the public that currently provide most of the functionality [249][250][251][252].…”

Section: Main Numerical Algorithmsmentioning

confidence: 99%

Review on novel methods for lattice gauge theories

2020

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Formulating gauge theories on a lattice offers a genuinely non-perturbative way of studying quantum field theories, and has led to impressive achievements. In particular, it significantly deepened our understanding of quantum chromodynamics. Yet, some very relevant problems remain inherently challenging, such as real time evolution, or the presence of a chemical potential, cases in which Monte Carlo simulations are hindered by a sign problem.In the last few years, a number of possible alternatives have been put forward, based on quantum information ideas, which could potentially open the access to areas of research that have so far eluded more standard methods. They include tensor network calculations, quantum simulations with different physical platforms and quantum computations, and constitute nowadays a vibrant research area. Experts from different fields, including experimental and theoretical high energy physics, condensed matter, and quantum information, are turning their attention to these interdisciplinary possibilities, and driving the progress of the field. The aim of this article is to review the status and perspectives of these new avenues for the exploration of lattice gauge theories.

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The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

Cited by 119 publications

References 199 publications

Probing Thermalization through Spectral Analysis with Matrix Product Operators

Probing Thermalization through Spectral Analysis with Matrix Product Operators

Drude weight increase by orbital and repulsive interactions in fermionic ladders

Review on novel methods for lattice gauge theories

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