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Numerical Methods for Many-Body Quantum Dynamics

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Numerical Methods for Many-Body Quantum Dynamics Numerical methods for many-body quantum dynamics Thesis by Christopher David White In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended 16 May 2019 ii c 2019 Christopher David White ORCID: 0000-0002-8372-2492 Some rights reserved. This thesis is distributed under a Creative Commons Attribution NonCommercial-ShareAlike License. iii ACKNOWLEDGEMENTS cum caritate et gratia magna parochiae et populo Sancti Thomas Apostoli Holivudensis. \congregavit nos in unum Christi amor; exsultemus, et in ipso jucundemur." iv ABSTRACT This thesis describes two studies of the dynamics of many-body quantum sys- tems with extensive numerical support. In Part I we first give a new algorithm for simulating the dynamics of one- dimensional systems that thermalize (that is, come to local thermal equilib- rium). The core of this algorithm is a new truncation for matrix product operators, which reproduces local properties faithfully without reproducing non-local properties (e.g. the information required for OTOCs). To the ex- tent that the dynamics depends only on local operators, timesteps interleaved with this truncation will reproduce that dynamics. We then apply this to algorithm to Floquet systems: first to clean, non- integrable systems with a high-frequency drive, where we find that the system is well-described by a natural diffusive phenomenology; and then to disordered systems with low-frequency drive, which display diffusion—not subdiffusion— at appreciable disorder strengths. In Part II, we study the utility of many-body localization as a medium for a thermodynamic engine. We first construct a small (\mesoscale") engine that gives work at high efficiency in the adiabatic limit, and show that thanks to the slow spread of information in many body localized systems, these mesoscale engines can be chained together without specially engineered insulation. Our construction takes advantage of precisely the fact that MBL systems do not thermalize. We then show that these engines still have high efficiency when run at finite speed, and we compare to competitor engines. v PUBLISHED CONTENT AND CONTRIBUTIONS This thesis contains material from: White, C.D., Zaletel, M.,, Mong, R.S.K., and Refael, G. \Quantum Dynamics of Thermalizing Systems." Phys. Rev. B 97, 035127 (2018). DOI: 10.1103/phys- revb.97.035127. CDW constructed the algorithm used, wrote the algorithm-specific code, con- structed the benchmarking protocol, carried out the simulations, and performed the analysis. Ye, B., Machado, F., White, C.D., Mong, R.S.K., and Yao, N. \Emergent hydrody- namics in Floquet quantum systems". arXiv:1902.01859. CDW participated in the conception of the project and the analysis of the re- sults (in particular the framing in terms of hydrodynamics). In addition, the project used CDW's algorithm-specific code. Yunger Halpern, N., White, C.D., Gopalakrishnan, S., and Refael, G. \Quantum engine based on many-body localization". Phys. Rev. B. 99, 024203. DOI: 10.1103/physrevb.99.024203 CDW participated in the conception of the project, carried out all of the nu- merical calculations and many of the analytical calculations, and participated in constructing the physical arguments. vi TABLE OF CONTENTS Acknowledgements . iii Abstract . iv Published Content and Contributions . v Table of Contents . vi Introduction . ix Chapter I: Background: MBL and MPS . 1 1.1 Many body localization . 1 1.1.1 Many-body localization in perturbation theory . 1 1.1.2 Long-time memory in MBL systems . 5 1.1.3 Level spacing . 7 1.1.4 Local dynamics at finite time . 10 1.2 Matrix product states . 10 1.2.1 Graphical notation for tensors and tensor contractions 14 1.2.2 Expectation values and correlations in MPSs . 15 1.2.3 Canonical forms and orthogonality centers . 16 1.3 Dynamics and matrix product states . 20 1.4 Matrix product operators . 21 1.4.1 MPO representation of an exponential interaction . 22 1.4.2 MPO representation of a power-law Hamiltonian . 23 1.4.3 MPDOs . 24 1.4.4 Matrix product density operator representations of Gibbs states . 26 1.5 Some essentials of quantum thermodynamics . 26 I Thermalizing systems and matrix product density operators 28 Chapter II: Simulating quantum dynamics of thermalizing systems . 29 2.1 Background and intuition . 30 2.1.1 Background: matrix product state methods . 30 2.1.2 Intuition: thermalization and computation . 33 2.2 Method: truncation of MPDOs . 36 2.2.1 Setting, notation, and tools . 37 2.2.2 Re-writing the MPDO to expose properties whose preser- vation we guarantee . 39 2.2.3 Modifying the MPDO . 41 2.2.4 Preservation of l-site operators . 44 2.3 Results . 46 2.3.1 Hamiltonian . 46 vii 2.3.2 Application: pure-state evolution . 47 2.3.3 Application: mixed-state evolution (near equilibrium) . 52 2.3.4 Application: mixed-state evolution (far from equilibrium) 52 2.4 Conclusion . 55 Chapter III: Heating, prethermalization, and hydrodynamics in a high- frequency Floquet system . 59 3.1 Model and Floquet phenomenology . 60 3.2 Benchmarking DMT . 65 3.2.1 Benchmarking Krylov subspace dynamics . 65 3.2.2 DMT and Krylov . 66 3.2.3 Entropy in DMT . 69 3.2.4 Effect of Trotter step size on DMT numerics . 69 3.3 Analysis and results . 71 3.3.1 Approach to Gibbs Ensemble . 73 3.3.2 Heating . 77 3.3.3 Extracting diffusion coefficients of a spatially uniform, static Hamiltonian . 78 3.3.4 Classical diffusion equation . 82 3.3.5 Hydrodynamics in large spin chains . 86 3.4 Discussion . 86 Chapter IV: Hydrodynamics in a disordered low-frequency Floquet system 88 4.1 Model . 89 4.2 Method . 90 4.2.1 Convergence testing . 91 4.2.2 Comparison with exact simulations . 93 4.2.3 Quantity of interest . 95 4.2.4 Disorder averaging . 96 4.3 Results . 97 4.4 Discussion . 99 II MBL-mobile: Many-body-localized engine 101 Chapter V: MBL and thermodynamic engines . 102 Chapter VI: The MBL-Mobile in its adiabatic limit . 105 6.1 Qubit toy model for the mesoscale engine . 105 6.2 Set-up for the mesoscale MBL engine . 107 6.3 Notation and definitions: . 110 6.4 Quantitative analysis of the adiabatic mesoscale engine: some easy limits . 111 6.5 Quantitative analysis of the adiabatic mesoscale engine: a more detailed calculation . 113 6.5.1 Partial-swap model of thermalization . 113 6.5.2 Average heat Q absorbed during stroke 2 . 114 h 2i 6.5.3 Average heat Q4 absorbed during stroke 4 . 116 6.5.4 Average per-cycleh i power W . 118 h toti viii 6.5.5 Efficiency ηMBL in the adiabatic approximation . 120 6.6 MBL engine in the thermodynamic limit . 121 6.7 Numerical simulations . 125 6.7.1 Hamiltonian . 125 6.7.2 Scaling factor . 126 6.7.3 Representing states and Hamiltonians . 129 6.7.4 Strokes 1 and 3: tuning . 129 6.7.5 Stroke 2: Thermalization with the cold bath . 130 6.7.6 Results . 131 Chapter VII: The MBL-Mobile at finite speed . 133 7.1 Diabatic corrections . 133 7.1.1 Fractional-Landau-Zener transitions . 134 7.1.2 Landau-Zener transitions . 135 7.1.3 APT transitions . 136 7.2 Numerical simulations . 137 7.2.1 Simulating finite-time tuning . 138 7.2.2 Enlightenment from numerical methods . 139 7.3 Precluding communication between subengines . 140 7.4 Lower bound on the cycle time from cold thermalization . 141 Chapter VIII: Racing the MBL-mobile . 144 8.1 Comparisons with competitor Otto engines . 144 8.1.1 Comparison with bandwidth engine . 144 8.1.2 Comparison with MBL engine tuned between same-strength disorder realizations . 145 8.1.3 Quantum-dot engine . 147 8.1.4 Anderson-localized engine . 147 8.2 Order-of-magnitude estimates . 148 8.3 Outlook . 149 Bibliography . 152 ix INTRODUCTION Numerical methods for many-body quantum systems go back to the earliest days of electronic computers. The physicists of Los Alamos during World War II relied heavily on numerical simulations; the first calculations were done by human \computers" or with mechanical calculators,1 but soon after the war they began using ENIAC, the first programmable electronic computer [84]. We need simulations because the phenomena we study are complex. This was certainly the case for the pioneers of computational physics, simulating weapons physics at Los Alamos, and it is the case for us now. We may have a microscopic Hamiltonian, but analytically solving the Schr¨odinger equation with that Hamiltonian is out of the question, and tightly-controlled approx- imations are frequently unavailable. So we are forced to rely on heuristic approximations justified by physical arguments. But these approximations need to be checked: do they describe the model's behaviour well? At the very least, do they describe some essential phenomenon at work in the system? Nu- merical simulations of a model catch hidden assumptions and subtle mistakes in your physical arguments|and they suggest further approximations to make and phenomena to investigate. (The work of Part II of this thesis proceeded in exactly this fashion, as an ongoing three-part conversation between physical argument, careful analytical work, and numerical simulation.) But numerical methods are not simply devices for checking your intuitions and arguments. Developing a new numerical method and teasing out a characteri- zation of where it succeeds, where it fails, and why it succeeds or fails provides important insight into the physics of the systems that method simulates Take the history of the density matrix renormalization group (DMRG) and the matrix product state (MPS).2 Steve White developed the method in 1993 [226, 227] as an extension of Wilson's numerical renormalization group (NRG) [230], and it quickly saw wide application [82, 100, 127, 180], not only for computing ground states but also for simulating low-energy dynamics.
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