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Local Operators and Quantum Chaos by Daniel Eric Parker a Dissertation Local Operators and Quantum Chaos by Daniel Eric Parker A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Joel Moore, Chair Professor Ehud Altman Professor David Limmer Summer 2020 The dissertation of Daniel Eric Parker, titled Local Operators and Quantum Chaos, is ap- proved: Chair Date Date Date University of California, Berkeley Local Operators and Quantum Chaos Copyright 2020 by Daniel Eric Parker i To my parents, Judith Fleishman and Thomas Parker, for all they have done for me. ii Contents Contents ii List of Figures iv List of Tables viii 1 Invitation: Tea. Earl Grey. Hot. 1 2 The Lanczos Algorithm 8 2.1 Krylov Spaces and Three-Term Recurrances . 8 2.2 Infinite Lanczos, Orthogonal Polynomials, and Gauss Quadrature . 13 3 A Universal Operator Growth Hypothesis 23 3.1 Introduction . 23 3.2 Synopsis . 24 3.3 Preliminaries: The Recursion Method . 25 3.4 The Hypothesis . 28 3.5 Exponential Growth of Complexities . 35 3.6 Growth Rate as a Bound on Chaos . 40 3.7 Application to Hydrodynamics . 46 3.8 Finite Temperature . 50 3.9 Conclusions . 55 Appendices 59 3.A Brief Review of the Recursion Method . 59 3.B Moments and Lanczos Coefficients in the SYK Model . 61 3.C Numerical Details for 1d Spin Chains . 66 3.D A Family of Exact Solution with Linear Growth . 68 3.E Derivation of the q-Complexity Bound . 70 3.F Geometric Origin of the Upper Bounds . 72 4 Local Matrix Product Operators 75 4.1 iMPOs Introduction . 75 iii 4.2 The Idea of Compression . 77 4.3 Review of MPOs . 79 4.4 Finite MPO Compression . 81 4.5 Local Infinite Matrix Product Operators . 87 4.6 Canonical forms for Infinite MPOs . 93 4.7 Compression of iMPOs . 99 4.8 Operator Entanglement and Error Bounds . 101 4.9 Relation to Control Theory . 105 4.10 iMPO Examples . 107 4.11 Unit Cell MPOs . 111 4.12 Properties of Compressed Hamiltonians . 120 4.13 Conclusions . 127 Appendices 129 4.A Proofs for Local MPOs . 129 4.B Proofs for Canonical forms . 134 4.C Exact estimates of Schmidt values . 139 4.D Elementary operations . 141 5 Conclusions 146 Bibliography 148 iv List of Figures 1.1 A cup of tea losing information. 1 1.2 Sketch of a highly-quantum system hooked up to a ammeter to measure electrical transport. 2 1.3 Schematic of the role of chaos in quantum dynamics. 3 1.4 Sketch of the computational resources required to simulate dynamics. 4 1.5 (Top) Sketch of the space of operators, with complexity of the operator increasing from left to right. (Bottom) The equivalent 1D chain. (Middle) The wavefunction on the 1d chain. 5 2.1 Lanczos on a random Hermitian matrix with N = 40. The initial vector is also chosen randomly. (Left) The true spectrum λi is represented by vertical lines, (n) while the eigenvalues θi of Tn are represented by red crosses. One can see that eigenvalues with the largest magnitude converge first. (Right) Error in the eigenvalues versus the number of Lanczos iterations n. The error falls off roughly exponentially. 12 2.2 (Left) Comparison of the full spectral density of the Chaotic Ising model with its approximation as a sum of δ-functions via Eq. (2.22) in the Krylov space iHtb K (H; ). (Right) Time-evolution of a state (t) = e− in the full Hilbert N j i j i space and the Krylov spaces KN (H; ). Throughout we take L = 12 (dimension 4096)b with open boundary conditions, and chosen randomly. 19 j i b 3.1 Artist's impression of the space of operators and its relation to the 1d chain de- fined by the Lanczos algorithm starting from a simple operator . The region of complex operators corresponds to that of large n on the 1d chain.O Under our hypothesis, the hopping amplitudes bn on the chain grow linearly asymptotically in generic thermalizing systems (with a log-correction in one dimension, see Sec- tion 3.4). This implies an exponential spreading (n) e2αt of the wavefunction t ∼ 'n on the 1d chain, which reflects the exponential growth of operator complexity under Heisenberg evolution, in a sense we make precise in Section 3.5. The form of the wavefunction 'n is only a sketch; see Fig. 3.5 for a realistic picture. 26 v 3.2 Lanczos coefficients in a variety of models demonstrating common asymptotic behaviors. \Ising" is H = X X + Z with = eiqj Z (q = 1=128 here i i i+1 i O j j and below) and has bn O(1). \X in XX" is H = XiXi+1 + YiYi+1 with ∼ P P i = X , which is a string rather than a bilinearb in the Majorana fermion O j j P representation, so this is effectively an interacting integrable model that has bn P ∼ pbn. XXX is H = X X +Y Y +Z Z with = eiqj (X Y Y X ) i i i+1 i i+1 i i+1 O j j j+1− j j+1 that appears to obey b pn. Finally, SYK is (3.18) where q = 4 and J = 1 and P n ∼ P = p2γ1 with bn n. The Lanczos coefficientsb have been rescaled vertically forO display purposes.∼ Numerical details are given in Appendices 3.B and 3.C. 29 3.3 Illustrationb of the spectral function and the analytical structure of C(t); t C. When the Lanczos coefficients have linear growth rate α, Φ(!) has exponential2 ! =!0 tails e−| j with ! = 2α/π; C(t) is analytical in a strip of half-width 1=! ∼ 0 0 and the singularities closest to the origin are at t = i=!0. See Appendix 3.A for further discussion. .± . 32 3.4 (a) Lanczos coefficients in a variety of strongly interacting spin-half chains: H1 = i XiXi+1 + 0:709Zi + 0:9045Xi, H2 = H1 + i 0:2Yi, H3 = H1 + i 0:2ZiZi+1. The initial operator is energy density wave with momentum q = 0:1. (b) Cross- P P P over to apparently linearO growth as interactions are added to a free model. Here H = X X 1:b05Z + h X , and 1:05X X + Z . The b 's are i i i+1 − i X i O/ i i i+1 i n bounded when hX = 0 but appears to have asymptotically linear growth for any P P hX = 0. Logarithmic corrections are notb clearly visible in the numerical data. Numerical6 details are given in Appendix 3.C. 34 3.5 The exact solution wavefunction (3.25) in the semi-infinite chain at various times. The wavefunction is defined only at n = 0; 1; 2 ::: , but has been extrapolated to intermediate values for display. 36 3.6 (a) The growth rate α versus the classical Lyapunov exponent λL=2 in the classical Feingold-Peres model of coupled tops, (3.46). α λL=2 in general, with equality around the c = 0 where the model is the most chaotic.≥ The growth rate appears to be discontinuous at the non-interacting points c = 1, similarly to Fig. 3.4- (b). (b) The first 40 Lanczos coefficients of quantum s±= 2;:::; 32 and classical (s = ) FP model, with c =0. ........................... 46 3.7 Numerical1 computation of the diffusion coefficient for the energy density operator = q in H = i XiXi+1 1:05Zi + 0:5Xi. (a) The Lanczos coefficients for qO= 0:E15 are fit to (3.50) with−α = 0:35 and η = 1:74. We found it actually better P notb to approximate G(N)(z) by G(N)(z), but instead by G(N+δ)(z) for some integer offset δ so that η 1 (in the example shown, δ = 12). Large η or negative values ≈ lead to numerical pathologies.e (b) The approximate Green'se function (3.52) at q = 0:15. The arrow shows the \leading" pole that governs diffusion. (c) The locations of the leading poles for a range of q. One can clearly see the diffusive dispersion relation z = iDq2=2 + O(q4). Fitting yields a diffusion coefficient D = 3:3(5). 51 vi 3.8 Exact Lyapunov exponent λL(T ) (3.97) and growth rate α(T ) with the Wightman inner product (3.99) of the SYK model in the large-q limit as a function of temper- ature (in units of coupling constant ). The conjectured bound λ (T ) 2α(T ) J L ≤ W is exactly saturated at all temperatures, while the universal bound λL(T ) 2πT only saturates in the zero temperature limit. .≤ . 55 3.B.1Change in the growth rate near integrability for the SYK model with q = 2 and q = 4 (3.85). The ratio of the q = 4 to q = 2 term is given by J, and the model becomes free at J =0................................. 64 3.C.1The size distribution of the Pauli strings in the Krylov vectors for the Hamil- On tonian H1 with parameters and initial operator as in Fig. 3.4. Though the distribution drops quickly after its peak, P (s) is supported onb [0; n=2 + 2]. 68 n b c 4.1 A finite-state machine that generates Eq. 4.3. 79 4.2 Compression of a finite MPO representing the Hamiltonian (4.34). (a) The bond dimensions for: W , the naive MPO representation of H1; WL, the left-canonical representation by Alg. 1; WC , the compressed MPO by Alg. 2, and WC0 , the result of the standardc MPS compression.
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