Scaling analysis of snapshot spectra in the world-line quantum Monte Carlo for the transverse-field Ising chain

Transcription

1 TNSAA Dec. 3-6, 2018, Kobe, Japan Scaling analysis of snapshot spectra in the world-line quantum Monte Carlo for the transverse-field Ising chain Kouichi Seki, Kouichi Okunishi Niigata University, Japan arxiv:

2 Entanglement entropy Quantum entanglement nontrivial superposition of quantum states among subsystems in a quantum many-body system Entanglement entropy A measure of the quantum entanglement Tracing out degrees of freedom in the subsystem B S A : Entanglement entropy of the subsystem A ρ A : Reduced density matrix of the subsystem A ψ(a, B) : Groundstate wave function of the whole system A, B : Index of subsystems ψ(a, B) Area Law of entanglement entropy Gapless system : S A = O(L d 1 log L) Gaped system : S A = O(L d 1 ) B A L 1

3 Computational methods for quantum systems Tensor network algorithm : Efficient optimization of the groundstate wavefunction based on the product of local tensors Good point : Bad point : We can handle frustrated systems We can refer the quantum entanglement It is difficult to calculate higher dimensional systems with high accuracy Quantum Monte Carlo method (QMC) : stochastic algorithm Estimation of bulk physical quantities by sampling averages in finite temperatures Good point: We can handle higher dimensional systems efficiently Bad point : It is difficult to simulate frustrated systems It is also difficult to extract information of entanglements These two methods are complementary to each other In this study, we focus on the QMC 2

4 World-line quantum Monte Carlo method quantum spin system N β : Trotter number β = 1/T Suzuki-Trotter decomposition Classical spins on 2D lattice (S z base) Including the discretization error N. Kawashima, K. Harada. JPSJ, 73, 1379 (2004). N β Continuous imaginary time limit 1. We map the d-dimensional quantum system to the (d+1)-dimensional classical system by the Suzuki-Trotter decomposition 2. In the Trotter number N β limit, spin configurations are represented as continuous world lines 3. We update world-line configurations by the Monte Carlo algorithm N β World lines (S z base) β red : up spin green : down spin A configuration of world-line is a classical object. However, the quantum fluctuation is embedded as scattering points/kinks in the world-lines. 3

5 Motivation This study : Analyzing SVD spectra of world-line snapshots in WL QMC may provide a new viewpoint of quantum fluctuations in QMC. We construct an analogue of the reduced density matrix for world-line snapshots. We perform scaling analysis for distributions of the snapshot spectra. cf. Tensor network algorithms : SVD is used for decomposing local tensors with keeping essential information for the total wavefunction. 4

6 Snapshot density matrix for classical Ising model 1. We generate snapshots of the classical 2D Ising model by the Monte Carlo method. 2. We regard " ± 1" spins on a 2D lattice as a matrix M x, y, which we call snapshot matrix. Snapshot of the 2D Ising model y Mapping Snapshot matrix x Snapshot density matrix ρ(x, x ) SVD of the snapshot matrix H. Matsueda, PRE 85, (2012). Y. Imura, et al. JPSJ (2014). Contracting on the y-axis 5

7 Snapshot matrix for world-lines 1. We generate world-line snapshots of the 1D quantum spin system with the loop algorithm. 2. We discretize the imaginary time of the world-line snapshots. 3. We regard the discretized snapshot as a 2D classical system in analogy with 2D classical system. 4. We map the discretized snapshot to a snapshot matrix M n, τ j. Snapshot on the 2D lattice Snapshot of the world-line (Sz base) Snapshot matrix Mapping Discretization of the imaginary time 6

8 Snapshot density matrix for world-lines Discretized snapshot matrix has discretization error We consider the N β limits, and define the snapshot density matrix by integration of the imaginary time index. β M z (n, τ) : Snapshot of the word-line in the S z base U l : Eigenvector of ρ z (n, m), n : Index of the real space direction ω l : Eigenvalue of ρ z (n, m), τ : Index of the imaginary time direction We analyze the eigenvalue distribution of the snapshot density matrix. L 7

9 Transverse-field Ising chain Groundstate S = 1/2 Γ = 0.5 : critical point, S z n Sz n+r r Γ < 0.5 : order 0.5 Γ > 0.5 : disorder Γ We analyzed the parameter dependence of snapshot spectra P(ω) Important parameter Scale of the real space : L Scale of the imaginary time : Γβ Aspect ratio of a snapshot : Q = Γβ L L : System size β = 1/T : Invers temperature β is length of imaginary time τ 8

10 Ordered phase : Γ = 0.4 Snapshot at β = 100 Temperature dependence of eigenvalue distribution τ maximum eigenvalue distribution x The maximum eigenvalue distribution is isolated at ω O L. The other eigenvalues are condensed in near ω 0. The classical order in the S z direction at the zero temperature. 9

11 Disordered phase : Γ = 4.0 Snapshot at β = 100 Temperature dependence of eigenvalue distribution τ x As temperature decreases, The maximum eigenvalue distribution is absorbed into the distribution in the small ω region. The peak of the zero ω condensation disappears. 10

12 Feature for the disordered phase The fixed aspect ratio: Q = Γβ L = 6.25 The shape of the eigenvalue distribution converges for Γ, L, β 1. The converged distribution depends only on aspect ratio Q. The distribution in the disordered regime is described by the universal curve characterized by Q. * universal but still different from the random matrix theory 11

13 Critical point : Γ = 0.5 Snapshot at β = 100 Eigenvalue distribution with fixed Q 0.39 τ x As L and β increase with fixing Q 0.39, the power-law region extends. We find that P(ω) can capture the critical behavior of the quantum system. How can we understand the exponent 2.33? 12

14 Origin of the power-law In the bulk and zero temperature limits, the snapshot density matrix can be expected to approach the correlation function by the self-averaging. ρ(x n, x m ) S z n Sz m x n x m η self-averaging long distance The snapshot density matrix can be diagonalized by Fourier transform because of the translation symmetry. ω(k)~ k 1 η, k = 2πi L, i = 0, ±1, ±2 (η < 1) Since quantum number k is uniformly distributed, the distribution of ω can be obtained as, P(ω)~ω α α = 2 η 1 η η = 1 4, α = Caution This derivation : sample average before diagonalization ρ(x n, x m ) Numerical result : sample average after diagonalization ρ(x n, x m ) Transverse-field Ising chain Y. Imura, et al. JPSJ (2014). 13

15 Summary We performed scaling analysis for the eigenvalue distribution of snapshots generated by the world-line Monte Carlo simulation for the transverse-field Ising chain. Ordered region : We found that the isolated maximum eigenvalue distribution represents the classical order at zero temperature. Disorder region : We found that the distribution in the disordered regime is described by the universal curve characterized by aspect ratio Q. Critical point : The distribution obeys the power-law P ω α. The exponent α is related to the anomalous dimension η. Future issue Extraction of the relationship between the snapshot spectrum and the quantum entanglement 14