Scaling analysis of snapshot spectra in the world-line quantum Monte - PowerPoint PPT Presentation

TNSAA 2018-2019 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:1706.09257

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 𝑇 𝐵 : Entanglement entropy of the subsystem A 𝜍 𝐵 : Reduced density matrix of the subsystem A 𝜔(𝐵, 𝐶) : Groundstate wave function of the whole system A, B : Index of subsystems Tracing out degrees of freedom 𝜔(𝐵, 𝐶) in the subsystem B Area Law of entanglement entropy • Gapless system : 𝑇 𝐵 = 𝑃(𝑀 𝑒−1 log 𝑀) B L A • Gaped system : 𝑇 𝐵 = 𝑃(𝑀 𝑒−1 ) 1

Computational methods for quantum systems Tensor network algorithm : • Efficient optimization of the groundstate wavefunction based on the product of local tensors ・ We can handle frustrated systems Good point : ・ We can refer the quantum entanglement ・ It is difficult to calculate higher dimensional systems Bad point : 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

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

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

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 𝑁 𝑦, 𝑧 , which we call “snapshot matrix”. Snapshot of the 2D Ising model Snapshot matrix y Mapping x Snapshot density matrix 𝜍(𝑦, 𝑦 ′ ) SVD of the snapshot matrix H. Matsueda, PRE 85, 031101 (2012). Contracting on the y-axis 5 Y. Imura, et al. JPSJ 83 . 114002 (2014).

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 𝑁 𝑜, 𝜐 𝑘 . Snapshot on the 2D lattice Snapshot of the world-line Snapshot matrix (Sz base) Mapping Discretization of the imaginary time 6

Snapshot density matrix for world-lines Discretized snapshot matrix has discretization error ↓ We consider the N 𝛾 → ∞ limits, and define the snapshot density matrix by integration of the imaginary time index. 𝛾 𝑀 𝑁 𝑨 (𝑜, 𝜐) : Snapshot of the word-line in the S z base 𝑉 𝑚 : Eigenvector of 𝜍 𝑨 (𝑜, 𝑛) , 𝑜 : Index of the real space direction 𝜕 𝑚 : Eigenvalue of 𝜍 𝑨 (𝑜, 𝑛) , 𝜐 : Index of the imaginary time direction We analyze the eigenvalue distribution of the snapshot density matrix. 7

Transverse-field Ising chain S = 1/2 Groundstate ≈ 𝑠 − 1 𝑨 𝑇 𝑜+𝑠 𝑨 Γ = 0.5 : critical point, 𝑇 𝑜 4 Γ Γ > 0.5 : disorder Γ < 0.5 : order 0 0.5 We analyzed the parameter dependence of snapshot spectra 𝑄(𝜕) Important parameter Scale of the real space : L L : System size Scale of the imaginary time : Γ𝛾 𝛾 = 1/𝑈 : Invers temperature Γ𝛾 𝛾 is length of imaginary time 𝜐 Aspect ratio of a snapshot : 𝑅 = 𝑀 8

Ordered phase : Γ = 0.4 Temperature dependence of Snapshot at 𝛾 = 100 eigenvalue distribution 𝜐 maximum eigenvalue distribution 𝑦 • The maximum eigenvalue distribution is isolated at 𝜕 ∼ 𝑃 𝑀 . • The other eigenvalues are condensed in near 𝜕 ∼ 0 . The classical order in the S z direction at the zero temperature. 9

Disordered phase : Γ = 4.0 Temperature dependence of Snapshot at 𝛾 = 100 eigenvalue distribution 𝜐 𝑦 As temperature decreases, • The maximum eigenvalue distribution is absorbed into the distribution in the small 𝜕 region. • The peak of the zero 𝜕 condensation disappears. 10

Feature for the disordered phase The fixed aspect ratio: Γ𝛾 𝑅 = 𝑀 = 6.25 • The shape of the eigenvalue distribution converges for Γ, 𝑀, 𝛾 ≫ 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

Critical point : Γ = 0.5 Snapshot at 𝛾 = 100 Eigenvalue distribution with fixed 𝑅 ≈ 0.39 𝜐 𝑦 As 𝑀 and 𝛾 increase with fixing 𝑅 ≈ 0.39 , the power-law region extends. We find that 𝑄(𝜕) can capture the critical behavior of the quantum system. How can we understand the exponent −2.33 ? 12

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. 𝑨 𝑇 𝑛 𝑨 ≈ |𝑦 𝑜 − 𝑦 𝑛 | −𝜃 𝜍(𝑦 𝑜 , 𝑦 𝑛 ) ≈ 𝑇 𝑜 self-averaging long distance The snapshot density matrix can be diagonalized by Fourier transform because of the translation symmetry. 2𝜌𝑗 𝜕(𝑙)~|𝑙| 1−𝜃 , 𝑙 = 𝑀 , i = 0, ±1, ±2 ⋯ (𝜃 < 1) Since quantum number 𝑙 is uniformly distributed, the distribution of 𝜕 can be obtained as, 𝛽 = 2 − 𝜃 1 − 𝜃 → 𝜃 = 1 4 , 𝛽 = 7 𝑄(𝜕)~𝜕 −𝛽 3 ≈ 2.33 Transverse-field Ising chain Caution • This derivation : sample average before diagonalization 𝜍(𝑦 𝑜 , 𝑦 𝑛 ) • Numerical result : sample average after diagonalization 𝜍(𝑦 𝑜 , 𝑦 𝑛 ) 13 Y. Imura, et al. JPSJ 83 . 114002 (2014).

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 𝑄 ∼ 𝜕 −𝛽 . • • The exponent 𝛽 is related to the anomalous dimension 𝜃 . Future issue • Extraction of the relationship between the snapshot spectrum and the quantum entanglement 14