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

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 TNSAA 2018-2019

• Dec. 3-6, 2018, Kobe, Japan

arXiv:1706.09257

Quantum entanglement

• nontrivial superposition of quantum states among subsystems in a

quantum many-body system

Entanglement entropy

• A measure of the quantum entanglement

1

Entanglement entropy

𝑇𝐵 : Entanglement entropy of the subsystem A 𝜍𝐵 : Reduced density matrix of the subsystem A 𝜔(𝐵, 𝐶) : Groundstate wave function of the whole system A, B : Index of subsystems

B A L

Tracing out degrees of freedom in the subsystem B 𝜔(𝐵, 𝐶)

Area Law of entanglement entropy

• Gapless system : 𝑇𝐵 = 𝑃(𝑀𝑒−1 log 𝑀)
• Gaped system : 𝑇𝐵 = 𝑃(𝑀𝑒−1)

2

Computational methods for quantum systems

Tensor network algorithm :

• Efficient optimization of the groundstate wavefunction based on the

product of local tensors

Good point : ・ We can handle frustrated systems ・ We can refer the quantum entanglement Bad point : ・ 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

3

World-line quantum Monte Carlo method

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 𝑂

𝛾 → ∞ limit, spin configurations are represented as

continuous world lines 3. We update world-line configurations by the Monte Carlo algorithm

Suzuki-Trotter decomposition

quantum spin system Classical spins on 2D lattice (Sz base) Including the discretization error

𝑂

𝛾 → ∞

Continuous imaginary time limit

World lines (Sz base)

𝑂

𝛾 : Trotter number

𝛾 = 1/𝑈 red : up spin green : down spin

• N. Kawashima, K. Harada. JPSJ, 73, 1379 (2004).

A configuration of world-line is a classical object. However, the quantum fluctuation is embedded as scattering points/kinks in the world-lines.

𝑂

𝛾

𝛾

4

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.

5

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”.

Mapping

Snapshot matrix SVD of the snapshot matrix Snapshot of the 2D Ising model Snapshot density matrix 𝜍(𝑦, 𝑦′)

Contracting on the y-axis x y

• H. Matsueda, PRE 85, 031101 (2012).
• Y. Imura, et al. JPSJ 83. 114002 (2014).

6

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, 𝜐𝑘 .

Snapshot matrix

Discretization of the imaginary time (Sz base) Snapshot on the 2D lattice Snapshot of the world-line Mapping

7

Snapshot density matrix for world-lines

Discretized snapshot matrix has discretization error ↓ We consider the N𝛾 → ∞ limits, and define the snapshot density matrix by integration of the imaginary time index.

𝑁𝑨(𝑜, 𝜐) : Snapshot of the word-line in the Sz base 𝑉𝑚 : Eigenvector of 𝜍𝑨(𝑜, 𝑛), 𝑜 : Index of the real space direction 𝜕𝑚 : Eigenvalue of 𝜍𝑨(𝑜, 𝑛), 𝜐 : Index of the imaginary time direction

We analyze the eigenvalue distribution of the snapshot density matrix.

𝑀 𝛾

Transverse-field Ising chain

8

S = 1/2

Scale of the real space : L Scale of the imaginary time : Γ𝛾 Aspect ratio of a snapshot : 𝑅 =

Γ𝛾 𝑀

Important parameter

L : System size 𝛾 = 1/𝑈 : Invers temperature 𝛾 is length of imaginary time 𝜐 0.5 Γ > 0.5 : disorder Γ = 0.5 : critical point, 𝑇𝑜

𝑨𝑇𝑜+𝑠 𝑨

≈ 𝑠−1

4

Γ

Γ < 0.5 : order

Groundstate We analyzed the parameter dependence of snapshot spectra 𝑄(𝜕)

9

Ordered phase : Γ = 0.4

• The maximum eigenvalue distribution is isolated at 𝜕 ∼ 𝑃 𝑀 .
• The other eigenvalues are condensed in near 𝜕 ∼ 0.

Snapshot at 𝛾 = 100 Temperature dependence of eigenvalue distribution

The classical order in the Sz direction at the zero temperature.

𝜐 𝑦

maximum eigenvalue distribution

10

Disordered phase : Γ = 4.0

Snapshot at 𝛾 = 100 Temperature dependence of 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.

11

Feature for the disordered phase

• The shape of the eigenvalue distribution converges for Γ, 𝑀, 𝛾 ≫ 1 .
• The converged distribution depends only on aspect ratio Q.

The fixed aspect ratio: 𝑅 =

Γ𝛾 𝑀 = 6.25

The distribution in the disordered regime is described by the universal curve characterized by Q.

* universal but still different from the random matrix theory

12

Critical point : Γ = 0.5

Eigenvalue distribution with fixed 𝑅 ≈ 0.39 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? Snapshot at 𝛾 = 100 𝜐 𝑦

13

Origin of the power-law

𝜍(𝑦𝑜, 𝑦𝑛) ≈ 𝑇𝑜

𝑨𝑇𝑛 𝑨

≈ |𝑦𝑜 − 𝑦𝑛|−𝜃 In the bulk and zero temperature limits, the snapshot density matrix can be expected to approach the correlation function by the self-averaging.

self-averaging long distance

The snapshot density matrix can be diagonalized by Fourier transform because of the translation symmetry. 𝜕(𝑙)~|𝑙|1−𝜃, 𝑙 =

2𝜌𝑗 𝑀 , i = 0, ±1, ±2 ⋯

(𝜃 < 1) 𝑄(𝜕)~𝜕−𝛽 𝛽 = 2 − 𝜃 1 − 𝜃 → 𝜃 = 1 4 , 𝛽 = 7 3 ≈ 2.33 Since quantum number 𝑙 is uniformly distributed, the distribution of 𝜕 can be obtained as,

Caution

• This derivation : sample average before diagonalization 𝜍(𝑦𝑜, 𝑦𝑛)
• Numerical result : sample average after diagonalization 𝜍(𝑦𝑜, 𝑦𝑛)

Transverse-field Ising chain

• Y. Imura, et al. JPSJ 83. 114002 (2014).

14

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