scholarly journals Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

2020 ◽  
Vol 2 (2) ◽  
Author(s):  
Jiaju Zhang ◽  
Pasquale Calabrese ◽  
Marcello Dalmonte ◽  
Mohammad Ali Rajabpour
Keyword(s):  
Correlation Functions ◽  
Spin Chain ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Density Matrices ◽  
Reduced Density Matrices ◽  
Trace Distance ◽  
The Difference ◽  
Reduced Density

We carry out a comprehensive comparison between the exact modular Hamiltonian and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional critical quantum spin chains. As a warm-up, we first illustrate how the trace distance provides a more informative mean of comparison between reduced density matrices when compared to any other Schatten nn-distance, normalized or not. In particular, as noticed in earlier works, it provides a way to bound other correlation functions in a precise manner, i.e., providing both lower and upper bounds. Additionally, we show that two close reduced density matrices, i.e. with zero trace distance for large sizes, can have very different modular Hamiltonians. This means that, in terms of describing how two states are close to each other, it is more informative to compare their reduced density matrices rather than the corresponding modular Hamiltonians. After setting this framework, we consider the ground states for infinite and periodic XX spin chain and critical Ising chain. We provide robust numerical evidence that the trace distance between the lattice BW reduced density matrix and the exact one goes to zero as \ell^{-2}ℓ−2 for large length of the interval \ellℓ. This provides strong constraints on the difference between the corresponding entanglement entropies and correlation functions. Our results indicate that discretized BW reduced density matrices reproduce exact entanglement entropies and correlation functions of local operators in the limit of large subsystem sizes. Finally, we show that the BW reduced density matrices fall short of reproducing the exact behavior of the logarithmic emptiness formation probability in the ground state of the XX spin chain.

CORRELATION FUNCTIONS IN QUANTUM SPIN CHAINS AND VERTEX MODELS

2001 ◽  
Vol 16 (11) ◽  
pp. 1875-1887
Author(s):  
VIERI MASTROPIETRO
Keyword(s):  
Correlation Functions ◽  
Spin Chain ◽  
Nearest Neighbor ◽  
Quantum Spin ◽  
Spin Chains ◽  
Vertex Model ◽  
Quantum Spin Chains ◽  
Neighbor Interaction ◽  
Vertex Models

Some correlation functions of critical models, like the anisotropic spin chain with nearest and next-to-nearest neighbor interaction, or the eight vertex model, are computed as a corollary of the study of the XYZ model in [2].

scholarly journals Deep Learning Quantum States for Hamiltonian Estimation

2021 ◽  
Vol 38 (11) ◽  
pp. 110301
Author(s):  
Xinran Ma ◽  
Z. C. Tu ◽  
Shi-Ju Ran
Keyword(s):  
Magnetic Fields ◽  
Ground States ◽  
Quantum Spin ◽  
Physical Parameters ◽  
Many Body ◽  
Density Matrices ◽  
Reduced Density Matrices ◽  
Training Samples ◽  
Quantum Spin Models ◽  
Reduced Density

Human experts cannot efficiently access physical information of a quantum many-body states by simply “reading” its coefficients, but have to reply on the previous knowledge such as order parameters and quantum measurements. We demonstrate that convolutional neural network (CNN) can learn from coefficients of many-body states or reduced density matrices to estimate the physical parameters of the interacting Hamiltonians, such as coupling strengths and magnetic fields, provided the states as the ground states. We propose QubismNet that consists of two main parts: the Qubism map that visualizes the ground states (or the purified reduced density matrices) as images, and a CNN that maps the images to the target physical parameters. By assuming certain constraints on the training set for the sake of balance, QubismNet exhibits impressive powers of learning and generalization on several quantum spin models. While the training samples are restricted to the states from certain ranges of the parameters, QubismNet can accurately estimate the parameters of the states beyond such training regions. For instance, our results show that QubismNet can estimate the magnetic fields near the critical point by learning from the states away from the critical vicinity. Our work provides a data-driven way to infer the Hamiltonians that give the designed ground states, and therefore would benefit the existing and future generations of quantum technologies such as Hamiltonian-based quantum simulations and state tomography.

A TEMPLATE FOR QUANTUM SPIN CHAIN SPECTRA

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 707-730 ◽  
Author(s):  
PAUL MARTIN ◽  
VLADIMIR RITTENBERG
Keyword(s):  
Spin Chain ◽  
Hecke Algebra ◽  
Quantum Spin ◽  
Spin Chains ◽  
Irreducible Representations ◽  
Quantum Spin Chains ◽  
Young Diagrams ◽  
Quantum Spin Chain ◽  
Fixed Length

We consider a series of N-state L(≥N) site quantum spin chains, characterised by the ordered partition of N into 2 parts, N=P+M. These (P/M) chains are invariant under an action of UqSU(P/M), and are built from a representation of the Hecke algebra HL-1(q). We establish that the intersection of the spectra of a (P/M) and (P'/M') chain of fixed length L is the spectrum of the (min(P,P')/min(M,M')) chain of that length. We establish that the spectrum of the (P/M) chain breaks into blocks corresponding to irreducible representations of HL-1(q) (or equivalently irreducible representations of UqSU(P/M)) characterised by Young diagrams with no rectangular subdiagrams of dimension (P+1)×(M+1) (height × width resp.). We give the corresponding quotient relations for the Hecke algebra. We discuss several implications of these results.

scholarly journals Dynamic Correlation Functions for Quantum Spin Chains

1983 ◽  
Vol 51 (3) ◽  
pp. 219-222 ◽  
Author(s):  
Gerhard Müller ◽  
Robert E. Shrock
Keyword(s):  
Correlation Functions ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Dynamic Correlation

THE TENSOR PRODUCT OF TENSOR OPERATORS OVER QUANTUM ALGEBRAS: SOME APPLICATIONS TO QUANTUM SPIN CHAINS

1994 ◽  
Vol 08 (25n26) ◽  
pp. 3655-3669
Author(s):  
M. SCHEUNERT
Keyword(s):  
Tensor Product ◽  
Correlation Functions ◽  
Recent Article ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Algebras ◽  
Quantum Spin Chains ◽  
Expectation Values ◽  
Point Correlation ◽  
Tensor Operators

The present work is a direct sequel to a recent article by the author, in which he has analysed the tensor product of tensor operators over quantum algebras. Here the results obtained there are summarized and then specialized and extended to prepare possible applications to quantum spin chains. In particular, certain invariant two-point operators are introduced (whose expectation values yield the invariant two-point correlation functions) and their multiplicative properties are derived.

scholarly journals Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics

2017 ◽  
Vol 3 (5) ◽  
Author(s):  
Michel Bauer ◽  
Denis Bernard ◽  
Tony Jin
Keyword(s):  
Spin Chain ◽  
Quantum Dynamics ◽  
Quantum Noise ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Heisenberg Spin ◽  
Heisenberg Spin Chain ◽  
Dissipative Effects ◽  
Macroscopic Fluctuation Theory

Motivated by the search for a quantum analogue of the macroscopic fluctuation theory, we study quantum spin chains dissipatively coupled to quantum noise. The dynamical processes are encoded in quantum stochastic differential equations. They induce dissipative friction on the spin chain currents. We show that, as the friction becomes stronger, the noise induced dissipative effects localize the spin chain states on a slow mode manifold, and we determine the effective stochastic quantum dynamics of these slow modes. We illustrate this approach by studying the quantum stochastic Heisenberg spin chain.

Solvable nineteen-vertex models and quantum spin chains of spin one

1994 ◽  
Vol 4 (8) ◽  
pp. 1151-1159 ◽  
Author(s):  
Makoto Idzumi ◽  
Tetsuji Tokihiro ◽  
Masao Arai
Keyword(s):  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Vertex Models
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