[Paper Review] Universal scaling relationship between classical and quantum correlations in critical quantum spin chains

Yan-Wei Dai, Xi-Hao Chen+3 more|arXiv (Cornell University)|May 9, 2018

Quantum many-body systems2 references3 citations

TL;DR

This study investigates classical and quantum correlations in one-dimensional critical quantum spin chains using infinite matrix product states (iMPS) and the infinite time-evolving block decimation (iTEBD) algorithm. It reveals a universal scaling relationship: the critical exponents of mutual information, classical correlation, and quantum discord are identical at quantum critical points, with η^I ≈ η^C ≈ η^D, and satisfy η = η^α / 2, linking them to spin-spin correlation exponents.

ABSTRACT

We numerically investigate classical and quantum correlations in one-dimensional quantum critical systems. The infinite matrix product state (iMPS) representation is employed in order to consider an infinite-size spin chain. By using the infinite time-evolving block decimation algorithm, iMPS ground state wave functions are obtained at critical points for the transverse-field spin-$1/2$ XY model. From the ground state wave functions, we calculate classical and quantum correlations and mutual information. All of the correlations are found to exhibit a power-law decay with the increments of the lattice distance for both the transition lines of the Ising universality class and the Gaussian universality class. Such power-law scaling behaviors of the correlations manifest the existence of diversing correlation lengths, which means scale invariance. The critical features of the correlations can be characterized by introducing a critical exponent of the power-law decaying correlations. Similar to the critical exponent $η$ of the spin-spin correlation for the universality classes in the transverse-field XY model, we calculate the critical exponents of the two-spin classical and quantum correlations as well as that of the corresponding mutual information. All of the correlations have the same critical exponents, i.e., $η^{I}=η^{C}=η^{D}$ at a critical point, where the superscripts $I$, $C$, and $D$ stand for mutual information, classical correlation, and quantum correlation, respectively. Furthermore, the critical exponent $η$ of the spin-spin correlation is shown to relate to $η= η^α/2$ with $α\in \{ I, C, D\}$.

Motivation & Objective

• To investigate the scaling behavior of classical and quantum correlations in one-dimensional quantum critical spin chains.
• To determine whether quantum mutual information and correlation measures can characterize universality classes in critical systems.
• To establish a universal relationship between critical exponents of different correlation types and the spin-spin correlation exponent η.
• To numerically estimate critical exponents for the Ising and Gaussian universality classes using iMPS and iTEBD.

Proposed method

• Employed infinite matrix product state (iMPS) representation to simulate infinite-size quantum spin chains.
• Applied the infinite time-evolving block decimation (iTEBD) algorithm to compute ground state wavefunctions at critical points.
• Calculated classical correlation, quantum discord, and quantum mutual information from the iMPS ground states.
• Used finite-size extrapolation via η^α(χ) = η^α_0 χ^a + η^α_∞ to estimate critical exponents in the thermodynamic limit.
• Compared the critical exponents of different correlation types to the spin-spin correlation exponent η.
• Validated results against exact solutions for the XX model and known values from conformal field theory.

Experimental results

Research questions

• RQ1 Do classical and quantum correlations exhibit universal scaling behavior in critical quantum spin chains?
• RQ2 Are the critical exponents of quantum mutual information, classical correlation, and quantum discord identical at quantum critical points?
• RQ3 Is there a universal relation between the critical exponents of different correlation measures and the spin-spin correlation exponent η?
• RQ4 Can the critical exponents of information-theoretic correlations be used to classify universality classes in quantum critical systems?

Key findings

• All correlations—mutual information, classical correlation, and quantum discord—exhibit power-law decay with increasing lattice distance at critical points.
• The critical exponents for mutual information (η^I), classical correlation (η^C), and quantum discord (η^D) are numerically indistinguishable: η^I ≈ η^C ≈ η^D.
• For the Ising universality class, the critical exponents are η^I ≈ η^C ≈ η^D ≈ 0.506(7), consistent with the exact value 1/2.
• For the Gaussian universality class (XX model), the exponents converge to η^I ≈ η^C ≈ η^D ≈ 1.008(8), matching the analytical value η^D = 1.
• A universal relation η = η^α / 2 holds across all correlation types, linking the spin-spin correlation exponent to those of classical and quantum correlations.
• The results confirm that quantum mutual information and correlation measures capture universal features of quantum criticality and can classify universality classes via their critical exponents.