Universal role of correlation entropy in critical phenomena

In statistical physics, if we divide successively an equilibrium system into two parts, we will face a situation that, to a certain length., the physics of a subsystem is no longer the same as the original one. The extensive property of the thermal entropy S(A boolean OR B) = S(A) + S(B) is then violated. This observation motivates us to introduce a concept of correlation entropy between two points, as measured by the mutual information in information theory, to study the critical phenomena. A rigorous relation is established to display some drastic features of the non-vanishing correlation entropy of a subsystem formed by any two distant particles with long-range correlation. This relation actually indicates a universal role played by the correlation entropy for understanding the critical phenomena. We also verify these analytical studies in terms of two well-studied models for both the thermal and quantum phase transitions: the two-dimensional Ising model and the one-dimensional transverse-field Ising model. Therefore, the correlation entropy provides us with a new physical intuition of the critical phenomena from the point of view of information theory.

[1]  M. Plenio,et al.  Quantifying Entanglement , 1997, quant-ph/9702027.

[2]  Eytan Barouch,et al.  Statistical Mechanics of the X Y Model. II. Spin-Correlation Functions , 1971 .

[3]  G. Vidal,et al.  Entanglement in quantum critical phenomena. , 2002, Physical review letters.

[4]  Shi-Jian Gu,et al.  Entanglement, quantum phase transition, and scaling in the XXZ chain , 2003 .

[5]  A. Osterloh,et al.  Scaling of entanglement close to a quantum phase transition , 2002, Nature.

[6]  D. Varshalovich,et al.  Quantum Theory of Angular Momentum , 1988 .

[7]  P. Jordan.  Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem , 1935 .

[8]  S. Gu,et al.  Local entanglement and quantum phase transition in 1D transverse field Ising model , 2006, quant-ph/0606133.

[9]  S. Sachdev.  Quantum Phase Transitions , 1999 .

[10]  Shang‐keng Ma.  Modern Theory of Critical Phenomena , 1976 .

[11]  Claude E. Shannon,et al.  The mathematical theory of communication , 1950 .

[12]  H. Stanley,et al.  Scaling, Universality, and Renormalization: Three Pillars of Modern Critical Phenomena , 1999 .

[13]  B. McCoy,et al.  The Two-Dimensional Ising Model , 1973 .

[14]  Chen Ning Yang,et al.  Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors , 1962 .

[15]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[16]  M. Continentino.  Quantum scaling in many-body systems , 2017 .

[17]  Hai-Qing Lin,et al.  Ground-state entanglement in the XXZ model , 2004, quant-ph/0408101.

[18]  M. Nielsen,et al.  Entanglement in a simple quantum phase transition , 2002, quant-ph/0202162.

[19]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[20]  F. Traversa,et al.  Two-point versus multipartite entanglement in quantum phase transitions. , 2005, Physical review letters.

[21]  Dong Yang,et al.  Effects of environmental parameters to total, quantum and classical correlations , 2006, quant-ph/0612223.

[22]  Chanchal K. Majumdar,et al.  On Next‐Nearest‐Neighbor Interaction in Linear Chain. II , 1969 .

[23]  V. Korepin,et al.  Universality of entropy scaling in one dimensional gapless models. , 2003, Physical Review Letters.

[24]  Shi-Jian Gu,et al.  Entanglement and quantum phase transition in the extended Hubbard model. , 2004, Physical review letters.

[25]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[26]  V. Vedral.  The role of relative entropy in quantum information theory , 2001, quant-ph/0102094.

[27]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[28]  N. Mermin,et al.  Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models , 1966 .

[29]  O. Klein.  Zur quantenmechanischen Begründung des zweiten Hauptsatzes der Wärmelehre , 1931 .

[30]  D. Sherrington,et al.  TWO KINDS OF BOSONS AND BOSE CONDENSATES. , 1970 .

[31]  R. Shankar.  Renormalization group approach to interacting fermions , 1994 .

[32]  A. Wehrl.  General properties of entropy , 1978 .

[33]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[34]  Eytan Barouch,et al.  Statistical Mechanics of the XY Model. III , 1970 .

[35]  K. Binder,et al.  A Guide to Monte Carlo Simulations in Statistical Physics , 2000 .