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Title: Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

Abstract

Here, we use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge \(c\) in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are even able to precisely reconstruct the first several \(\mathbb {Z}_2\)-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension \(\Delta _\sigma = 0.518154(15)\), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

Authors:
 [1];  [2];  [3];  [4];  [5];  [6]
  1. European Organization for Nuclear Research (CERN), Geneva (Switzerland)
  2. Brown Univ., Providence, RI (United States)
  3. Yale Univ., New Haven, CT (United States)
  4. European Organization for Nuclear Research (CERN), Geneva (Switzerland); École Normale Supérieure, Paris (France); Univ. Pierre et Marie Curie, Paris (France)
  5. Institute for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
  6. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)
Publication Date:
Research Org.:
Institute for Advanced Study, Princeton, NJ (United States); Univ. of California, Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF)
OSTI Identifier:
1599339
Grant/Contract Number:  
SC0009988; PHY11-25915; 05-BLAN-NT09-573739; PITN-GA-2009-237920; FG02-11ER41742; AC02-05CH1123
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of Statistical Physics
Additional Journal Information:
Journal Volume: 157; Journal Issue: 4-5; Journal ID: ISSN 0022-4715
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Critical phenomena; Conformal invariance; Ising Model; Critical exponents; Central charge; Stress tensor

Citation Formats

El-Showk, Sheer, Paulos, Miguel F., Poland, David, Rychkov, Slava, Simmons-Duffin, David, and Vichi, Alessandro. Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. United States: N. p., 2014. Web. doi:10.1007/s10955-014-1042-7.
El-Showk, Sheer, Paulos, Miguel F., Poland, David, Rychkov, Slava, Simmons-Duffin, David, & Vichi, Alessandro. Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. United States. https://doi.org/10.1007/s10955-014-1042-7
El-Showk, Sheer, Paulos, Miguel F., Poland, David, Rychkov, Slava, Simmons-Duffin, David, and Vichi, Alessandro. 2014. "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents". United States. https://doi.org/10.1007/s10955-014-1042-7. https://www.osti.gov/servlets/purl/1599339.
@article{osti_1599339,
title = {Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents},
author = {El-Showk, Sheer and Paulos, Miguel F. and Poland, David and Rychkov, Slava and Simmons-Duffin, David and Vichi, Alessandro},
abstractNote = {Here, we use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge \(c\) in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are even able to precisely reconstruct the first several \(\mathbb {Z}_2\)-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension \(\Delta _\sigma = 0.518154(15)\), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.},
doi = {10.1007/s10955-014-1042-7},
url = {https://www.osti.gov/biblio/1599339}, journal = {Journal of Statistical Physics},
issn = {0022-4715},
number = 4-5,
volume = 157,
place = {United States},
year = {Thu Jun 26 00:00:00 EDT 2014},
month = {Thu Jun 26 00:00:00 EDT 2014}
}

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