Emptiness formation probability, Toeplitz determinants, and conformal field theory
International audience We revisit the study of the emptiness formation probability, the probability of forming a sequence of $\ell$ spins with the same ferromagnetic orientation in the ground-state of a quantum spin chain. We focus on two different examples, exhibiting strikingly different behavior:...
Published in: | Journal of Statistical Mechanics: Theory and Experiment |
---|---|
Main Author: | |
Other Authors: | , |
Format: | Article in Journal/Newspaper |
Language: | English |
Published: |
HAL CCSD
2014
|
Subjects: | |
Online Access: | https://hal.archives-ouvertes.fr/hal-01586223 https://doi.org/10.1088/1742-5468/2014/05/P05010 |
id |
ftccsdartic:oai:HAL:hal-01586223v1 |
---|---|
record_format |
openpolar |
spelling |
ftccsdartic:oai:HAL:hal-01586223v1 2023-05-15T14:59:12+02:00 Emptiness formation probability, Toeplitz determinants, and conformal field theory Stéphan, Jean-Marie Department of Physics, University of Virginia University of Virginia Charlottesville 2014-05-01 https://hal.archives-ouvertes.fr/hal-01586223 https://doi.org/10.1088/1742-5468/2014/05/P05010 en eng HAL CCSD IOP Publishing info:eu-repo/semantics/altIdentifier/arxiv/1303.5499 info:eu-repo/semantics/altIdentifier/doi/10.1088/1742-5468/2014/05/P05010 hal-01586223 https://hal.archives-ouvertes.fr/hal-01586223 ARXIV: 1303.5499 doi:10.1088/1742-5468/2014/05/P05010 ISSN: 1742-5468 Journal of Statistical Mechanics: Theory and Experiment https://hal.archives-ouvertes.fr/hal-01586223 Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2014, 2014 (5), pp.P05010. ⟨10.1088/1742-5468/2014/05/P05010⟩ [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] info:eu-repo/semantics/article Journal articles 2014 ftccsdartic https://doi.org/10.1088/1742-5468/2014/05/P05010 2020-12-25T09:14:43Z International audience We revisit the study of the emptiness formation probability, the probability of forming a sequence of $\ell$ spins with the same ferromagnetic orientation in the ground-state of a quantum spin chain. We focus on two different examples, exhibiting strikingly different behavior: the XXZ and Ising chains. One has a conserved number of particles, the other does not. In the latter we show that the sequence of fixed spins can be viewed as an additional boundary in imaginary time. We then use conformal field theory (CFT) techniques to derive all universal terms in its scaling, and provide checks in free fermionic systems. These are based on numerical simulations or, when possible, mathematical results on the asymptotic behavior of Toeplitz and Toeplitz+Hankel determinants. A perturbed CFT analysis uncovers an interesting $\ell^{-1}\log \ell$ correction, that also appears in the closely related spin full counting statistics. The XXZ case turns out to be more challenging, as scale invariance is broken. We use a simple qualitative picture in which the ferromagnetic sequence of spins freezes all degrees of freedom inside of a certain "arctic" region, that we determine numerically. We also provide numerical evidence for the existence of universal logarithmic terms, generated by the massless field theory living outside of the arctic region. Article in Journal/Newspaper Arctic Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Arctic Journal of Statistical Mechanics: Theory and Experiment 2014 5 P05010 |
institution |
Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] |
spellingShingle |
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Stéphan, Jean-Marie Emptiness formation probability, Toeplitz determinants, and conformal field theory |
topic_facet |
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] |
description |
International audience We revisit the study of the emptiness formation probability, the probability of forming a sequence of $\ell$ spins with the same ferromagnetic orientation in the ground-state of a quantum spin chain. We focus on two different examples, exhibiting strikingly different behavior: the XXZ and Ising chains. One has a conserved number of particles, the other does not. In the latter we show that the sequence of fixed spins can be viewed as an additional boundary in imaginary time. We then use conformal field theory (CFT) techniques to derive all universal terms in its scaling, and provide checks in free fermionic systems. These are based on numerical simulations or, when possible, mathematical results on the asymptotic behavior of Toeplitz and Toeplitz+Hankel determinants. A perturbed CFT analysis uncovers an interesting $\ell^{-1}\log \ell$ correction, that also appears in the closely related spin full counting statistics. The XXZ case turns out to be more challenging, as scale invariance is broken. We use a simple qualitative picture in which the ferromagnetic sequence of spins freezes all degrees of freedom inside of a certain "arctic" region, that we determine numerically. We also provide numerical evidence for the existence of universal logarithmic terms, generated by the massless field theory living outside of the arctic region. |
author2 |
Department of Physics, University of Virginia University of Virginia Charlottesville |
format |
Article in Journal/Newspaper |
author |
Stéphan, Jean-Marie |
author_facet |
Stéphan, Jean-Marie |
author_sort |
Stéphan, Jean-Marie |
title |
Emptiness formation probability, Toeplitz determinants, and conformal field theory |
title_short |
Emptiness formation probability, Toeplitz determinants, and conformal field theory |
title_full |
Emptiness formation probability, Toeplitz determinants, and conformal field theory |
title_fullStr |
Emptiness formation probability, Toeplitz determinants, and conformal field theory |
title_full_unstemmed |
Emptiness formation probability, Toeplitz determinants, and conformal field theory |
title_sort |
emptiness formation probability, toeplitz determinants, and conformal field theory |
publisher |
HAL CCSD |
publishDate |
2014 |
url |
https://hal.archives-ouvertes.fr/hal-01586223 https://doi.org/10.1088/1742-5468/2014/05/P05010 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_source |
ISSN: 1742-5468 Journal of Statistical Mechanics: Theory and Experiment https://hal.archives-ouvertes.fr/hal-01586223 Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2014, 2014 (5), pp.P05010. ⟨10.1088/1742-5468/2014/05/P05010⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/arxiv/1303.5499 info:eu-repo/semantics/altIdentifier/doi/10.1088/1742-5468/2014/05/P05010 hal-01586223 https://hal.archives-ouvertes.fr/hal-01586223 ARXIV: 1303.5499 doi:10.1088/1742-5468/2014/05/P05010 |
op_doi |
https://doi.org/10.1088/1742-5468/2014/05/P05010 |
container_title |
Journal of Statistical Mechanics: Theory and Experiment |
container_volume |
2014 |
container_issue |
5 |
container_start_page |
P05010 |
_version_ |
1766331321902170112 |