Emptiness formation probability, Toeplitz determinants, and conformal field theory
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 chai...
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ftdatacite:10.48550/arxiv.1303.5499 2023-05-15T14:59:12+02:00 Emptiness formation probability, Toeplitz determinants, and conformal field theory Stéphan, Jean-Marie 2013 https://dx.doi.org/10.48550/arxiv.1303.5499 https://arxiv.org/abs/1303.5499 unknown arXiv https://dx.doi.org/10.1088/1742-5468/2014/05/p05010 arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Statistical Mechanics cond-mat.stat-mech Strongly Correlated Electrons cond-mat.str-el Mathematical Physics math-ph FOS Physical sciences article-journal Article ScholarlyArticle Text 2013 ftdatacite https://doi.org/10.48550/arxiv.1303.5499 https://doi.org/10.1088/1742-5468/2014/05/p05010 2022-04-01T13:27:50Z 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. : 34 pages, 11 figures. v2: additional discussions, typos fixed. To appear in J. Stat. Mech Text Arctic DataCite Metadata Store (German National Library of Science and Technology) Arctic |
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topic |
Statistical Mechanics cond-mat.stat-mech Strongly Correlated Electrons cond-mat.str-el Mathematical Physics math-ph FOS Physical sciences |
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Statistical Mechanics cond-mat.stat-mech Strongly Correlated Electrons cond-mat.str-el Mathematical Physics math-ph FOS Physical sciences Stéphan, Jean-Marie Emptiness formation probability, Toeplitz determinants, and conformal field theory |
topic_facet |
Statistical Mechanics cond-mat.stat-mech Strongly Correlated Electrons cond-mat.str-el Mathematical Physics math-ph FOS Physical sciences |
description |
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. : 34 pages, 11 figures. v2: additional discussions, typos fixed. To appear in J. Stat. Mech |
format |
Text |
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 |
arXiv |
publishDate |
2013 |
url |
https://dx.doi.org/10.48550/arxiv.1303.5499 https://arxiv.org/abs/1303.5499 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_relation |
https://dx.doi.org/10.1088/1742-5468/2014/05/p05010 |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1303.5499 https://doi.org/10.1088/1742-5468/2014/05/p05010 |
_version_ |
1766331335081721856 |