Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Account
Menu
Find a journal Publish with us Track your research
Search
Cart
  1. Home
  2. Journal of High Energy Physics
  3. Article

Entanglement entropy and phase space density: lowest Landau levels and 1/2 BPS states

  • Regular Article - Theoretical Physics
  • Open access
  • Published: 08 June 2022
  • Volume 2022, article number 46, (2022)
  • Cite this article
Download PDF

You have full access to this open access article

Journal of High Energy Physics Aims and scope Submit manuscript
Entanglement entropy and phase space density: lowest Landau levels and 1/2 BPS states
Download PDF
  • Sumit R. Das1,
  • Shaun Hampton2 &
  • Sinong Liu  ORCID: orcid.org/0000-0002-8338-81993 

  • 236 Accesses

  • 4 Citations

  • 1 Altmetric

  • Explore all metrics

An Erratum to this article was published on 17 August 2023

This article has been updated

A preprint version of the article is available at arXiv.

Abstract

We consider the entanglement entropy of an arbitrary subregion in a system of N non-relativistic fermions in 2+1 dimensions in Lowest Landau Level (LLL) states. Using the connection of these states to those of an auxiliary 1 + 1 dimensional fermionic system, we derive an expression for the leading large-N contribution in terms of the expectation value of the phase space density operator in 1 + 1 dimensions. For appropriate subregions the latter can replaced by its semiclassical Thomas-Fermi value, yielding expressions in terms of explicit integrals which can be evaluated analytically. We show that the leading term in the entanglement entropy is a perimeter law with a shape independent coefficient. Furthermore, we obtain analytic expressions for additional contributions from sharp corners on the entangling curve. Both the perimeter and the corner pieces are in good agreement with existing calculations for special subregions. Our results are relevant to the integer quantum Hall effect problem, and to the half-BPS sector of \( \mathcal{N} \) = 4 Yang Mills theory on S3. In this latter context, the entanglement we consider is an entanglement in target space. We comment on possible implications to gauge-gravity duality.

Article PDF

Download to read the full article text

Similar content being viewed by others

Entanglement content of quantum particle excitations. Part I. Free field theory

Article Open access 05 October 2018

Entanglement Entropy of Ground States of the Three-Dimensional Ideal Fermi Gas in a Magnetic Field

Article Open access 07 November 2023

Entanglement and edge effects in superpositions of many-body Fock states with spatial constraints

Article 11 May 2020
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Change history

  • 17 August 2023

    An Erratum to this paper has been published: https://doi.org/10.1007/JHEP08(2023)095

References

  1. L. Spruch, Pedagogic notes on Thomas-Fermi theory (and on some improvements): atoms, stars, and the stability of bulk matter, Rev. Mod. Phys. 63 (1991) 151 [INSPIRE].

  2. D.S. Dean, P. Le Doussal, S.N. Majumdar and G. Schehr, Noninteracting fermions at finite temperature in a d-dimensional trap: universal correlations, Phys. Rev. A 94 (2016) 063622 [arXiv:1609.04366] [INSPIRE].

  3. E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. J. Polchinski, Classical limit of (1 + 1)-dimensional string theory, Nucl. Phys. B 362 (1991) 125 [INSPIRE].

  5. A. Jevicki and B. Sakita, The quantum collective field method and its application to the planar limit, Nucl. Phys. B 165 (1980) 511 [INSPIRE].

  6. S.R. Das, A. Dhar, G. Mandal and S.R. Wadia, Bosonization of nonrelativistic fermions and W-infinity algebra, Mod. Phys. Lett. A 7 (1992) 71 [hep-th/9111021] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. A. Dhar, G. Mandal and S.R. Wadia, Classical Fermi fluid and geometric action for c = 1, Int. J. Mod. Phys. A 8 (1993) 325 [hep-th/9204028] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  8. A. Dhar, G. Mandal and S.R. Wadia, Nonrelativistic fermions, coadjoint orbits of W∞ and string field theory at c = 1, Mod. Phys. Lett. A 7 (1992) 3129 [hep-th/9207011] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. A. Dhar, G. Mandal and S.R. Wadia, W∞ coherent states and path integral derivation of bosonization of nonrelativistic fermions in one-dimension, Mod. Phys. Lett. A 8 (1993) 3557 [hep-th/9309028] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. I. Klich and L. Levitov, Quantum noise as an entanglement meter, Phys. Rev. Lett. 102 (2009) 100502 [arXiv:0804.1377] [INSPIRE].

  11. H.F. Song, S. Rachel, C. Flindt, I. Klich, N. Laflorencie and K. Le Hur, Bipartite fluctuations as a probe of many-body entanglement, Phys. Rev. B 85 (2012) 035409 [arXiv:1109.1001] [INSPIRE].

  12. I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36 (2003) L205.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. A. Petrescu et al., Fluctuations and entanglement spectrum in quantum hall states, J. Stat. Mech. 2014 (2014) P10005 [arXiv:1405.7816].

  14. P. Calabrese, M. Mintchev and E. Vicari, Exact relations between particle fluctuations and entanglement in Fermi gases, EPL 98 (2012) 20003 [arXiv:1111.4836] [INSPIRE].

    Article  ADS  Google Scholar 

  15. P. Calabrese, P.L. Doussal and S.N. Majumdar, Random matrices and entanglement entropy of trapped Fermi gases, Phys. Rev. A 91 (2015) 012303.

  16. N.R. Smith, P. Le Doussal, S.N. Majumdar and G. Schehr, Counting statistics for noninteracting fermions in a d-dimensional potential, Phys. Rev. E 103 (2021) L030105 [arXiv:2008.01045] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  17. F.D. Cunden, S.N. Majumdar and N. O’Connell, Free fermions and α-determinantal processes, J. Phys. A 52 (2019) 165202 [arXiv:1811.11556].

  18. S.R. Das, S. Hampton and S. Liu, Quantum quench in non-relativistic fermionic field theory: harmonic traps and 2d string theory, JHEP 08 (2019) 176 [arXiv:1903.07682] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. S.R. Das, S. Hampton and S. Liu, Quantum quench in c = 1 matrix model and emergent space-times, JHEP 04 (2020) 107 [arXiv:1910.00123] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  20. D. Tong, Lectures on the quantum Hall effect, arXiv:1606.06687 [INSPIRE].

  21. S. Corley, A. Jevicki and S. Ramgoolam, Exact correlators of giant gravitons from dual N = 4 SYM theory, Adv. Theor. Math. Phys. 5 (2002) 809 [hep-th/0111222] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Berenstein, A toy model for the AdS/CFT correspondence, JHEP 07 (2004) 018 [hep-th/0403110] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  23. Y. Takayama and A. Tsuchiya, Complex matrix model and fermion phase space for bubbling AdS geometries, JHEP 10 (2005) 004 [hep-th/0507070] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  24. A. Ghodsi, A.E. Mosaffa, O. Saremi and M.M. Sheikh-Jabbari, LLL vs. LLM: half BPS sector of N = 4 SYM equals to quantum Hall system, Nucl. Phys. B 729 (2005) 467 [hep-th/0505129] [INSPIRE].

  25. A. Jevicki, Nonperturbative collective field theory, Nucl. Phys. B 376 (1992) 75 [INSPIRE].

  26. A. Donos, A. Jevicki and J.P. Rodrigues, Matrix model maps in AdS/CFT, Phys. Rev. D 72 (2005) 125009 [hep-th/0507124] [INSPIRE].

  27. A.V. Ryzhov, Quarter BPS operators in N = 4 SYM, JHEP 11 (2001) 046 [hep-th/0109064] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  28. G. Mandal and N.V. Suryanarayana, Counting 1/8-BPS dual-giants, JHEP 03 (2007) 031 [hep-th/0606088] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  29. S.R. Das, A. Kaushal, G. Mandal and S.P. Trivedi, Bulk entanglement entropy and matrices, J. Phys. A 53 (2020) 444002 [arXiv:2004.00613] [INSPIRE].

  30. S.R. Das, A. Kaushal, S. Liu, G. Mandal and S.P. Trivedi, Gauge invariant target space entanglement in D-brane holography, JHEP 04 (2021) 225 [arXiv:2011.13857] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. E.A. Mazenc and D. Ranard, Target space entanglement entropy, arXiv:1910.07449 [INSPIRE].

  32. H.R. Hampapura, J. Harper and A. Lawrence, Target space entanglement in matrix models, JHEP 10 (2021) 231 [arXiv:2012.15683] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. S. Sugishita, Target space entanglement in quantum mechanics of fermions and matrices, JHEP 08 (2021) 046 [arXiv:2105.13726] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. A. Frenkel and S.A. Hartnoll, Entanglement in the quantum Hall matrix model, JHEP 05 (2022) 130 [arXiv:2111.05967] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [arXiv:1005.3035] [INSPIRE].

  36. N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].

    Article  ADS  Google Scholar 

  37. T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. S.R. Das, Geometric entropy of nonrelativistic fermions and two-dimensional strings, Phys. Rev. D 51 (1995) 6901 [hep-th/9501090] [INSPIRE].

    Article  ADS  Google Scholar 

  39. S.R. Das, Degrees of freedom in two-dimensional string theory, Nucl. Phys. B Proc. Suppl. 45BC (1996) 224 [hep-th/9511214] [INSPIRE].

  40. S.A. Hartnoll and E. Mazenc, Entanglement entropy in two dimensional string theory, Phys. Rev. Lett. 115 (2015) 121602 [arXiv:1504.07985] [INSPIRE].

  41. J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. M.T. Grisaru, R.C. Myers and O. Tafjord, SUSY and goliath, JHEP 08 (2000) 040 [hep-th/0008015] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. A. Hashimoto, S. Hirano and N. Itzhaki, Large branes in AdS and their field theory dual, JHEP 08 (2000) 051 [hep-th/0008016] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  45. G. Mandal, Fermions from half-BPS supergravity, JHEP 08 (2005) 052 [hep-th/0502104] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  46. L. Grant, L. Maoz, J. Marsano, K. Papadodimas and V.S. Rychkov, Minisuperspace quantization of ‘bubbling AdS’ and free fermion droplets, JHEP 08 (2005) 025 [hep-th/0505079] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  47. L. Maoz and V.S. Rychkov, Geometry quantization from supergravity: the case of ‘bubbling AdS’, JHEP 08 (2005) 096 [hep-th/0508059] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  48. N.V. Suryanarayana, Half-BPS giants, free fermions and microstates of superstars, JHEP 01 (2006) 082 [hep-th/0411145] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  49. V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, The library of Babel: on the origin of gravitational thermodynamics, JHEP 12 (2005) 006 [hep-th/0508023] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  50. K. Skenderis and M. Taylor, Anatomy of bubbling solutions, JHEP 09 (2007) 019 [arXiv:0706.0216] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  51. L. Charles and B. Estienne, Entanglement entropy and Berezin-Toeplitz operators, Commun. Math. Phys. 376 (2019) 521.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. I.D. Rodríguez and G. Sierra, Entanglement entropy of integer quantum hall states, Phys. Rev. B 80 (2009) 153303 [arXiv:0811.2188].

  53. I.D. Rodriguez and G. Sierra, Entanglement entropy of integer quantum Hall states in polygonal domains, J. Stat. Mech. 1012 (2010) P12033 [arXiv:1007.5356] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  54. B. Sirois, L.M. Fournier, J. Leduc and W. Witczak-Krempa, Geometric entanglement in integer quantum Hall states, Phys. Rev. B 103 (2021) 115115 [arXiv:2009.02337] [INSPIRE].

  55. P. Bueno, R.C. Myers and W. Witczak-Krempa, Universality of corner entanglement in conformal field theories, Phys. Rev. Lett. 115 (2015) 021602 [arXiv:1505.04804] [INSPIRE].

  56. P. Bueno and R.C. Myers, Corner contributions to holographic entanglement entropy, JHEP 08 (2015) 068 [arXiv:1505.07842] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. T. Faulkner, R.G. Leigh and O. Parrikar, Shape dependence of entanglement entropy in conformal field theories, JHEP 04 (2016) 088 [arXiv:1511.05179] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  58. M. Alishahiha, A.F. Astaneh, P. Fonda and F. Omidi, Entanglement entropy for singular surfaces in hyperscaling violating theories, JHEP 09 (2015) 172 [arXiv:1507.05897] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. B. Estienne, J.-M. Stéphan and W. Witczak-Krempa, Cornering the universal shape of fluctuations, Nature Commun. 13 (2022) 287 [arXiv:2102.06223] [INSPIRE].

    Article  ADS  Google Scholar 

  60. S. Cremonini, R. de Mello Koch and A. Jevicki, Matrix model maps and reconstruction of AdS SUGRA interactions, Phys. Rev. D 77 (2008) 105005 [arXiv:0712.4366] [INSPIRE].

  61. M. Hanada, Bulk geometry in gauge/gravity duality and color degrees of freedom, Phys. Rev. D 103 (2021) 106007 [arXiv:2102.08982] [INSPIRE].

  62. M. Hanada, Large-N limit as a second quantization, arXiv:2103.15873 [INSPIRE].

  63. V.P. Nair, Entanglement for quantum Hall states and a generalized Chern-Simons form, Phys. Rev. D 101 (2020) 125021 [arXiv:2001.04957] [INSPIRE].

  64. D. Karabali, Entanglement entropy for integer quantum Hall effect in two and higher dimensions, Phys. Rev. D 102 (2020) 025016 [arXiv:2006.10920] [INSPIRE].

  65. S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids, JHEP 05 (2008) 016 [arXiv:0802.3231] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  66. J.R. Fliss et al., Interface contributions to topological entanglement in Abelian Chern-Simons theory, JHEP 09 (2017) 056 [arXiv:1705.09611] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  67. A. Jevicki, O. Karim, J.P. Rodrigues and H. Levine, Loop space Hamiltonians and numerical methods for large N gauge theories, Nucl. Phys. B 213 (1983) 169 [INSPIRE].

  68. R.d.M. Koch, A. Jevicki, X. Liu, K. Mathaba and J.P. Rodrigues, Large N optimization for multi-matrix systems, JHEP 01 (2022) 168 [arXiv:2108.08803] [INSPIRE].

  69. P. Caputa, R. de Mello Koch and K. Zoubos, Extremal versus non-extremal correlators with giant gravitons, JHEP 08 (2012) 143 [arXiv:1204.4172] [INSPIRE].

  70. C.R. Graham and A. Karch, Minimal area submanifolds in AdS × compact, JHEP 04 (2014) 168 [arXiv:1401.7692] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  71. A. Mollabashi, N. Shiba and T. Takayanagi, Entanglement between two interacting CFTs and generalized holographic entanglement entropy, JHEP 04 (2014) 185 [arXiv:1403.1393] [INSPIRE].

    Article  ADS  Google Scholar 

  72. A. Karch and C.F. Uhlemann, Holographic entanglement entropy and the internal space, Phys. Rev. D 91 (2015) 086005 [arXiv:1501.00003] [INSPIRE].

  73. M.R. Mohammadi Mozaffar and A. Mollabashi, On the entanglement between interacting scalar field theories, JHEP 03 (2016) 015 [arXiv:1509.03829] [INSPIRE].

  74. V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  75. V. Balasubramanian, B. Craps, T. De Jonckheere and G. Sárosi, Entanglement versus entwinement in symmetric product orbifolds, JHEP 01 (2019) 190 [arXiv:1806.02871] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. J. Erdmenger and M. Gerbershagen, Entwinement as a possible alternative to complexity, JHEP 03 (2020) 082 [arXiv:1910.05352] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. F. Alet, M. Hanada, A. Jevicki and C. Peng, Entanglement and confinement in coupled quantum systems, JHEP 02 (2021) 034 [arXiv:2001.03158] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  78. V. Balasubramanian, A. Lawrence, A. Rolph and S. Ross, Entanglement shadows in LLM geometries, JHEP 11 (2017) 159 [arXiv:1704.03448] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. H. Lin and Y. Zhu, Entanglement and mixed states of Young tableau states in gauge/gravity correspondence, Nucl. Phys. B 972 (2021) 115572 [arXiv:2107.14219] [INSPIRE].

  80. A. Tsuchiya and K. Yamashiro, Target space entanglement in a matrix model for the bubbling geometry, JHEP 04 (2022) 086 [arXiv:2201.06871] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. B. Lacroix-A-Chez-Toine, S.N. Majumdar and G. Schehr, Rotating trapped fermions in two dimensions and the complex Ginibre ensemble: exact results for the entanglement entropy and number variance, Phys. Rev. A 99 (2019) 021602 [arXiv:1809.05835] [INSPIRE].

  82. M. Kulkarni, S.N. Majumdar and G. Schehr, Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap, Phys. Rev. A 103 (2021) 033321.

  83. N.R. Smith, P. Le Doussal, S.N. Majumdar and G. Schehr, Counting statistics for non-interacting fermions in a rotating trap, arXiv:2112.13355.

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, University of Kentucky, Lexington, KY, 40506, USA

    Sumit R. Das

  2. Institut de Physique Theorique, Universite Paris Saclay, CEA, CNRS, Orme des Merisiers, Gif sur Yvette, 91191, CEDEX, France

    Shaun Hampton

  3. Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warsaw, Poland

    Sinong Liu

Authors
  1. Sumit R. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaun Hampton
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sinong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sinong Liu.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2201.08330

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Das, S.R., Hampton, S. & Liu, S. Entanglement entropy and phase space density: lowest Landau levels and 1/2 BPS states. J. High Energ. Phys. 2022, 46 (2022). https://doi.org/10.1007/JHEP06(2022)046

Download citation

  • Received: 28 February 2022

  • Accepted: 09 May 2022

  • Published: 08 June 2022

  • DOI: https://doi.org/10.1007/JHEP06(2022)046

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Field Theories in Lower Dimensions
  • Gauge-Gravity Correspondence
  • Matrix Models
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us
  • Track your research

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Journal finder
  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support
  • Cancel contracts here

Not affiliated

Springer Nature

© 2024 Springer Nature