Entanglement Spectroscopy and probing the Li-Haldane Conjecture in Topological Quantum Matter

Torsten V. Zache; Christian Kokail; Bhuvanesh Sundar; Peter Zoller

doi:10.22331/q-2022-04-27-702

Entanglement Spectroscopy and probing the Li-Haldane Conjecture in Topological Quantum Matter

Torsten V. Zache^1,2, Christian Kokail^1,2, Bhuvanesh Sundar^1,3, and Peter Zoller^1,2

¹Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck 6020, Austria
²Center for Quantum Physics, University of Innsbruck, Innsbruck 6020, Austria
³JILA, Department of Physics, University of Colorado, Boulder CO 80309, USA

Published:	2022-04-27, volume 6, page 702
Eprint:	arXiv:2110.03913v2
Doi:	https://doi.org/10.22331/q-2022-04-27-702
Citation:	Quantum 6, 702 (2022).

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Abstract

Topological phases are characterized by their entanglement properties, which is manifest in a direct relation between entanglement spectra and edge states discovered by Li and Haldane. We propose to leverage the power of synthetic quantum systems for measuring entanglement via the Entanglement Hamiltonian to probe this relationship experimentally. This is made possible by exploiting the quasi-local structure of Entanglement Hamiltonians. The feasibility of this proposal is illustrated for two paradigmatic examples realizable with current technology, an integer quantum Hall state of non-interacting fermions on a 2D lattice and a symmetry protected topological state of interacting fermions on a 1D chain. Our results pave the road towards an experimental identification of topological order in strongly correlated quantum many-body systems.

Popular summary

Topological Quantum Matter — in contrast to ordinary phases of matter — can not be detected by probing local observables, such as the magnetisation of a magnet. Instead, topological phases are characterised by their quantum correlations, as well as excitations supported at the boundary of the system. The spectrum of these edge excitations is directly related to the structure of the entanglement in the bulk, a relation known as the Li-Haldane conjecture. In this article, we propose to leverage the power of quantum simulators to probe the Li-Haldane conjecture experimentally. Our approach is based on the quasi-locality of the Entanglement Hamiltonian, which enables the application of recently developed protocols to measure the entanglement spectrum. We demonstrate the feasibility of our proposal with numerical simulations of examples in one and two spatial dimensions.

► BibTeX data

@article{Zache2022entanglement,
  doi = {10.22331/q-2022-04-27-702},
  url = {https://doi.org/10.22331/q-2022-04-27-702},
  title = {Entanglement {S}pectroscopy and probing the {L}i-{H}aldane {C}onjecture in {T}opological {Q}uantum {M}atter},
  author = {Zache, Torsten V. and Kokail, Christian and Sundar, Bhuvanesh and Zoller, Peter},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {702},
  month = apr,
  year = {2022}
}

► References

[1] N. R. Cooper, J. Dalibard, and I. B. Spielman. ``Topological bands for ultracold atoms''. Rev. Mod. Phys. 91, 015005 (2019).
https://doi.org/10.1103/RevModPhys.91.015005

[2] Monika Aidelsburger, Sylvain Nascimbene, and Nathan Goldman. ``Artificial gauge fields in materials and engineered systems''. Comptes Rendus Physique 19, 394–432 (2018).
https://doi.org/10.1016/j.crhy.2018.03.002

[3] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani. ``Observation of chiral edge states with neutral fermions in synthetic hall ribbons''. Science 349, 1510–1513 (2015).
https://doi.org/10.1126/science.aaa8736

[4] Sylvain de Léséleuc, Vincent Lienhard, Pascal Scholl, Daniel Barredo, Sebastian Weber, Nicolai Lang, Hans Peter Büchler, Thierry Lahaye, and Antoine Browaeys. ``Observation of a symmetry-protected topological phase of interacting bosons with rydberg atoms''. Science 365, 775–780 (2019).
https://doi.org/10.1126/science.aav9105

[5] K. J. Satzinger, Y.-J Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth, C. Gidney, I. Aleiner, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, J. Basso, A. Bengtsson, A. Bilmes, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. R. Derk, D. Eppens, C. Erickson, L. Faoro, E. Farhi, A. G. Fowler, B. Foxen, M. Giustina, A. Greene, J. A. Gross, M. P. Harrigan, S. D. Harrington, J. Hilton, S. Hong, T. Huang, W. J. Huggins, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, T. Khattar, S. Kim, P. V. Klimov, A. N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, A. Locharla, E. Lucero, O. Martin, J. R. McClean, M. McEwen, K. C. Miao, M. Mohseni, S. Montazeri, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, T. E. O’Brien, A. Opremcak, B. Pató, A. Petukhov, N. C. Rubin, D. Sank, V. Shvarts, D. Strain, M. Szalay, B. Villalonga, T. C. White, Z. Yao, P. Yeh, J. Yoo, A. Zalcman, H. Neven, S. Boixo, A. Megrant, Y. Chen, J. Kelly, V. Smelyanskiy, A. Kitaev, M. Knap, F. Pollmann, and P. Roushan. ``Realizing topologically ordered states on a quantum processor''. Science 374, 1237–1241 (2021).
https://doi.org/10.1126/science.abi8378

[6] G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin. ``Probing topological spin liquids on a programmable quantum simulator''. Science 374, 1242–1247 (2021).
https://doi.org/10.1126/science.abi8794

[7] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbène, N. R. Cooper, I. Bloch, and N. Goldman. ``Measuring the chern number of hofstadter bands with ultracold bosonic atoms''. Nature Physics 11, 162–166 (2015).
https://doi.org/10.1038/nphys3171

[8] N. Fläschner, B. S. Rem, M. Tarnowski, D. Vogel, D.-S. Lühmann, K. Sengstock, and C. Weitenberg. ``Experimental reconstruction of the berry curvature in a floquet bloch band''. Science 352, 1091–1094 (2016).
https://doi.org/10.1126/science.aad4568

[9] Pimonpan Sompet, Sarah Hirthe, Dominik Bourgund, Thomas Chalopin, Julian Bibo, Joannis Koepsell, Petar Bojović, Ruben Verresen, Frank Pollmann, Guillaume Salomon, et al. ``Realising the symmetry-protected haldane phase in fermi-hubbard ladders'' (2021). url: doi.org/10.48550/arXiv.2103.10421.
https://doi.org/10.48550/arXiv.2103.10421

[10] Pascal Scholl, Michael Schuler, Hannah J. Williams, Alexander A. Eberharter, Daniel Barredo, Kai-Niklas Schymik, Vincent Lienhard, Louis-Paul Henry, Thomas C. Lang, Thierry Lahaye, Andreas M. Läuchli, and Antoine Browaeys. ``Quantum simulation of 2d antiferromagnets with hundreds of rydberg atoms''. Nature 595, 233–238 (2021).
https://doi.org/10.1038/s41586-021-03585-1

[11] Sepehr Ebadi, Tout T. Wang, Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Dolev Bluvstein, Rhine Samajdar, Hannes Pichler, Wen Wei Ho, Soonwon Choi, Subir Sachdev, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``Quantum phases of matter on a 256-atom programmable quantum simulator''. Nature 595, 227–232 (2021).
https://doi.org/10.1038/s41586-021-03582-4

[12] Alexei Kitaev and John Preskill. ``Topological entanglement entropy''. Phys. Rev. Lett. 96, 110404 (2006).
https://doi.org/10.1103/PhysRevLett.96.110404

[13] Michael Levin and Xiao-Gang Wen. ``Detecting topological order in a ground state wave function''. Phys. Rev. Lett. 96, 110405 (2006).
https://doi.org/10.1103/PhysRevLett.96.110405

[14] Hui Li and F. D. M. Haldane. ``Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states''. Phys. Rev. Lett. 101, 010504 (2008).
https://doi.org/10.1103/PhysRevLett.101.010504

[15] Christian Kokail, Rick van Bijnen, Andreas Elben, Benoı̂t Vermersch, and Peter Zoller. ``Entanglement hamiltonian tomography in quantum simulation''. Nature Physics 17, 936–942 (2021).
https://doi.org/10.1038/s41567-021-01260-w

[16] Christian Kokail, Bhuvanesh Sundar, Torsten V. Zache, Andreas Elben, Benoı̂t Vermersch, Marcello Dalmonte, Rick van Bijnen, and Peter Zoller. ``Quantum variational learning of the entanglement hamiltonian''. Phys. Rev. Lett. 127, 170501 (2021).
https://doi.org/10.1103/PhysRevLett.127.170501

[17] Joseph J. Bisognano and Eyvind H. Wichmann. ``On the duality condition for a hermitian scalar field''. Journal of Mathematical Physics 16, 985–1007 (1975).
https://doi.org/10.1063/1.522605

[18] Joseph J. Bisognano and Eyvind H. Wichmann. ``On the duality condition for quantum fields''. Journal of Mathematical Physics 17, 303–321 (1976).
https://doi.org/10.1063/1.522898

[19] N. Regnault. ``Entanglement spectroscopy and its application to the quantum hall effects'' (2015). url: doi.org/10.48550/arXiv.1510.07670.
https://doi.org/10.48550/arXiv.1510.07670

[20] Anushya Chandran, M. Hermanns, N. Regnault, and B. Andrei Bernevig. ``Bulk-edge correspondence in entanglement spectra''. Phys. Rev. B 84, 205136 (2011).
https://doi.org/10.1103/PhysRevB.84.205136

[21] Xiao-Liang Qi, Hosho Katsura, and Andreas W. W. Ludwig. ``General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states''. Phys. Rev. Lett. 108, 196402 (2012).
https://doi.org/10.1103/PhysRevLett.108.196402

[22] Horacio Casini, Marina Huerta, and Robert C. Myers. ``Towards a derivation of holographic entanglement entropy''. Journal of High Energy Physics 2011, 36 (2011).
https://doi.org/10.1007/JHEP05(2011)036

[23] John Cardy and Erik Tonni. ``Entanglement hamiltonians in two-dimensional conformal field theory''. Journal of Statistical Mechanics: Theory and Experiment 2016, 123103 (2016).
https://doi.org/10.1088/1742-5468/2016/12/123103

[24] Brian Swingle and T. Senthil. ``Geometric proof of the equality between entanglement and edge spectra''. Phys. Rev. B 86, 045117 (2012).
https://doi.org/10.1103/PhysRevB.86.045117

[25] M. Dalmonte, B. Vermersch, and P. Zoller. ``Quantum simulation and spectroscopy of entanglement hamiltonians''. Nature Physics 14, 827–831 (2018).
https://doi.org/10.1038/s41567-018-0151-7

[26] Francesco Parisen Toldin and Fakher F. Assaad. ``Entanglement hamiltonian of interacting fermionic models''. Phys. Rev. Lett. 121, 200602 (2018).
https://doi.org/10.1103/PhysRevLett.121.200602

[27] G. Giudici, T. Mendes-Santos, P. Calabrese, and M. Dalmonte. ``Entanglement hamiltonians of lattice models via the bisognano-wichmann theorem''. Phys. Rev. B 98, 134403 (2018).
https://doi.org/10.1103/PhysRevB.98.134403

[28] Jiaju Zhang, Pasquale Calabrese, Marcello Dalmonte, and M. A. Rajabpour. ``Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains''. SciPost Phys. Core 2, 7 (2020).
https://doi.org/10.21468/SciPostPhysCore.2.2.007

[29] Viktor Eisler, Giuseppe Di Giulio, Erik Tonni, and Ingo Peschel. ``Entanglement hamiltonians for non-critical quantum chains''. Journal of Statistical Mechanics: Theory and Experiment 2020, 103102 (2020).
https://doi.org/10.1088/1742-5468/abb4da

[30] W. Zhu, Zhoushen Huang, and Yin-Chen He. ``Reconstructing entanglement hamiltonian via entanglement eigenstates''. Phys. Rev. B 99, 235109 (2019).
https://doi.org/10.1103/PhysRevB.99.235109

[31] Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar. ``Sample-efficient learning of interacting quantum systems''. Nature Physics 17, 931–935 (2021).
https://doi.org/10.1038/s41567-021-01232-0

[32] Aaron Szasz, Johannes Motruk, Michael P. Zaletel, and Joel E. Moore. ``Chiral spin liquid phase of the triangular lattice hubbard model: A density matrix renormalization group study''. Phys. Rev. X 10, 021042 (2020).
https://doi.org/10.1103/PhysRevX.10.021042

[33] Ruben Verresen, Mikhail D. Lukin, and Ashvin Vishwanath. ``Prediction of toric code topological order from rydberg blockade''. Phys. Rev. X 11, 031005 (2021).
https://doi.org/10.1103/PhysRevX.11.031005

[34] N. Goldman, J. C. Budich, and P. Zoller. ``Topological quantum matter with ultracold gases in optical lattices''. Nature Physics 12, 639–645 (2016).
https://doi.org/10.1038/nphys3803

[35] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein. ``Synthetic gauge fields in synthetic dimensions''. Phys. Rev. Lett. 112, 043001 (2014).
https://doi.org/10.1103/PhysRevLett.112.043001

[36] D Jaksch and P Zoller. ``Creation of effective magnetic fields in optical lattices: the hofstadter butterfly for cold neutral atoms''. New Journal of Physics 5, 56–56 (2003).
https://doi.org/10.1088/1367-2630/5/1/356

[37] David Tong. ``Lectures on the quantum hall effect'' (2016). url: doi.org/10.48550/arXiv.1606.06687.
https://doi.org/10.48550/arXiv.1606.06687

[38] K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete. ``Discriminating states: The quantum chernoff bound''. Phys. Rev. Lett. 98, 160501 (2007).
https://doi.org/10.1103/PhysRevLett.98.160501

[39] Yeong-Cherng Liang, Yu-Hao Yeh, Paulo E M F Mendonça, Run Yan Teh, Margaret D Reid, and Peter D Drummond. ``Quantum fidelity measures for mixed states''. Reports on Progress in Physics 82, 076001 (2019).
https://doi.org/10.1088/1361-6633/ab1ca4

[40] Anushya Chandran, Vedika Khemani, and S. L. Sondhi. ``How universal is the entanglement spectrum?''. Phys. Rev. Lett. 113, 060501 (2014).
https://doi.org/10.1103/PhysRevLett.113.060501

[41] Andreas Elben, Jinlong Yu, Guanyu Zhu, Mohammad Hafezi, Frank Pollmann, Peter Zoller, and Benoît Vermersch. ``Many-body topological invariants from randomized measurements in synthetic quantum matter''. Science Advances 6, eaaz3666 (2020).
https://doi.org/10.1126/sciadv.aaz3666

[42] Niklas Mueller, Torsten V Zache, and Robert Ott. ``Thermalization of gauge theories from their entanglement spectrum'' (2021). url: doi.org/10.48550/arXiv.2107.11416.
https://doi.org/10.48550/arXiv.2107.11416

[43] Horacio Casini, Marina Huerta, and José Alejandro Rosabal. ``Remarks on entanglement entropy for gauge fields''. Phys. Rev. D 89, 085012 (2014).
https://doi.org/10.1103/PhysRevD.89.085012

[44] Wilbur Shirley, Kevin Slagle, and Xie Chen. ``Universal entanglement signatures of foliated fracton phases''. SciPost Phys. 6, 15 (2019).
https://doi.org/10.21468/SciPostPhys.6.1.015

[45] Edward Witten. ``Aps medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory''. Rev. Mod. Phys. 90, 045003 (2018).
https://doi.org/10.1103/RevModPhys.90.045003

[46] Ingo Peschel, Matthias Kaulke, and Örs Legeza. ``Density-matrix spectra for integrable models''. Annalen der Physik 8, 153–164 (1999).
https://doi.org/10.1002/andp.19995110203

[47] Ingo Peschel and Ming-Chiang Chung. ``Density matrices for a chain of oscillators''. Journal of Physics A: Mathematical and General 32, 8419–8428 (1999).
https://doi.org/10.1088/0305-4470/32/48/305

[48] Michael Pretko and T. Senthil. ``Entanglement entropy of $u$(1) quantum spin liquids''. Phys. Rev. B 94, 125112 (2016).
https://doi.org/10.1103/PhysRevB.94.125112

[49] Gerald V. Dunne. ``Aspects of chern-simons theory'' (1999). url: doi.org/10.48550/arXiv.hep-th/9902115.
https://doi.org/10.48550/arXiv.hep-th/9902115

[50] Ingo Peschel. ``Calculation of reduced density matrices from correlation functions''. Journal of Physics A: Mathematical and General 36, L205–L208 (2003).
https://doi.org/10.1088/0305-4470/36/14/101

[51] Sara Murciano, Vittorio Vitale, Marcello Dalmonte, and Pasquale Calabrese. ``Negativity hamiltonian: An operator characterization of mixed-state entanglement''. Phys. Rev. Lett. 128, 140502 (2022).
https://doi.org/10.1103/PhysRevLett.128.140502

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