- Letter
- Open Access
Topological order in random interacting Ising-Majorana chains stabilized by many-body localization
Phys. Rev. Research 4, L032016 – Published 29 July, 2022
DOI: https://doi.org/10.1103/PhysRevResearch.4.L032016
Abstract
We numerically explore -symmetric random interacting Ising-Majorana chains at high energy. A very rich phase diagram emerges with two topologically distinct many-body localization (MBL) regimes separated by a much broader thermal phase than previously found. This is a striking consequence of the avalanche theory. We further find MBL spin-glass order always associated to a many-body spectral pairing, presumably signaling a strong zero mode operator which opens fascinating perspectives for MBL-protected topological qubits.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (59)
- B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Quasiparticle Lifetime in a Finite System: A Nonperturbative Approach, Phys. Rev. Lett. 78, 2803 (1997).
- Ph. Jacquod and D. L. Shepelyansky, Emergence of Quantum Chaos in Finite Interacting Fermi Systems, Phys. Rev. Lett. 79, 1837 (1997).
- I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interacting Electrons in Disordered Wires: Anderson Localization and Low-T Transport, Phys. Rev. Lett. 95, 206603 (2005).
- D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Metalinsulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys. 321, 1126 (2006).
- M. Žnidarič, T. Prosen, and P. Prelovšek, Many-body localization in the Heisenberg XXZ magnet in a random field, Phys. Rev. B 77, 064426 (2008).
- A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
- J. H. Bardarson, F. Pollmann, and J. E. Moore, Unbounded Growth of Entanglement in Models of Many-Body Localization, Phys. Rev. Lett. 109, 017202 (2012).
- D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103(R) (2015).
- F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Phys. 19, 498 (2018).
- D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
- E. V. H. Doggen, F. Schindler, K. S. Tikhonov, A. D. Mirlin, T. Neupert, D. G. Polyakov, and I. V. Gornyi, Many-body localization and delocalization in large quantum chains, Phys. Rev. B 98, 174202 (2018).
- T. Chanda, P. Sierant, and J. Zakrzewski, Time dynamics with matrix product states: Many-body localization transition of large systems revisited, Phys. Rev. B 101, 035148 (2020).
- P. Sierant, D. Delande, and J. Zakrzewski, Thouless Time Analysis of Anderson and Many-Body Localization Transitions, Phys. Rev. Lett. 124, 186601 (2020).
- D. A. Abanin, J. H. Bardarson, G. De Tomasi, S. Gopalakrishnan, V. Khemani, S. A. Parameswaran, F. Pollmann, A. C. Potter, M. Serbyn, and R. Vasseur, Distinguishing localization from chaos: Challenges in finite-size systems, Ann. Phys. 427, 168415 (2021).
- M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
- J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Many-body localization in a quantum simulator with programmable random disorder, Nat. Phys. 12, 907 (2016).
- Jae-yoon Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
- P. Roushan et al., Spectroscopic signatures of localization with interacting photons in superconducting qubits, Science 358, 1175 (2017).
- J. Z. Imbrie, On many-body localization for quantum spin chains, J. Stat. Phys. 163, 998 (2016).
- D. Pekker, G. Refael, E. Altman, E. Demler, and V. Oganesyan, Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality, Phys. Rev. X 4, 011052 (2014).
- J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Many-Body Localization in a Disordered Quantum Ising Chain, Phys. Rev. Lett. 113, 107204 (2014).
- R. Sahay, F. Machado, B. Ye, C. R. Laumann, and N. Y. Yao, Emergent Ergodicity at the Transition between Many-Body Localized Phases, Phys. Rev. Lett. 126, 100604 (2021).
- S. Moudgalya, D. A. Huse, and V. Khemani, Perturbative instability towards delocalization at phase transitions between MBL phases, arXiv:2008.09113.
- T. B. Wahl, F. Venn, and B. Béri, Local integrals of motion detection of localization-protected topological order, Phys. Rev. B 105, 144205 (2022).
- D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Localization-protected quantum order, Phys. Rev. B 88, 014206 (2013).
- P. Fendley, Parafermionic edge zero modes in -invariant spin chains, J. Stat. Mech.: Theory Exp. (2012) P11020.
- D. S. Fisher, Critical behavior of random transverse-field Ising spin chains, Phys. Rev. B 51, 6411 (1995).
- W. De Roeck and F. Huveneers, Stability and instability towards delocalization in many-body localization systems, Phys. Rev. B 95, 155129 (2017).
- Y. Bahri, R. Vosk, E. Altman, and A. Vishwanath, Localization and topology protected quantum coherence at the edge of hot matter, Nat. Commun. 6, 7341 (2015).
- D. V. Else, P. Fendley, J. Kemp, and C. Nayak, Prethermal Strong Zero Modes and Topological Qubits, Phys. Rev. X 7, 041062 (2017).
- H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. part I, Phys. Rev. 60, 252 (1941).
- The SZM operator pairs states in different symmetry (here the parity) sectors, leading to identical spectra up to exponentially small finite-size corrections, see also [47].
- A. Rahmani, X. Zhu, M. Franz, and I. Affleck, Phase diagram of the interacting Majorana chain model, Phys. Rev. B 92, 235123 (2015).
- A. M. Lobos, R. M. Lutchyn, and S. Das Sarma, Interplay of Disorder and Interaction in Majorana Quantum Wires, Phys. Rev. Lett. 109, 146403 (2012).
- F. Crépin, G. Zaránd, and P. Simon, Nonperturbative phase diagram of interacting disordered Majorana nanowires, Phys. Rev. B 90, 121407(R) (2014).
- N. M. Gergs, L. Fritz, and D. Schuricht, Topological order in the Kitaev/Majorana chain in the presence of disorder and interactions, Phys. Rev. B 93, 075129 (2016).
- J. F. Karcher, M. Sonner, and A. D. Mirlin, Disorder and interaction in chiral chains: Majoranas versus complex fermions, Phys. Rev. B 100, 134207 (2019).
- A. Yu Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001).
- S.-H. Lin, B. Sbierski, F. Dorfner, C. Karrasch, and F. Heidrich-Meisner, Many-body localization of spinless fermions with attractive interactions in one dimension, SciPost Physics 4, 002 (2018).
- F. Pietracaprina, N. Macé, D. J. Luitz, and F. Alet, Shift-invert diagonalization of large many-body localizing spin chains, SciPost Physics 5, 045 (2018).
- This corresponds to 32 Majorana fermions, and Hilbert spaces of maximum size 65 536.
- For odd system sizes we cut at the bond .
- V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
- Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles, Phys. Rev. Lett. 110, 084101 (2013).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L032016 for additional data for the correlations.
- N. Laflorencie (unpublished) (2022).
- P. Fendley, Strong zero modes and eigenstate phase transitions in the XYZ/interacting majorana chain, J. Phys. A: Math. Theor. 49, 30LT01 (2016).
- J. Kemp, N. Y Yao, C. R Laumann, and P. Fendley, Long coherence times for edge spins, J. Stat. Mech.: Theory Exp. (2017) 063105.
- R. Vasseur, A. J. Friedman, S. A. Parameswaran, and A. C. Potter, Particle-hole symmetry, many-body localization, and topological edge modes, Phys. Rev. B 93, 134207 (2016).
- O. Giraud, N. Macé, É. Vernier, and F. Alet, Probing Symmetries of Quantum Many-Body Systems through Gap Ratio Statistics, Phys. Rev. X 12, 011006 (2022).
- We observe a critical power-law decay at with which contrasts with free-fermions where [59].
- In the MBL PM regime, typical and average localization lengths show very similar behaviors.
- Where the typical localization length is [27].
- S. A. Parameswaran and R. Vasseur, Many-body localization, symmetry and topology, Rep. Prog. Phys. 81, 082501 (2018).
- L. Wang, K. S. D. Beach, and A. W. Sandvik, High-precision finite-size scaling analysis of the quantum-critical point of s=1/2 heisenberg antiferromagnetic bilayers, Phys. Rev. B 73, 014431 (2006).
- D. Rainis and D. Loss, Majorana qubit decoherence by quasiparticle poisoning, Phys. Rev. B 85, 174533 (2012).
- A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Avalanches and many-body resonances in many-body localized systems, Phys. Rev. B 105, 174205 (2022).
- D. Sels, Markovian baths and quantum avalanches, arXiv:2108.10796 (2021).
- D. S. Fisher and A. P. Young, Distributions of gaps and end-to-end correlations in random transverse-field Ising spin chains, Phys. Rev. B 58, 9131 (1998).