The interplay between interaction, disorder, and dissipation has shown a rich phenomenology. Here, we investigate a disordered XXZ spin chain in contact with a bath which, alone, would drive the system toward a highly delocalized and coherent Dicke state. We show that there exist regimes for which the natural orbitals of the single-particle density matrix of the steady state are all localized in the presence of strong disorders, for either weak interaction or strong interaction. We show that the averaged steady-state occupation in the eigenbasis of the open system Hamiltonian could follow an exponential decay for intermediate disorder strength in the presence of weak interactions, while it is more evenly spread for strong disorder or for stronger interactions. Last, we show that strong dissipation increases the coherence of the steady states, thus reducing the signatures of localization. We capture such signatures of localization also with a concatenated inverse participation ratio that simultaneously takes into account how localized are the eigenstates of the Hamiltonian and how close is the steady state to an incoherent mixture of different energy eigenstates.
REFERENCES
1.
P. W.
Anderson
, “Absence of diffusion in certain random lattices
,” Phys. Rev.
109
, 1492
–1505
(1958
). 2.
D.
Basko
, I.
Aleiner
, and B.
Altshuler
, “Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states
,” Ann. Phys. (N.Y.)
321
, 1126
–1205
(2006
). 3.
V.
Oganesyan
and D. A.
Huse
, “Localization of interacting fermions at high temperature
,” Phys. Rev. B
75
, 155111
(2007
). 4.
A.
Pal
and D. A.
Huse
, “Many-body localization phase transition
,” Phys. Rev. B
82
, 174411
(2010
). 5.
T.
Giamarchi
and H. J.
Schulz
, “Localization and interaction in one-dimensional quantum fluids
,” Europhys. Lett.
3
, 1287
–1293
(1987
). 6.
T.
Giamarchi
and H. J.
Schulz
, “Anderson localization and interactions in one-dimensional metals
,” Phys. Rev. B
37
, 325
–340
(1988
). 7.
R.
Nandkishore
and D. A.
Huse
, “Many-body localization and thermalization in quantum statistical mechanics
,” Annu. Rev. Condens. Matter Phys.
6
, 15
–38
(2015
). 8.
D. A.
Abanin
and Z.
Papić
, “Recent progress in many-body localization: Recent progress in many-body localization
,” Ann. Phys. (Berlin)
529
, 1700169
(2017
). 9.
D. A.
Abanin
, E.
Altman
, I.
Bloch
, and M.
Serbyn
, “Colloquium: Many-body localization, thermalization, and entanglement
,” Rev. Mod. Phys.
91
, 021001
(2019
). 10.
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
–845
(2015
). 11.
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
–845
(2015
). 12.
P.
Bordia
, H. P.
Lüschen
, S. S.
Hodgman
, M.
Schreiber
, I.
Bloch
, and U.
Schneider
, “Coupling identical one-dimensional many-body localized systems
,” Phys. Rev. Lett.
116
, 140401
(2016
). 13.
J.-Y.
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
–1552
(2016
). 14.
P.
Bordia
, H.
Lüschen
, S.
Scherg
, S.
Gopalakrishnan
, M.
Knap
, U.
Schneider
, and I.
Bloch
, “Probing slow relaxation and many-body localization in two-dimensional quasiperiodic systems
,” Phys. Rev. X
7
, 041047
(2017
).15.
P.
Bordia
, H.
Lüschen
, U.
Schneider
, M.
Knap
, and I.
Bloch
, “Periodically driving a many-body localized quantum system
,” Nat. Phys.
13
, 460
–464
(2017
). 16.
H. P.
Lüschen
, P.
Bordia
, S. S.
Hodgman
, M.
Schreiber
, S.
Sarkar
, A. J.
Daley
, M. H.
Fischer
, E.
Altman
, I.
Bloch
, and U.
Schneider
, “Signatures of many-body localization in a controlled open quantum system
,” Phys. Rev. X
7
, 011034
(2017
).17.
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
–911
(2016
). 18.
P.
Roushan
, C.
Neill
, J.
Tangpanitanon
, V. M.
Bastidas
, A.
Megrant
, R.
Barends
, Y.
Chen
, Z.
Chen
, B.
Chiaro
, A.
Dunsworth
, A.
Fowler
, B.
Foxen
, M.
Giustina
, E.
Jeffrey
, J.
Kelly
, E.
Lucero
, J.
Mutus
, M.
Neeley
, C.
Quintana
, D.
Sank
, A.
Vainsencher
, J.
Wenner
, T.
White
, H.
Neven
, D. G.
Angelakis
, and J.
Martinis
, “Spectroscopic signatures of localization with interacting photons in superconducting qubits
,” Science
358
, 1175
–1179
(2017
). 19.
Q.
Guo
, C.
Cheng
, Z.-H.
Sun
, Z.
Song
, H.
Li
, Z.
Wang
, W.
Ren
, H.
Dong
, D.
Zheng
, Y.-R.
Zhang
, R.
Mondaini
, H.
Fan
, and H.
Wang
, “Observation of energy-resolved many-body localization
,” Nat. Phys.
17
, 234
–239
(2021
).20.
R.
Nandkishore
, S.
Gopalakrishnan
, and D. A.
Huse
, “Spectral features of a many-body-localized system weakly coupled to a bath
,” Phys. Rev. B
90
, 064203
(2014
). 21.
S.
Johri
, R.
Nandkishore
, and R. N.
Bhatt
, “Many-body localization in imperfectly isolated quantum systems
,” Phys. Rev. Lett.
114
, 117401
(2015
). 22.
R.
Nandkishore
, “Many-body localization proximity effect
,” Phys. Rev. B
92
, 245141
(2015
). 23.
M.
Žnidarič
, A.
Scardicchio
, and V. K.
Varma
, “Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system
,” Phys. Rev. Lett.
117
, 040601
(2016
). 24.
M. V.
Medvedyeva
, T.
Prosen
, and M.
Žnidarič
, “Influence of dephasing on many-body localization
,” Phys. Rev. B
93
, 094205
(2016
). 25.
E.
Levi
, M.
Heyl
, I.
Lesanovsky
, and J. P.
Garrahan
, “Robustness of many-body localization in the presence of dissipation
,” Phys. Rev. Lett.
116
, 237203
(2016
). 26.
M. H.
Fischer
, M.
Maksymenko
, and E.
Altman
, “Dynamics of a many-body-localized system coupled to a bath
,” Phys. Rev. Lett.
116
, 160401
(2016
). 27.
K.
Hyatt
, J. R.
Garrison
, A. C.
Potter
, and B.
Bauer
, “Many-body localization in the presence of a small bath
,” Phys. Rev. B
95
, 035132
(2017
). 28.
B.
Everest
, I.
Lesanovsky
, J. P.
Garrahan
, and E.
Levi
, “Role of interactions in a dissipative many-body localized system
,” Phys. Rev. B
95
, 024310
(2017
). 29.
D. J.
Luitz
, F.
Huveneers
, and W.
De Roeck
, “How a small quantum bath can thermalize long localized chains
,” Phys. Rev. Lett.
119
, 150602
(2017
). 30.
R.
Nandkishore
and S.
Gopalakrishnan
, “Many body localized systems weakly coupled to baths: Many body localized systems weakly coupled to baths
,” Ann. Phys. (Berlin)
529
, 1600181
(2017
). 31.
M.
Žnidarič
, J. J.
Mendoza-Arenas
, S. R.
Clark
, and J.
Goold
, “Dephasing enhanced spin transport in the ergodic phase of a many-body localizable system
,” Ann. Phys. (Berlin)
529
, 1600298
(2017
). 32.
M.
Žnidarič
and M.
Ljubotina
, “Interaction instability of localization in quasiperiodic systems
,” Proc. Natl. Acad. Sci. U.S.A.
115
, 4595
–4600
(2018
). 33.
D.
Karlsson
, M.
Hopjan
, and C.
Verdozzi
, “Disorder and interactions in systems out of equilibrium: The exact independent-particle picture from density functional theory
,” Phys. Rev. B
97
, 125151
(2018
). 34.
I.
Vakulchyk
, I.
Yusipov
, M.
Ivanchenko
, S.
Flach
, and S.
Denisov
, “Signatures of many-body localization in steady states of open quantum systems
,” Phys. Rev. B
98
, 020202
(2018
). 35.
X.
Xu
, C.
Guo
, and D.
Poletti
, “Interplay of interaction and disorder in the steady state of an open quantum system
,” Phys. Rev. B
97
, 140201
(2018
). 36.
L.-N.
Wu
, A.
Schnell
, G. D.
Tomasi
, M.
Heyl
, and A.
Eckardt
, “Describing many-body localized systems in thermal environments
,” New J. Phys.
21
, 063026
(2019
). 37.
A.
Rubio-Abadal
, J.-Y.
Choi
, J.
Zeiher
, S.
Hollerith
, J.
Rui
, I.
Bloch
, and C.
Gross
, “Many-body delocalization in the presence of a quantum bath
,” Phys. Rev. X
9
, 041014
(2019
).38.
E.
Wybo
, M.
Knap
, and F.
Pollmann
, “Entanglement dynamics of a many-body localized system coupled to a bath
,” Phys. Rev. B
102
, 064304
(2020
). 39.
S.
Diehl
, A.
Micheli
, A.
Kantian
, B.
Kraus
, H. P.
Büchler
, and P.
Zoller
, “Quantum states and phases in driven open quantum systems with cold atoms
,” Nat. Phys.
4
, 878
–883
(2008
). 40.
P.
Schindler
, M.
Müller
, D.
Nigg
, J. T.
Barreiro
, E. A.
Martinez
, M.
Hennrich
, T.
Monz
, S.
Diehl
, P.
Zoller
, and R.
Blatt
, “Quantum simulation of dynamical maps with trapped ions
,” Nat. Phys.
9
, 361
–367
(2013
). 41.
P.
Jordan
and E.
Wigner
, “Über das paulische äquivalenzverbot
,” Z. Phys.
47
, 631
–651
(1928
). 42.
E.
Lieb
, T.
Schultz
, and D.
Mattis
, “Two soluble models of an antiferromagnetic chain
,” Ann. Phys. (N.Y.)
16
, 407
–466
(1961
). 43.
S.
Bera
, H.
Schomerus
, F.
Heidrich-Meisner
, and J. H.
Bardarson
, “Many-body localization characterized from a one-particle perspective
,” Phys. Rev. Lett.
115
, 046603
(2015
). 44.
S.
Bera
, T.
Martynec
, H.
Schomerus
, F.
Heidrich-Meisner
, and J. H.
Bardarson
, “One-particle density matrix characterization of many-body localization
,” Ann. Phys. (Berlin)
529
, 1600356
(2017
). 45.
T. L. M.
Lezama
, S.
Bera
, H.
Schomerus
, F.
Heidrich-Meisner
, and J. H.
Bardarson
, “One-particle density matrix occupation spectrum of many-body localized states after a global quench
,” Phys. Rev. B
96
, 060202
(2017
). 46.
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 Phys.
4
, 002
(2018
). 47.
D. J.
Luitz
, N.
Laflorencie
, and F.
Alet
, “Many-body localization edge in the random-field Heisenberg chain
,” Phys. Rev. B
91
, 081103
(2015
). 48.
M.
Serbyn
, Z.
Papić
, and D. A.
Abanin
, “Criterion for many-body localization-delocalization phase transition
,” Phys. Rev. X
5
, 041047
(2015
).49.
J.
Šuntajs
, J.
Bonča
, T.
Prosen
, and L.
Vidmar
, “Quantum chaos challenges many-body localization
,” Phys. Rev. E
102
, 062144
(2020
). 50.
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
). 51.
R. K.
Panda
, A.
Scardicchio
, M.
Schulz
, S. R.
Taylor
, and M.
Žnidarič
, “Can we study the many-body localisation transition?
,” Europhys. Lett.
128
, 67003
(2019
). 52.
M.
Kiefer-Emmanouilidis
, R.
Unanyan
, M.
Fleischhauer
, and J.
Sirker
, “Slow delocalization of particles in many-body localized phases
,” Phys. Rev. B
103
, 024203
(2021
). 53.
V.
Gorini
, A.
Kossakowski
, and E. C. G.
Sudarshan
, “Completely positive dynamical semigroups of N-level systems
,” J. Math. Phys.
17
, 821
–825
(1976
). 54.
G.
Lindblad
, “On the generators of quantum dynamical semigroups
,” Commun. Math. Phys.
48
, 119
–130
(1976
). 55.
S.
Diehl
, A.
Tomadin
, A.
Micheli
, R.
Fazio
, and P.
Zoller
, “Dynamical phase transitions and instabilities in open atomic many-body systems
,” Phys. Rev. Lett.
105
, 015702
(2010
). 56.
A.
Tomadin
, S.
Diehl
, and P.
Zoller
, “Nonequilibrium phase diagram of a driven and dissipative many-body system
,” Phys. Rev. A
83
, 013611
(2011
). 57.
W.
Yi
, S.
Diehl
, A. J.
Daley
, and P.
Zoller
, “Driven-dissipative many-body pairing states for cold fermionic atoms in an optical lattice
,” New J. Phys.
14
, 055002
(2012
). 58.
C.-E.
Bardyn
, M. A.
Baranov
, C. V.
Kraus
, E.
Rico
, A.
İmamoğlu
, P.
Zoller
, and S.
Diehl
, “Topology by dissipation
,” New J. Phys.
15
, 085001
(2013
). 59.
S.
Lloyd
, “Universal quantum simulators
,” Science
273
, 1073
–1078
(1996
). 60.
I.
Yusipov
, T.
Laptyeva
, S.
Denisov
, and M.
Ivanchenko
, “Localization in open quantum systems
,” Phys. Rev. Lett.
118
, 070402
(2017
). 61.
O. S.
Vershinina
, I. I.
Yusipov
, S.
Denisov
, M. V.
Ivanchenko
, and T. V.
Laptyeva
, “Control of a single-particle localization in open quantum systems
,” Europhys. Lett.
119
, 56001
(2017
). 62.
We assume no degeneracies.
63.
R.
Fan
, P.
Zhang
, H.
Shen
, and H.
Zhai
, “Out-of-time-order correlation for many-body localization
,” Sci. Bull.
62
, 707
–711
(2017
). 64.
See https://www.nscc.sg/.
© 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
