A cornerstone of the theory of phase transitions is the observation that many-body systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analog of such a theory. Specifically, we consider local dissipative dynamics preparing an equilibrium steady-state of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive fluctuations in a local order parameter. We show that for all such processes, there exist asymptotically stationary symmetry-breaking states, i.e., states that become stationary in the thermodynamic limit and give a finite value to the order parameter. We give results both for discrete and continuous symmetries and explicitly show how to construct the symmetry-breaking states. Our results show in a simple way that, in large systems, local dissipative dynamics satisfying detailed balance cannot uniquely and efficiently prepare states with extensive fluctuations with respect to local operators. We discuss the implications of our results for quantum simulators and dissipative state preparation.

1.
R. K.
Pathria
and
P. D.
Beale
,
Statistical Mechanics
(
Elsevier Science
,
2011
).
2.
M.
Kliesch
,
C.
Gogolin
,
M. J.
Kastoryano
,
A.
Riera
, and
J.
Eisert
, “
Locality of temperature
,”
Phys. Rev. X
4
,
031019
(
2014
); e-print arXiv:1309.0816.
3.
R. B.
Griffiths
, “
Spontaneous magnetization in idealized ferromagnets
,”
Phys. Rev.
152
,
240
246
(
1966
).
4.
F. J.
Dyson
,
E. H.
Lieb
, and
B.
Simon
, “
Phase transitions in quantum spin systems with isotropic and nonisotropic interactions
,”
J. Stat. Phys.
18
,
335
383
(
1978
).
5.
T.
Koma
and
H.
Tasaki
, “
Symmetry breaking and finite size effects in quantum many-body systems
,”
J. Stat. Phys.
76
,
745
803
(
1994
); e-print arXiv:cond-mat/9708132.
6.
J.
Eisert
and
T.
Prosen
, “
Noise-driven quantum criticality
” (
2010
); e-print arXiv:1012.5013.
7.
M.
Höning
,
M.
Moos
, and
M.
Fleischhauer
, “
Critical exponents of steady-state phase transitions in fermionic lattice models
,”
Phys. Rev. A
86
,
013606
(
2012
); e-print arXiv:1108.2263.
8.
K.
Temme
, “
Lower bounds to the spectral gap of davies generators
,”
J. Math. Phys.
54
,
122110
(
2013
); e-print arXiv:1305.5591.
9.
M. J.
Kastoryano
and
J.
Eisert
, “
Rapid mixing implies exponential decay of correlations
,”
J. Math. Phys.
54
,
102201
(
2013
); e-print arXiv:1303.6304.
10.
Z.
Cai
and
T.
Barthel
, “
Algebraic versus exponential decoherence in dissipative many-particle systems
,”
Phys. Rev. Lett.
111
,
150403
(
2013
); e-print arXiv:1304.6890.
11.
M.
Znidaric
, “
Relaxation times of dissipative many-body quantum systems
,”
Phys. Rev. E
92
,
042143
(
2015
); e-print arXiv:1507.07773.
12.
S.
Diehl
,
E.
Rico
,
M. A.
Baranov
, and
P.
Zoller
, “
Topology by dissipation in atomic quantum wires
,”
Nat. Phys.
7
,
971
(
2011
); e-print arXiv:1105.5947.
13.
I.
Bloch
,
J.
Dalibard
, and
S.
Nascimbene
, “
Quantum simulations with ultracold quantum gases
,”
Nat. Phys.
8
,
267
(
2012
).
14.
H.
Weimer
,
M.
Müller
,
I.
Lesanovsky
,
P.
Zoller
, and
H. P.
Büchler
, “
A Rydberg quantum simulator
,”
Nat. Phys.
6
,
382
388
(
2010
); e-print arXiv:0907.1657.
15.
G.
Lindblad
, “
On the generators of quantum dynamical semigroups
,”
Commun. Math. Phys.
48
,
119
130
(
1976
).
16.
E. H.
Lieb
and
D. W.
Robinson
, “
The finite group velocity of quantum spin systems
,”
Commun. Math. Phys.
28
,
251
257
(
1972
).
17.
D.
Poulin
, “
Lieb-robinson bound and locality for general markovian quantum dynamics
,”
Phys. Rev. Lett.
104
,
190401
(
2010
); e-print arXiv:1003.3675.
18.
T.
Barthel
and
M.
Kliesch
, “
Quasi-locality and efficient simulation of markovian quantum dynamics
,”
Phys. Rev. Lett.
108
,
230504
(
2011
); e-print arXiv:1111.4210.
19.
B.
Nachtergaele
,
A.
Vershynina
, and
V. A.
Zagrebnov
, “
Lieb-Robinson bounds and existence of the thermodynamic limit for a class of irreversible quantum dynamics
,”
AMS Contemp. Math.
552
,
161
175
(
2011
); e-print arXiv:1103.1122.
20.
M.
Kliesch
,
C.
Gogolin
, and
J.
Eisert
, “
Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems
,” in
Many-Electron Approaches in Physics, Chemistry and Mathematics
, edited by
V.
Bach
and
L. D.
Site
(
Springer: Mathematical Physics Studies
,
2014
), pp.
301
318
.e-print arXiv:1306.0716.
21.
E. B.
Davies
, “
Markovian master equations
,”
Commun. Math. Phys.
39
,
91
110
(
1974
).
22.
R.
Alicki
, “
On the detailed balance condition for non-hamiltonian systems
,”
Rep. Math. Phys.
10
,
249
258
(
1976
).
23.
A.
Kossakowski
,
A.
Frigerio
,
V.
Gorini
, and
M.
Verri
, “
Quantum detailed balance and kms condition
,”
Commun. Math. Phys.
57
,
97
110
(
1977
).
24.
V.
Jakšić
and
C.-A.
Pillet
, “
On entropy production in quantum statistical mechanics
,”
Commun. Math. Phys.
217
,
285
293
(
2001
).
25.
F.
Fagnola
and
V.
Umanit
, “
Generators of detailed balance quantum markov semigroups
,”
Infinite Dimens. Anal., Quantum Probab. Relat. Top.
10
,
335
363
(
2007
).
26.
F.
Fagnola
and
V.
Umanit
, “
Generators of kms symmetric Markov semigroups on B(h) symmetry and quantum detailed balance
,”
Commun. Math. Phys.
298
,
523
547
(
2010
).
27.
M. J.
Kastoryano
and
F. G. S. L.
Brandao
, “
Quantum gibbs samplers: The commuting case
,”
Commun. Math. Phys.
344
,
915
957
(
2016
); e-print arXiv:1409.3435.
28.

Unlike in most studies using the notion of detailed balance, we do not require the steady-state to be faithful (full-rank) because the proofs of our results do not require this assumption.

29.

However, it would be sufficient to assume the weaker notion of asymptotic reversibility for the reference state with long-range correlations instead of full detailed balance to obtain our results.

30.
V. V.
Albert
and
L.
Jiang
, “
Symmetries and conserved quantities in Lindblad master equations
,”
Phys. Rev. A
89
,
022118
(
2014
); e-print arXiv:1310.1523.
31.
L.
Landau
,
J. F.
Perez
, and
W. F.
Wreszinski
, “
Energy gap, clustering, and the goldstone theorem in statistical mechanics
,”
J. Stat. Phys.
26
,
755
766
(
1981
).
32.
M. J.
Kastoryano
and
K.
Temme
, “
Quantum logarithmic sobolev inequalities and rapid mixing
,”
J. Math. Phys.
54
,
052202
(
2013
); e-print arXiv:1207.3261.
33.
T. S.
Cubitt
,
A.
Lucia
,
S.
Michalakis
, and
D.
Perez-Garcia
, “
Stability of local quantum dissipative systems
,”
Commun. Math. Phys.
337
,
1275
1315
(
2015
); e-print arXiv:1303.4744.
34.
K.
Fujii
,
M.
Negoro
,
N.
Imoto
, and
M.
Kitagawa
, “
Measurement-free topological protection using dissipative feedback
,”
Phys. Rev. X
4
,
041039
(
2014
); e-print arXiv:1401.6350.
35.
M.
Herold
,
M.
Kastoryano
,
E. T.
Campbell
, and
J.
Eisert
, “
Fault tolerant dynamic decoders for topological quantum memories
,”
npj Quantum Inf.
1
,
15010
(
2015
); e-print arXiv:1406.2338.
36.
R.
Koenig
and
F.
Pastawski
, “
Generating topological order: No speedup by dissipation
,”
Phys. Rev. B
90
,
045101
(
2013
); e-print arXiv:1310.1037.
37.
F.
Verstraete
,
M. M.
Wolf
, and
J. I.
Cirac
, “
Quantum computation and quantum-state engineering driven by dissipation
,”
Nat. Phys.
5
,
633
636
(
2009
); e-print arXiv:0803.1447.
38.
M. J.
Kastoryano
,
M. M.
Wolf
, and
J.
Eisert
,
Phys. Rev. Lett.
103
,
110501
(
2013
); e-print arXiv:1205.0985.
39.
P. D.
Johnson
,
F.
Ticozzi
, and
L.
Viola
, “
General fixed points of quasi-local frustration-free quantum semigroups: From invariance to stabilization
,”
Quantum Inf. Comp.
16
,
657
699
(
2016
).
40.
C.-E.
Bardyn
,
M. A.
Baranov
,
C. V.
Kraus
,
E.
Rico
,
A.
Imamoglu
,
P.
Zoller
, and
S.
Diehl
, “
Topology by dissipation
,”
New J. Phys.
15
,
085001
(
2013
); e-print arXiv:1302.5135.
41.
R.
Gutierrez
,
J. P.
Garrahan
, and
I.
Lesanovsky
, “
Non-equilibrium fluctuations and metastability in the dynamics of dissipative multi-component Rydberg gases
,”
New J. Phys.
18
,
093054
(
2016
); e-print arXiv:1603.00828.
42.
N.
Schuch
,
D.
Perez-Garcia
, and
J. I.
Cirac
, “
Classifying quantum phases using matrix product states and projected entangled pair states
,”
Phys. Rev. B
84
,
165139
(
2011
); e-print arXiv:1010.3732.
43.
X.
Chen
,
Z.-C.
Gu
, and
X.-G.
Wen
, “
Classification of gapped symmetric phases in 1d spin systems
,”
Phys. Rev. B
83
,
035107
(
2011
); e-print arXiv:1103.3323.
44.
A. M.
Turner
,
F.
Pollmann
, and
E.
Berg
, “
Topological phases of one-dimensional fermions: An entanglement point of view
,”
Phys. Rev. B
83
,
075102
(
2011
); e-print arXiv:1409.8616.
45.
C. H.
Bennett
and
G.
Grinstein
, “
Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems
,”
Phys. Rev. Lett.
55
,
657
(
1985
).
46.
G.
Grinstein
, “
Can complex structures be generically stable in a noisy world?
,”
IBM J. Res. Dev.
48
,
5
12
(
2004
).
You do not currently have access to this content.