The existence of quasi-long range order is demonstrated in nonequilibrium steady states in isotropic XY spin chains including of two types of additional terms that generate a gap in the energy spectrum. The system is driven out of equilibrium by initializing a domain-wall magnetization profile through application of external magnetic field and switching off the magnetic field at the same time the energy gap is activated. An energy gap is produced by either applying a staggered magnetic field in the transverse direction or introducing a modulation to the XY coupling. The magnetization, spin current and spin-spin correlation functions are computed in the thermodynamic limit at long times after the quench. For both types of systems, we find the persistence of power-law correlations despite the ground state correlation functions exhibiting exponential decay. It is discussed how these power-law correlations appear related to the periodic nature of the perturbation which generates the energy gap.

1. Persistence of power-law correlations in nonequilibrium steady states of gapped
quantum spin chains
Jarrett L. Lancaster and Joseph P. Godoy
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2. Introduction: Quench Dynamics
Theoretical view
Time evolution from arbitrary initial state
|Ψ(t) = e−i ˆHt/
|Ψ0
Experimental view
System begins (t → −∞) in ground state of ˆH0
ˆH0 |Ψ0 = E0 |Ψ0
Time evolution generated by
ˆH = ˆH0 + Θ(t) ˆV (t)
ˆV (t) contains rapidly-tunable fields or interactions.
Example: Domain-wall initial state in one-dimensional
spin chain:
Ψ0| ˆSz
j |Ψ0 = −
2
sgn(j)
For t > 0
current
central
subsystem
NESS
_m0
m0
j
Sz
j
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3. Inhomogeneous Quench Dynamics in Spin Chains
0 5 10 15 20 25 30 35 40 45 50
0
0.1
0.2
Jt = 0
κ
2
/(8n)
1/2
0 5 10 15 20 25 30 35 40 45 50
0
0.1
0.2
|Cxx
(j=0,n,t)|
Jt = 50
κ2
/(8n)1/2
0 5 10 15 20 25 30 35 40 45 50
0
0.1
0.2
distance from origin n
Jt = 5000
κ2
/(8n)1/2
Cxx
(n)
t→∞
−−−→ Cxx
0 (n) cos (πn/2)
∼ A0
cos πn
2
√
n
(n → ∞)
JLL and A. Mitra, Phys. Rev. E 81, 061134 (2010)
Experimental realization in terms of hardcore bosons
ˆH = −J
j
ˆa†
j ˆaj+1 + h.c
ˆa†
j
2
= 0
x x
r(x) r(x)
+ time-of-flight
expansion
free expansion
ˆa†
j ˆaj+r ∼ r− 1
2 ei πr
2
L. Vidmar et al, Phys. Rev. Lett. 115, 175301 (2015)
Lancaster and Godoy (HPU) Power-law correlations 3 / 11

4. Motivation
Isotropic XY model with cavity-induced, global-range
interactions
ˆH = −J
j
ˆSx
j
ˆSx
j+1 + ˆSy
j
ˆSy
j+1 + h
j
ˆSz
j
−
L


j,odd
ˆSz
j −
j,odd
ˆSz
j


2
Surprisingly rich phase diagram with order parameter
ˆx =
1
L


j,odd
ˆSz
j −
j,odd
ˆSz
j


.
Can we calculate the correlation function exactly in similar
models?
Quench from XY phase: algebraic decay
Cxx
(n)
t→∞
−−−→ (A + B cos (πn)) n− 1
2
(n → ∞)
Several interesting features:
same power law as XY ground state
different amplitudes (A ± B) for even/odd
correlations
F. Igl´oi, B. Blaß, G. Ro´osz and H. Rieger, Phys. Rev. B
98, 184415 (2018)
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5. Domain Wall Time Evolution
Domain wall defines initial state
|Ψ0 = |· · · ↑↑↑↓↓↓ · · ·
=
j<0
c†
j |0
Jordan-Wigner transformation
ˆSz
j → c†
jcj −
1
2
, ˆS+
j = c†
j exp iπ
j−1
n=1
c†
ncn
We consider models generating time evolution
which are quadratic in quasiparticles:
ˆHf =
k
kγ†
kγk,
cj =
m
Vjmγm
Two-point correlation functions
Czz
(n) = ˆSz
j
ˆSz
j+n
Cxx
(n) = ˆSx
j
ˆSx
j+n
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6. Models
1 Isotropic XY chain with “staggered” magnetic field
ˆHs = ˆHxy + m
j
(−1)j ˆSz
j
2 Dimerized XY chain
ˆHd = −J
j
1 + (−1)j
δ ˆSx
j
ˆSx
j+1 + ˆSy
j
ˆSy
j+1
Correlations in ground state of ˆHs
Cxx
0 (n) ∼ A0( ˜m)e−α( ˜m)n
Czz
0 (n) ∼ B0( ˜m)e−β( ˜m)n
as n → ∞, where ˜m ≡ m/J quantifies energy gap.
0 20 40 60 80 100
10
-15
10
-10
10
-5
10
0
Similar exponential decay in correlation functions for ground state of ˆHd.
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7. Extracting Long-Time Behavior
kk
kk-
-
F F
F F
+ +
_ _
nj
j
k
k
Domain-wall initial state equivalent to different
chemical potentials.
System quasiparticles are mixture of
Jordan-Wigner fermions
γk = cos
θk
2
ck − sin
θk
2
c†
k
kk- F F
_
+
nj
j
k
Initial state encoded in small-q momentum correlations
Ψ0| c†
k+ q
2
ck− q
2
|Ψ0
iΘ(k−
F − |k|)
q + i0+
+
−iΘ(k+
F − |k|)
q − i0+
T. Sabetta and G. Misguich, Phys. Rev. B 88, 245114 (2013)
Lancaster and Godoy (HPU) Power-law correlations 7 / 11

8. Nonequilibrium correlations
Power-law correlations persist despite nonzero energy
gap:
Czz
NESS(n) ∼ −
1
(πn)2



1
1+ ˜m2 (n odd)
1 − (−1)
n
2 ˜m
√
1+ ˜m2
2
(n even)
Correlation function behavior
Power-law decay
Different amplitudes for even/odd n
Transverse correlation function also inherits oscillations
due to nonzero spin current.
10
0
10
1
10
2
10
-4
10
-3
10-2
10
-1
Cxx
NESS(n) ∼ A + B cos
πn
4
cos πn
2
n
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9. Fisher-Hartwig conjecture
Calculation of Cxx
(n) requires the evaluation of a large
pfaffian or determinant.
Correlation function as determinant
Cxx
NESS(n) =
1
4
0 q
(j)
1 · · · q
(j)
n−1
q
(j+1)
−1 0 · · · q
(j+1)
n−2
...
...
...
...
q
(j+n−1)
−n+1 q
(j+n−1)
−n+2 · · · 0
q(j)
n =
π
π
dp
2π
e−ipn
˜q(j)
(p)
˜q(j)
(p) =
cos p − (−1)j
˜m
cos2 p + ˜m2
sgn(p)
× sgn
π
2
− p sgn
π
2
+ p
Asymptotic form for n → ∞ can sometimes be
extracted from Fisher-Hartwig conjecture which
depends on ˜q(j)
(p).
Position dependence (appearance of j) spoils
Toeplitz form.
Na¨ıve application yields partially correct answer
Cxx
F.H.(n) ∼ n−1
correct power law
× e−αn
erroneous exponential decay
Are generalizations of FH conjecture possible
for cases such as this?
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10. Physical interpretations
Spatially modulated field hj = (−1)j
m generates
an energy gap, which should affect low-energy
physics.
(k) → ±J cos2 k + ˜m2 |k| <
π
2
Doubling of unit cell halves Brillouin zone
k k
e(k) e(k)
p p- p/2p/2-
extended zone scheme reduced zone scheme
Fisher-Hartwig conjecture provides link between
jump discontinuities in (effective) momentum
distribution/Wigner function and power-law decay
Cxx
NESS ∝ n−(β2
1 +···+β2
l )
where βj is contribution of jump discontinuity
indexed by j.
Energy gap leads to distribution factor acquiring
factor
sgn
π
2
+ k sgn
π
2
− k
Doubling discontinuities squares correlations
1
√
n
→
1
√
n
2
=
1
n
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11. Summary/outlook
Persistence of power-law decay
Two point correlation functions retain power-law decay
in nonequilibrium steady state
Exact results for noninteracting model
Similar complexity (different even/odd decay
amplitudes) to interacting model of Igl´oi et al
Conclusion
Exact results provide useful benchmarks and motivate
possible extensions of Fisher-Hartwig conjecture to cases
with spatial inhomogeneities
JLL and J. P. Godoy, Phys. Rev. Research 1, 033104 (2019)
100
101
102
10-4
10
-3
10-2
10
-1
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