Ideas About Quantum Metastability
Juan P. Garrahan
(Physics & Astronomy, University of Nottingham)
M. van Horssen, E. Levi (Nottingham)
K. Macieszczak, I. Lesanovsky, M. Guta (Maths)
£:
PLAN
!
Problem: relaxation & non-ergodicity in quantum systems
(both closed = unitary & open = dissipative)
vs. slow relaxation in classical glasses
!
!
1. Slow relaxation in closed quantum systems due to dynamical constraints
!
2. Signatures of many-body localisation in the absence of disorder
!
3. Towards a theory of metastability in (open) quantum systems
Generic Q. many-body systems “equilibrate” (become stationary):
{Short, Popescu, Linden, Reimann,
Eisert, Goldstein, many others}
no degenerate E gaps → dephasing in E basis → diagonal ensemble
(t) !
E.g.
= lim
t!
1
t
Z
t
0
L
X
H=
k=1
{Lesanovsky-Olmos-JPG, PRL 2010}
k
Ut 0
+
†
0
U
dt
0 t0
L
X
k=1
nk + V
|hA(t)i
L
X
nm nk
|m
k6=m
k|6
Tr(A )| small
(Rydbergs)
Generic Q. many-body systems “equilibrate” (become stationary):
{Short, Popescu, Linden, Reimann,
Eisert, Goldstein, many others}
no degenerate E gaps → dephasing in E basis → diagonal ensemble
(t) !
= lim
t!
1
t
Z
t
0
Ut 0
†
0
U
dt
0 t0
|hA(t)i
Most Q. many-body systems also “thermalise”:
A
B
A
= TrB | ih |
!
closed
thermalisation
Tr(A )| small
{Deutsch, Srednicki, Goldstein, Eisert, Rigol, Gogolin, many others}
A
= TrB
th
=e
th
H
set by h
0 |H| 0 i
ergodicity (independence of initial conditions)
eigenstate thermalisation hypothesis (ETH):
hE|Oloc l |Ei = smooth F(E)
Ergodic & thermal → non-ergodic & many-body localised (MBL)
{Basko-Aleiner-Altshuler, Huse+, Prosen+,
Abanin+, Moore+, Altman+, many others}
Cf. Anderson localisation, but with interactions
MBL transition driven by quenched disorder
(rnd h small = thermal, rnd h large = MBL)
{Schreiber et al 2015}
MBL = breakdown of ETH
(excited states = Area law)
MBL often thought of as a “glass transition”
What can we say about slow relaxation in Q.S. and MBL from
what we know of classical glasses?
Classical glasses
- Slowdown & eventual arrest
- Timescales not divergent
- No disorder
Relaxation is intermittent and spatially heterogeneous
t
⇥
t
⇥
t
⇥
e.g. 50:50 L-J mixture {Hedges 2009}
Due to (effective) local constraints on dynamics
{JPG-Chandler, PRL 2002}
1. Quantum slow relaxation due to dynamical constraints
{van Horssen-JPG, 2016}
Steric (excluded volume) constraint on hopping:
k
i
classical = constrained lattice gas
j
{Kob-Andersen, Jackle+}
l
H=
X
h ji
(1
nk n )
n
(
+
j
+
+
j
)
(1
î
) n (1
nj )
nj (1
n)
óo
constraint (“1-vacancy assisted” or 1-TLG)
Rokhsar-Kivelson pt:
Away from RK:
6=
=
1
2
1
2
! H=
class. Master op. W
Ç
! Large-deviations: active
>
1
2
!
t |Pi
å
= W|Pi
Ç
inactive
<
1
2
å
1. Quantum slow relaxation due to dynamical constraints
{van Horssen-JPG, 2016}
H=
X
(1
nk n )
n
(
h ji
| (t)i = e
+
j
+
+
j
(L-1)
tH
| (0)i
)
(1
1
3
î
) n (1
…
nj )
nj (1
L-1
n)
(1)
…
…
(L)
2
4
1
= 0.2
metastability = memory of initial conditions
óo
3
2
4
L
(2)
L-1
= 0.8 L
N = 24
M = 20
1. Quantum slow relaxation due to dynamical constraints
{van Horssen-JPG, 2016}
(1
nk n )
n
h ji
(
+
j
+
+
j
(L-1)
| (t)i = e
tH
| (0)i
(1
1
3
(L)
T=
106
l=7
l=8
l=9
l = 10
l = 11
l = 12
l = 13
L-1
n)
(1)
L
3
2
-2
4
-4
0
(2)
L-1
L
-6
-2
l=6
l=7
l=8
l=9
l = 10
l = 11
l = 12
-4
4
-12
-14
0
N = 24
102 M = 20
-8
-10
-8
0
nj (1
4
1
-6
10
…
2
E0
108
nj )
0.5
λ
1
0
0.5
λ
1st order singularity in g.s.
0.5
s
óo
…
2
change at 1
RK ⇡
2
) n (1
…
1010
τΦ
)
î
dE0 /dλ
H=
X
1
cf. classical active-inactive transitions
1
2. Dynamics of MBL in a quantum glass (w/o disorder)
H=
e
s
X
= / (1
{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}
Ä
n +1 e
)
s
1
ä
! sRK = 0
):
Classical
0.08 East glass model (T = 0.08
{Jackle+, Sollich-Evans, Aldous-Diaconis, JPG-Chandler, Chleboun-Faggionato-Martinelli, Blondel-Toninelli, many others}
θ(s)
0.04
11 ⇄ 01
10 ⇄ 00
θ(s)
0.04
s<0 active to s>0 inactive transition ≡ quantum phase transition in g.s. of H
0.00
0.00
0.08
0.6
0.6
-0.2 -0.1
0.6
0.4
0.4
θ(s)
K(s)
0.4
0.04
0.2
0.00
0
0
s
0.1
K(s)
N!
K(s)
{JPG-JackLecomte-Pitardvan Duijvendijkvan Wijland
PRL 2007}
0.2
0.2
0
-0.2 -0.1
0
s
0.1
0.2
2. Dynamics of MBL in a quantum glass (w/o disorder)
H=
e
s
X
{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}
Ä
n +1 e
= / (1
)
s
1
ä
! sRK = 0
| (t)i = e
tH
| (0)i
Signatures of MBL dynamics for s>0 (inactive):
Propagation of
excitations
loglog
t t
log t
(c)(c)
1
(c) 1
(a)(a)
(a)
1
0.80.8
0.8
0.60.6
0.6
0.40.4
0.4
0.20.2
0.2
0 0
0
Ĩ Ĩ
Ĩ
t t
t
(b)(b)
(b)
N =N8 = 8
N =N10= 10
N
N=
=N8
12= 12
N
=
10
N =N14= 14
N = 12
= 14 0 0
-2N -2
-2
0s s
s
2 2
2
2. Dynamics of MBL in a quantum glass (w/o disorder)
H=
e
= / (1
n +1 e
)
s
1
ä
tH
| (t)i = e
! sRK = 0
4
| (0)i
4
4 (a)
2
2
4
2 (b)
2
1
s = −1 ss =
= −1
−1
ss =
= −1
1
M̄
M̄ww(t)
(t)
M̄w (t)
M̄w (t)
M̄w (t)
time averaged magnetisation w=1 w=1
0
0
0
w=1
0
w = 2 0w = 2
w=2
dependence on initial conditions - ETH
(active)
v.
no-ETH
(inactive)
w
=
3
-2
w=3
-2
-1-2
2
w=3
w=4 w=4
w=4
w = 5 -2-4
w=
=5
5
w = 36 w
-1
w
=
w 10
=6
60
w10
= 37 10
10
w
=
w=7
7
-2
-4
4
)
10-1
s = −1 ss =
= −1
−1
0
0
-2
-2
-4
-4 0
-1
10
10
10-1
4
2 (b)
2
1
1
0
10
10
100
N=8
tJ
2
1
10
10
101
tJ
tJ
3
2
10
2
10
10
10
w=1 w=1
0
w=1
0
w=2 w=2
w=2
w = 3 -2
-1 w
w=
= 33
w=4 w=4
w=4
w = 5 -2-4
w
=
55
w
=
w = 36 w =
-1 6
3 10
w
=
60
10
w10
= 7 w 10
w=
= 77
10
-1
Var
M̄M̃
(normalised)
final
Var
final (normalised)
4
4 (a)
2
2
M̄w (t)
M̄w (t)
M̄
M̄ww(t)
(t)
2
2
s
Ä
Signatures of MBL dynamics for s>0 (inactive):
4
0
X
{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}
-4
-4 0
-1
10
10
10-1
0.8
0.6
10001
10
10
tJ
10112
10
10
tJ
tJ
10223−1
ss =
10
10
10 1
=
N=8
N = 10
N = 12
N = 14
5
10
0
5
(d)
0
0.4
-5
-5
0.2
0
-3
02
10
10
-10
-2
-1
14
10
10
tJtJ
s
0
s
1
(c)
02
62
10
10
-10
3
01
83
10
10
s=1
0
10
102
1
10
104
2
106
10
tJtJ
w=1
w=2
w=3
w 1=24 2
EE
w =E 5
w=6
w=7
8
1010
s=1
⟨M⟨M
⟩mc
⟩mc
⟨M⟨M
final
final⟩⟩E
E
3
3
3
2. Dynamics of MBL in a quantum glass (w/o disorder)
X
H=
s
e
{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}
Ä
n +1 e
= / (1
s
1
ä
| (t)i = e
! sRK = 0
)
tH
| (0)i
Signatures of MBL dynamics for s>0 (inactive):
4
S̄Sww(t)
3
w=2
w=4
w=6
(b)
4
10
s=1
2
S̄Sww (t)
(t)
{Serbyn-Papic-Abanin,
2
Moore+, others}
0
s
10 6
3
N = 12
4
⌧τ
10
S(t)
S̄(t)
1 log t
growth
ofs = −1
0
100 tJ 102
entanglement
4 w=2
w=4
entropy
3 w=6
-2
N = 14
∝ log(t)
6
N = 10
102
s=0
0
s = −1
2
10
6
10
N
N =8
N =6
14
1
1
2
s=1
0
10 8
100 102 104 106 108
tJ
(c)
s=1
0
10 0
10 2
10 4
tJ
10 6
(d)
10 8
2. Dynamics of MBL in a quantum glass (w/o disorder)
H=
e
s
X
{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}
Ä
n +1 e
= / (1
)
s
1
ä
tH
| (t)i = e
! sRK = 0
| (0)i
Signatures of MBL dynamics for s>0 (inactive):
e
0
(k) ⇠ e1/ k
MBL or quasi-MBL {cf. Yao-Laumann-Cirac-Lukin-Moore, De Roeck-Huveneers}
N =8
N = 10
N = 12
(a)
D(k; t)(N = 12)
10
4
s=1
(k) ⇠ e1/ k
1
1
log t !
s = −1
0.5
j
⇠
D(k; t)(N = 12)
5
103
s = 0.5
s=1
102
k
101
0.5
e
1
s = −1
0
10
0
10
2
10
tJ
(b)
4
10
6
10
8
2
4
6
8
2π/k
(c)
10
12
100
τ
D(k, t) = Tr Fk (t)Fk (0)
X
z kj
F(k) =
e
j
1
10
⌧
relaxation of correlations
3. Towards a theory of quantum metastability
→ in A+B (closed) = difficult
B
A
{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}
→ in A only (open) = easier by analogy with classical
quantum master equation: {Lindblad, Gorini-et-al}
Xî
¶
1
†
[H, ] +
J
J
J
t =
2
stochastic
wave function:
{Dalibard-et-al, Belavkin}
He
|
t0 i
|
Jk1 |
t1 i
|
He
Jk2 |
t1 i
t2 i
|
†
J ,
t3 i
©ó
{Gaveau-Shulman,
Bovier et al.}
⌘ L( )
He = H
X
2
!
J
†
t2 i
E.g. 2-level system at T = 0:
|1
H=
L1J =
⇥⇤
p
⇥
quantum jump trajectory
⇥
|0
Note: if ρ = diagonal, H = 0 and J = rank 1 (e.g. J = |C0 ihC| ) then QME → classical ME
J
3. Towards a theory of quantum metastability
→ in A+B (closed) = difficult
B
A
{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}
→ in A only (open) = easier by analogy with classical
quantum master equation: {Lindblad, Gorini-et-al}
Xî
¶
1
†
[H, ] +
J
J
J
t =
2
†
J ,
©ó
{Gaveau-Shulman,
Bovier et al.}
⌘ L( )
Examples of metastability in open quantum dynamics
(i) 3-level system (electron shelving, blinking q. dot ...)
|1
⇥1
1
⇥2
|0
|2
H = 1 |0ih1| +
p
J=
1 |0ih1|
2 |0ih2| + c.c.
2
⌧
1,
1
intermittency
{Hegerfeldt-Plenio}
3. Towards a theory of quantum metastability
{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}
→ in A+B (closed) = difficult
A
B
→ in A only (open) = easier by analogy with classical
quantum master equation: {Lindblad, Gorini-et-al}
Xî
¶
1
†
[H, ] +
J
J
J
t =
2
†
J ,
©ó
{Gaveau-Shulman,
Bovier et al.}
⌘ L( )
Examples of metastability in open quantum dynamics
(ii) dissipative transverse field Ising model
H=
J =
p
+V
X
z
h ,ji
z
j
stationary state transition
· · ··#)
· · #)
!
(!!
(!!
· · ··!)
· · !)
ferro (##(##
→!
para
space
X
{Ates-Olmos-JPG-Lesanovsky PRA 2012}
time
intermittency
3. Towards a theory of quantum metastability
{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}
→ in A+B (closed) = difficult
A
B
→ in A only (open) = easier by analogy with classical
quantum master equation: {Lindblad, Gorini-et-al}
Xî
¶
1
†
[H, ] +
J
J
J
t =
2
metastable states from spectrum :
Re( )
small
{ 0
gap
2
..
.
3
ss
L(Rk ) =
1
= 0 , L1 = 1 , R1 =
=
⇡
J ,
†
R
,
L
(Lk ) =
k k
(t) = et L
R2
†
k Lk
⌘ L( )
, Tr(Lk R ) =
k
but Rk>1 < 0
0
ss +
ss
ss
©ó
{Gaveau-Shulman,
Bovier et al.}
+
m
X
k=2
m
X
k=2
et
k
Rk Tr(Lk
Rk Tr(Lk
0)
0) + · · ·
O(t
m)
metastable state ∈ manifold dim=m-1
dimensional reduction (dim H)2 ! (m 1)
3. Towards a theory of quantum metastability
→ in A+B (closed) = difficult
A
B
{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}
→ in A only (open) = easier by analogy with classical
quantum master equation: {Lindblad, Gorini-et-al}
Xî
¶
1
†
[H, ] +
J
J
J
t =
2
metastable states from spectrum :
Re( )
small
{ 0
gap
2
..
.
3
ss
R2
L(Rk ) =
1
†
J ,
†
R
,
L
(Lk ) =
k k
= 0 , L1 = 1 , R1 =
ss
©ó
k Lk
{Gaveau-Shulman,
Bovier et al.}
⌘ L( )
, Tr(Lk R ) =
but Rk>1 < 0
k
3. Towards a theory of quantum metastability
{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}
|1
⇥1
⇥2
1
|2
˜1 =
˜2 =
|0
ss
m x
+
c
R2
ss
2
min
+
c
R2
ss
2
m x,min
c2
R2
0
⌦1
= max/min evals of L2
2⌦1
3⌦1
⇢˜1
⇢ss
distance to metastable
manifold
⇢˜21.0
1
Á
10
3Ê
2
10
10
104
105
W
⌦1t1 t
0.81.0
0.8
W êW =1ê25
0.6
C(t)
metastable
manifold
is 1d
simplex
1
W22êW1=1ê25
W22êW
êW
W
1=1ê50
1=1ê50
W
1=1ê100
W22êW
êW
1=1ê100
0.6
0.40.4
0.20.2
1
1
Á
Û
·
Á 102
·
10 Û
10
autocorrelation of |1ih1|
10
2
10
Ê
3
‡ 4Ú
Ê
10
3
10
W11t t
‡105 4 ⌦
Ú
5 W1t
10
10
|2ih2|
3. Towards a theory of quantum metastability
{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}
|1
⇥1
⇥2
1
|2
˜1 =
˜2 =
|0
m x,min
c2
ss
Wss ⇡ R2
L
A
..
.
K
eigenstates
of
Ls
sA
2)
s
0
⌦1
2⌦1
3⌦1
0
B
B
B
B
@
ss
O(
R2
= max/min evals of L2
Connection to s-ensemble?
K
ss
m x
+
c
R2
ss
2
min
+
c
R2
ss
2
1
L2
0
s J† J
s J† J
..
.
2
..
.
A
A
=
=
=
=
···
1
C
C
C
··· C
.. A
.
˜1 + O(t
˜1 + O(t
˜2 + O(t
˜2 + O(t
almost
degenerate P.T.
A ,
2)
2)
2)
2)
SUMMARY
!
!
1. Slow relaxation through dynamical constraints
Constrained hopping, fast-slow crossover at RK point
!
!
2. Signatures of MBL dynamics in absence of disorder
Quantum East model, thermal-MBL (quasi-MBL) transition at RK point
!
!
3. Theory of metastability of open quantum systems
Metastable states and effective dynamics from spectrum of dynamical generator
Reduction to low-dimensional metastable manifold