Juan P. Garrahan (Physics & Astronomy, University of Nottingham)

Ideas About 
Quantum Metastability

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):no degenerate E gaps → dephasing in E basis → diagonal ensemble {Short, Popescu, Linden, Reimann, Eisert, Goldstein, many others}

�(t) �! � = limt!�

1

t

Z t

0Ut0�0U

†t0dt0 |hA(t)i � Tr(A�)| small

{Lesanovsky-Olmos-JPG, PRL 2010}

H = �LX

k=1��k + �

LX

k=1nk + V

LX

k 6=m

nmnk|m� k|6

E.g. (Rydbergs)

Generic Q. many-body systems “equilibrate” (become stationary):no degenerate E gaps → dephasing in E basis → diagonal ensemble

�(t) �! � = limt!�

1

t

Z t

0Ut0�0U

†t0dt0 |hA(t)i � Tr(A�)| small

Most Q. many-body systems also “thermalise”: {Deutsch, Srednicki, Goldstein, Eisert, Rigol, Gogolin, 
many others}

A

B

closed

�th = e��H

�A = TrB |�ih�| �! �A = TrB �th

� set by h�0|H|�0i

thermalisation ↔ ergodicity (independence of initial conditions)eigenstate thermalisation hypothesis (ETH):

hE|Oloc�l|Ei = smooth F(E)

{Short, Popescu, Linden, Reimann, Eisert, Goldstein, many others}

Ergodic & thermal → non-ergodic & many-body localised (MBL){Basko-Aleiner-Altshuler, Huse+, Prosen+,

Abanin+, Moore+, Altman+, many others}

Cf. Anderson localisation, but with interactions

(excited states = Area law)MBL = breakdown of ETH

(rnd h small = thermal, rnd h large = MBL)MBL transition driven by quenched disorder

{Schreiber et al 2015}

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?

- Slowdown & eventual arrest

- Timescales not divergent

- No disorder

e.g. 
50:50 L-J mixture 


{Hedges 2009}

t� ⇥� t � ⇥� t� ⇥�

Classical glasses

Relaxation is intermittent and spatially heterogeneous

Due to (effective) local constraints on dynamics {JPG-Chandler, PRL 2002}

1. Quantum slow relaxation due to dynamical constraints

{van Horssen-JPG, 2016}

H = �X

h�ji(1� nkn�)n

�(�+� ��j + �

+j ��� )� (1� �)î

n�(1� nj)� nj(1� n�)ó

o

constraint (“1-vacancy assisted” or 1-TLG)

Rokhsar-Kivelson pt: � =1

2�! H = � class. Master op. W �! �t |Pi =W|Pi

Away from RK: � 6=1

2�! Large-deviations:

� >

1

2

åactive

� <

1

2

åinactive

Steric (excluded volume) constraint on hopping:

classical = constrained lattice gas{Kob-Andersen, Jackle+}

k

l

i j

1. Quantum slow relaxation due to dynamical constraints

H = �X

h�ji(1� nkn�)n

�(�+� ��j + �

+j ��� )� (1� �)î

n�(1� nj)� nj(1� n�)ó

o

|�(t)i = e��tH|�(0)i … …1

2

3

4

L-1

L

(1)

(2)

(L-1)

(L)

…

1234 L

L-1� = 0.8� = 0.2

metastability = memory of 
initial conditions

{van Horssen-JPG, 2016}

N = 24M = 20

1. Quantum slow relaxation due to dynamical constraints

H = �X

h�ji(1� nkn�)n

�(�+� ��j + �

+j ��� )� (1� �)î

n�(1� nj)� nj(1� n�)ó

o

|�(t)i = e��tH|�(0)i … …1

2

3

4

L-1

L

(1)

(2)

(L-1)

(L)

…

1234 L

L-1

λ

0 0.5 1

E0

-8

-6

-4

-2

0

2

l = 6l = 7l = 8l = 9l = 10l = 11l = 12

λ

0 0.5 1

dE0/d

λ

-14

-12

-10

-8

-6

-4

-2

λ

0 0.5 1

gap(H

)

0

0.5

1

1.5

2

1st order singularity in g.s. cf. classical active-inactive transitions

s

0 0.5 1

τΦ

102

104

106

108

1010

l = 7

l = 8

l = 9

l = 10

l = 11

l = 12

l = 13�

�

T =�

N = 24M = 20

change at �RK ⇡

1

2

{van Horssen-JPG, 2016}

s0 inactive transition ≡ quantum phase transition in g.s. of H0.00

0.04

0.08

θ(s)

-0.2 -0.1 0 0.1 0.2s

0

0.2

0.4

0.6

K(s)

0.00

0.04

0.08

θ(s)

-0.2 -0.1 0 0.1 0.2s

0

0.2

0.4

0.6

K(s)

0.00

0.04

0.08

θ(s)

-0.2 -0.1 0 0.1 0.2s

0

0.2

0.4

0.6

K(s)

N!�

{JPG-Jack-Lecomte-Pitard-van Duijvendijk-van Wijland
PRL 2007}

2. Dynamics of MBL in a quantum glass (w/o disorder)

{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}

H = �X

�n�+1Äe�s��� � 1ä

e�s = �/(1� �) �! sRK = 0

Classical East glass model :(T =�) 11 ⇄ 01 10 ⇄ 00{Jackle+, Sollich-Evans, Aldous-Diaconis, JPG-Chandler, 
Chleboun-Faggionato-Martinelli, Blondel-Toninelli, 
many others}

H = �X

�n�+1Äe�s��� � 1ä

e�s = �/(1� �) �! sRK = 0|�(t)i = e��tH|�(0)i

Signatures of MBL dynamics for s>0 (inactive):

s

-2 0 2

Ĩ

0

0.2

0.4

0.6

0.8

1

N = 8N = 10N = 12N = 14

(a)

(b)

t

log t

(c)

Propagation of excitations

s

-2 0 2

Ĩ

0

0.2

0.4

0.6

0.8

1

N = 8N = 10N = 12N = 14

(a)

(b)

t

log t

(c)

s

-2 0 2

Ĩ

0

0.2

0.4

0.6

0.8

1

N = 8N = 10N = 12N = 14

(a)

(b)

t

log t

(c)

2. Dynamics of MBL in a quantum glass (w/o disorder)

{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}

H = �X

�n�+1Äe�s��� � 1ä

e�s = �/(1� �) �! sRK = 0|�(t)i = e��tH|�(0)i

Signatures of MBL dynamics for s>0 (inactive):

time averaged magnetisation 
dependence on initial conditions - ETH (active) v. no-ETH (inactive)

E

0 1 2 3

-10

-5

0

5

10

⟨M⟩mc⟨Mfinal⟩E

tJ

100

102

104

106

108

M̄w(t)

-2

-1

0

1

2 s = 1

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

s

-3 -2 -1 0 1 2 3

VarM̃

final(normalised)

0

0.2

0.4

0.6

0.8N = 8N = 10N = 12N = 14

E

0 1 2 3

-10

-5

0

5

10

⟨M⟩mc⟨Mfinal⟩E

(a) (b)

(c)

s = 1(d)

Es

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

Var

¯ Mfinal(normalised)

E

0 1 2 3

-10

-5

0

5

10

⟨M⟩mc⟨Mfinal⟩E

tJ

100

102

104

106

108

M̄w(t)

-2

-1

0

1

2 s = 1

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

s

-3 -2 -1 0 1 2 3

VarM̃

final(normalised

)

0

0.2

0.4

0.6

0.8N = 8N = 10N = 12N = 14

E

0 1 2 3

-10

-5

0

5

10

⟨M⟩mc⟨Mfinal⟩E

(a) (b)

(c)

s = 1(d)

Es

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

tJ

10-1

100

101

102

103

M̄w(t)

-4

-2

0

2

4 s = −1

w = 1w = 2w = 3w = 4w = 5w = 6w = 7

Var

¯ Mfinal(normalised)

2. Dynamics of MBL in a quantum glass (w/o disorder)

{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}

H = �X

�n�+1Äe�s��� � 1ä

e�s = �/(1� �) �! sRK = 0|�(t)i = e��tH|�(0)i

Signatures of MBL dynamics for s>0 (inactive):

log t growth of

entanglement entropy

tJ

100

102

104

106

108

S(t)

0

1

2

3

4

N = 6

N = 8

N = 10

N = 12

N = 14∝ log(t)

s = 1

N

6 10 14

τ

100

102

104

106

s = −1

s = 0

s = 1

tJ10

010

210

410

610

8

Sw(t)

0

1

2

3

4

s = 1

w = 2w = 4w = 6

tJ100

102

Sw(t)

0

1

2

3

4

s = −1

w = 2w = 4w = 6

tJ

100

102

104

106

108

S(t)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5s = -1s = 0s = 1

(a)

(b)

(c)

s-2 0 2

S̃/N

0.24

0.26

0.28

0.3

0.32

(d)

S̄(t)

S̄(t)

⌧

S̄eq/N

S̄w(t)

S̄w(t)

{Serbyn-Papic-Abanin, Moore+, others}

2. Dynamics of MBL in a quantum glass (w/o disorder)

{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}

H = �X

�n�+1Äe�s��� � 1ä

e�s = �/(1� �) �! sRK = 0|�(t)i = e��tH|�(0)i

Signatures of MBL dynamics for s>0 (inactive):

2π/k2 4 6 8 10 12

τ

100

101

102

103

104

105

s = −1

s = 0.5

s = 1

N = 8N = 10N = 12

0

0.5

1s = −1D(k; t)(N = 12)

tJ10

010

210

410

610

80

0.5

1s = 1D(k; t)(N = 12)

k

e�1

e�1

⌧

(a)

(b) (c)

D(k, t) = Tr Fk(t)Fk(0)

F(k) =X

j�zj e

�kj

relaxation of correlations

MBL or quasi-MBL 
{cf. Yao-Laumann-Cirac-Lukin-Moore, 


De Roeck-Huveneers}

�(k) ⇠ e1/k�

� ⇠ � log t ! �(k) ⇠ e1/k�

2. Dynamics of MBL in a quantum glass (w/o disorder)

{van Horssen-Levi-JPG, PRB 92, 100305(R) 2015}

3. Towards a theory of quantum metastability

{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}

Note: if ρ = diagonal, H = 0 and = rank 1 (e.g. ) then QME → classical MEJ� J� = |C0ihC|

E.g. 2-level system at T = 0:

quantum jump trajectory

H = � ⇥⇤

L1 =⇥�⇥�

|1�

�

|0�

�J =p���

{Dalibard-et-al, Belavkin}

stochastic wave function:

He� = H�

�

2

X

�J�†J�

!|�t3 i|�t2 i|�t1 i

|�t0 i

He� He�

Jk1 |�t1 i Jk2 |�t2 i

AB

→ in A+B (closed) = difficult

→ in A only (open) = easier by analogy with classical {Gaveau-Shulman, Bovier et al.}{Lindblad, Gorini-et-al}quantum master equation:

�t� = ��[H,�] +X

�

îJ� � J�† � 12¶J�†J�,�©ó⌘ L(�)

3. Towards a theory of quantum metastability

{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}

Examples of metastability in open quantum dynamics

AB

→ in A+B (closed) = difficult

→ in A only (open) = easier by analogy with classical {Gaveau-Shulman, Bovier et al.}{Lindblad, Gorini-et-al}quantum master equation:

�t� = ��[H,�] +X

�

îJ� � J�† � 12¶J�†J�,�©ó⌘ L(�)

(i) 3-level system (electron shelving, blinking q. dot ...)

|0�

|1� |2�

⇥1⇥2�1

�2 ⌧ �1, �1H = �1|0ih1|+�2|0ih2|+ c.c.J =p�1 |0ih1|

intermittency

{Hegerfeldt-Plenio}

3. Towards a theory of quantum metastability

{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}

intermittency

spac

e

time

(ii) dissipative transverse field Ising model

H = �X

���� + VX

h�,ji�z� �

zj

J� =p����

stationary state transition

(## · · · #) �! (!! · · ·!)(## · · · #) �! (!! · · ·!)ferro → para

Examples of metastability in open quantum dynamics

AB

→ in A+B (closed) = difficult

→ in A only (open) = easier by analogy with classical {Gaveau-Shulman, Bovier et al.}{Lindblad, Gorini-et-al}quantum master equation:

�t� = ��[H,�] +X

�

îJ� � J�† � 12¶J�†J�,�©ó⌘ L(�)

{Ates-Olmos-JPG-Lesanovsky PRA 2012}

metastable states from spectrum :

�1 = 0 , L1 = 1 , R1 = �ss but Rk>1 < 0�ss

R2

Re(�)

0�2

�3...

small

gap{

3. Towards a theory of quantum metastability

{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}

L(Rk) = �kRk , L†(Lk) = �kLk , Tr(LkR�) = �k�

AB

→ in A+B (closed) = difficult

→ in A only (open) = easier by analogy with classical {Gaveau-Shulman, Bovier et al.}{Lindblad, Gorini-et-al}quantum master equation:

�t� = ��[H,�] +X

�

îJ� � J�† � 12¶J�†J�,�©ó⌘ L(�)

�(t) =etL�0

=�ss +mX

k=2et�kRkTr(Lk�0) + · · ·

⇡�ss +mX

k=2RkTr(Lk�0) O(t�m)

metastable state ∈ manifold dim=m-1 (dimH)2 ! (m� 1)dimensional reduction

metastable states from spectrum :

�1 = 0 , L1 = 1 , R1 = �ss but Rk>1 < 0�ss

R2

Re(�)

0�2

�3...

small

gap{

3. Towards a theory of quantum metastability

{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}

L(Rk) = �kRk , L†(Lk) = �kLk , Tr(LkR�) = �k�

AB

→ in A+B (closed) = difficult

→ in A only (open) = easier by analogy with classical {Gaveau-Shulman, Bovier et al.}{Lindblad, Gorini-et-al}quantum master equation:

�t� = ��[H,�] +X

�

îJ� � J�† � 12¶J�†J�,�©ó⌘ L(�)

|0�

|1� |2�

⇥1⇥2�1

0

�⌦1

�2⌦1

�3⌦1

�ssR2

�̃1 = �ss + cm�x2 R2

�̃2 = �ss + cmin2 R2

cm�x,min2 =max/min evals of L2

C(t)

1 10 102 103 104 105 W1t

0.2

0.4

0.6

0.8

1.0

· ‡Û ÚÁ Ê

W2êW1=1ê100W2êW1=1ê50W2êW1=1ê25

⌦1t1 10 102 103 104 105 W1t

0.2

0.4

0.6

0.8

1.0

· ‡Û ÚÁ Ê

W2êW1=1ê100W2êW1=1ê50W2êW1=1ê25 |1ih1|� |2ih2|autocorrelation of

⇢̃1

⇢̃2

⇢ss

⌦1t1 10 102 103 104 105 W1tÁ Ê

distance to metastable manifold

metastable

manifold

is 1d

simplex

3. Towards a theory of quantum metastability

{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}

|0�

|1� |2�

⇥1⇥2�1

0

�⌦1

�2⌦1

�3⌦1

�ssR2

�̃1 = �ss + cm�x2 R2

�̃2 = �ss + cmin2 R2

cm�x,min2 =max/min evals of L2

3. Towards a theory of quantum metastability

{Macieszczak-Guta-Lesanovsky-JPG, arXiv:1512.05801}

sA s�

K

χ

s

K

ss

K

��

�A

O(�2)

eigenstates of

Connection to s-ensemble?

1 L2

Ws ⇡

�ss

R2...

0BBBB@

0 �s J†J · · ·

�s J†J �2 · · ·...

.... . .

1CCCCA

�A = �̃1 +O(t�2)�� = �̃2 +O(t�2)�A = �̃1 +O(t�2)�� = �̃2 +O(t�2)

almost degenerate P.T.

�A , ��

Ls

Ls

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