Structure of nonequilibrium quantum
phase transitions
Jonas Larson
with Julia Hannukainen
Stockholm University
Uppsala 16/3-2017
1
Motivation
✦ Equilibrium phase transitions (continuous):
!
• Universal - few critical exponents determine the physics (microscopic
details irrelevant).
• Spontaneous symmetry breaking - in one phase the state does not
possess the same symmetry as the full Hamiltonian.
• Excitations - energy gap closes at the critical point. Excitations in
‘symmetry broken phase’ either gapped (Higgs) or continuous (Goldstone).
• Mermin-Wagner theorem - the type of symmetry + the dimensionality
determine which type of transition that is allowed.
!
✦ Non-equilibrium phase transitions: (Especially) cold atom
experiments. Drive them and engineer desired coupling to reservoir !
non-trivial steady states.
!
✦ How do the above general results translate to these new phase
transitions?
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
3
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
4
Phase transitions!
Phase transition
✦ Action S[ ¯, ] , giving partition function
Z
!
¯
Z = D( ¯, ) e S[ ,
]
!
Mean-field solution: minimizing the action for order parameter .
✦ PT ! change in order parameter
<
c
S
>
c
Sudden jump - first
order PT.
Phase transitions!
Phase transition
✦ Action S[ ¯, ] , giving partition function
Z
!
¯
Z = D( ¯, ) e S[ ,
]
!
Mean-field solution: minimizing the action for order parameter .
✦ Symmetries S[g ¯, g ] = S[ ¯, ].
!
S[
] = S[ ]
Z2 symmetry
< c
S
>
S[ei
c
] = S[ ]
U (1) symmetry
um
t
n
Q ua
Phase transitions!
2nd order phase transition
✦ Continuous phase transition.
!
✦ Exists a symmetry [Û , Ĥ] = 0 .
!
✦ Quantum mechanics: Û and Ĥ common eigenbasis - energy
eigenstates well defined symmetry.
!
✦ Thermodynamic limit: spontaneous symmetry breaking - ground
state | 0 ( )i does not have a defined symmetry!
!
✦ “Potential barrier” becomes infinite
!
✦ If
Û continuous
!
Û discontinuous !
! degeneracy.
Goldstone (gapless) excitations.
Higgs (gapped) excitations.
Phase transitions!
um
t
n
Q ua
2nd order phase transition
✦ Ground state energy E0 ( ) continuous,
system parameter.
!
✦ Derivatives @ n E0 ( ) can be discontinuous for some critical coupling
!
c.
✦ When and why?
• Thermodynamic limit - system ‘size’ infinite.
!
• Competing terms supporting different properties
!
!
Ĥ = Ĥ1 + Ĥ2 ,
✦ Roughly,
i
Ĥ1 , Ĥ2 6= 0
< 1 ground state properties from Ĥ1,
• At
=
•|
)i and
0(
h
c
⌘ 1 spectrum gapless.
‘non-analytic’ at
> 1 from Ĥ2.
c.
• The PT driven by quantum fluctuations at T = 0 (classical PT, thermal
fluctuations).
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
9
Phase transitions!
Phase transition (2nd) - scale invariance
At the critical point
!
!
!
!
!
!
!
!
!
!
c
Hclass =
X
si sj
hiji
System the same independent of “zooming” - length scale diverges!!
h
X
i
si
Phase transitions!
Phase transition (2nd) - universality
✦ Systematic method (renormalization group) - eliminates short length
scales (‘high energies’).
!
✦ Effective low energy model - microscopic details irrelevant.
!
✦ Models belong to different ‘classes’ - universality (depend on
macroscopic properties like symmetries).
!
✦ Critical regime:
!
!
!
!
⇠/|
c|
⌧ /|
E/|
c|
...
c|
⌫
z⌫
,
Characteristic length
,
Characteristic time
,
Gap closing
Critical exponents ⌫, , z, ... , different universality classes.
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
12
Open quantum systems!
Density operator
!
!
!
!
!
!
!
!
!
System S
System B
H = HA ⌦ H B
System A
✦ Only access to A, local observables ÔA.
!
✦ What is the state of subsystem A?
| i=
X
ij
cij |
A
i i
⌦|
B
j i
Open quantum systems!
Density operator
!
!
!
!
!
!
!
!
!
hÔA i
=
=
=
=
=
P
P
⇤
B
A
A
B
ijnm cij cnm h i |h j |ÔA | n i| m i
⇤
A
A
ijn cij cni h j |ÔA | n i
P P
l
⇤
A
A
ijn cij cni h j |ÔA |lihl| n i
P P
⇤
A
A
ijn cij cni hl| n ih j |ÔA |li
l
P
⇢A ÔA |li
l hl|ˆ
Reduced density operator ⇢ˆA =
!
State of system S: ⇢ˆ ,
X
ijn
h
= TrA ⇢ˆA ÔA
c⇤ij cni |
A
A
n ih j |
i
= TrB [ˆ
⇢] , with ⇢ˆ = | ih | .
state of subsystem A: ⇢ˆA . In general ⇢ˆ 6= ⇢ˆA ⌦ ⇢ˆB.
Open quantum systems!
Density operator
✦ In general ⇢ˆ 6= | ih |.
!
Bl
✦ Physical:
h
c
o
sp
re
e
h
Tr [ˆ
⇢] = 1,
!
!
!
⇢ˆ† = ⇢ˆ,
||ˆ
⇢|| 0
✦ If ⇢ˆ = | ih | (pure state), two-level state | i = cos ✓|0i + sin ✓ei' |1i.
!
✦ Bloch vector R̄ = (x, y, z),
!
↵ = Tr [ˆ↵ ⇢ˆ], with ˆ↵ Pauli matrices.
Pure state |R̄| = 1 , in general |R̄| < 1 .
✦ Random multi-qubit state ⇢ˆ
Majority of states not pure!
! reduced single qubit state ⇢ˆ1 ⇡ I/2 .
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
16
Open quantum systems!
Density operator
Combined system
! Ĥ = ĤS + ĤB + ĤSB
!
e iĤt
⇢ˆSB (0)
⇢ˆSB (t)
!
!
TrB
TrB
!
!
⇢ˆS (t)
⇢ˆS!(0)
?
!
t
!
Bath
System
Generator “?”, operators defined on HS . Generally not possible!!
!
Open quantum systems!
Density operator
✦ Weak coupling, “big” bath
1. Bath time-scale short: no memory, Markovian.
2. System negligible influence on bath, Born.
!
!
!
3. Rotating wave approximation.
h
i
!
@t ⇢ˆ(t) = i ⇢ˆ(t), ĤS + L̂ [ˆ
⇢(t)] ,
⌘
X ⇣
†
†
†
L̂ [ˆ
⇢(t)] =
gi 2Âi ⇢ˆ(t)Âi Âi Âi ⇢ˆ(t) ⇢ˆ(t)Âi Âi
i
Lindblad master equation
!
Âi Lindblad jump operators. Non-unitary evolution, ⇢ˆ(t) not generally pure.
Phase transitions in open quantum systems!
✦ Quantum PT’s, non-analyticity in ground state | 0 ( )iat critical
coupling c .!
!
✦ Transition due to quantum fluctuations.!
!
✦ Open quantum system!
!
!
!
!
!
h
i
@t ⇢ˆ(t) = i ⇢ˆ(t), ĤS + L̂ [ˆ
⇢(t)] ,
L̂ [ˆ
⇢(t)] =
X
i
gi
⇣
2Âi ⇢ˆ(t)†i
†i Âi ⇢ˆ(t)
✦ No ground state, no energy spectrum!!!
⇢ˆ(t)†i Âi
⌘
!
What do we mean by a PT here, what is the
relevant state?
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
20
Phase transitions in open quantum systems!
✦ One (obvious) physically relevant state is the steady state!
!
h
i
@t ⇢ˆss (t) = 0 ! i ⇢ˆss , ĤS + L̂[ˆ
⇢ss ] = 0
!
!
✦ May be non-equilibrium, but time-independent. !
!
✦ Closed system, all energy eigenstates also steady states.!
!
✦ If:!
1. [Âi , ĤS ] = 0, 8Âi , Âi hermitian, energy eigenstates also steady states.!
2. ĤS critical and L̂ [| 0 ( < c )ih 0 ( < c )|] = 0, the environment is
expected to support the symmetric phase.!
!
✦ If neither of the two
!
! new physics???!
✦ Phase transition if ⇢ˆss non-analytic for some
c.
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
22
New type of phase transition!
Model
✦ Simplest (non-trivial) scenario
• @t ⇢ˆ = i[ˆ
⇢, Ĥs ]
⇢†
• @t ⇢ˆ = (2ˆ
trivial
† ˆ
⇢
⇢, Ĥs ] + (2ˆ
⇢†
• @t ⇢ˆ = i[ˆ
⇢ˆÂ† Â)
† ˆ
⇢
trivial
⇢ˆÂ† Â)
critical
✦ One such model (which is also solvable!)
!
[Ŝi , Ŝj ] = i"ijk Ŝk
Large spin-S (preserved)
ĤS = ! Ŝx
Coherent drive
 = Ŝ
Spontaneous decay to state |S, Si
!
New type of phase transition!
Model
✦ Equation to solve
!
!
@ ⇢ˆ = i[ˆ
⇢, ! Ŝx ] + (2Ŝ ⇢ˆŜ+
Ŝ+ Ŝ ⇢ˆ
⇢ˆŜ+ Ŝ )
✦ Limiting cases
!
!
!
!

= 0 ! |S, Six
!
Hamiltonian dominates
!
= 0 ! |S, Siz

Lindbladian dominates
✦ Phase transition somewhere between?
New type of phase transition!
Mean-field solution
✦ Mean-field (normal ordering): S↵ = hŜ↵ i, hŜ↵ Ŝ i = hŜ↵ ihŜ i.

!
Ṡx = 2 Sx Sz ,
S
!
⇣ 
⌘
!
Ṡy = Sz 2 Sy ! ,
S
!
!
!
Ṡz = !Sy
2
 2
Sx + Sy2
S
✦ Conserved spin: S 2 = Sx2 + Sy2 + Sz2 .
!
✦ Steady state solutions
(Sx , Sy , Sz ) =
±
r
1
=
!
2
!
1 1
, ,0 ,
⇣
p
(Sx , Sy , Sz ) = 0, , ± 1
⌘
.
New type of phase transition!
1
0.5
S ( )/S,
(a)
= x,y,z
Mean-field solution
Classical steady state
solutions: x (blue), y (green),
z (black).
!
!
0
Blue curve: Hopf (like)
bifurcation (stability, purely
imaginary eigenvalues).
0.5
1
0
!
0.5
1
1.5
2
Black curve: Strange
bifurcation (lower branch
stable, upper unstable).
New type of phase transition!
Mean-field solution
✦ Understanding the bifurcation?
!
✦ Rewrite eom’s in canonical variables , z = cos ✓
ż =
2 1
˙=
!p
z
2
z
1
z2
+!
cos
p
1
z 2 sin ,
✦ Not possible to assign a (local)
’potential’ for these
!
Sz
!
!
!
0
✦ Phase space = sphere, think of it as
one attractive and one repulsive fixed
point.
!
✦ Not possible on a plane.
0
S
y
0
Sx
New type of phase transition!
Universality - mean-field
✦ Universality - critical exponents.
!
• “Magnetisation”
!
Sz / |
|1/2
c
(Quantum ⌫ = 1/2)
!
⌫ = 1/2.
✦ ‘Critical slowing down’. Linear stability around the mean-field solution eigenvalues gives relaxation time
!
!
!
T /|
c
✦ Seems to be universal!
|
1/2
,
= 1/2
New type of phase transition!
Universality - quantum
✦ Quantum treatment. Analytical solution
!
!
!
⇢ˆss
2S
1 X ⇤
=
(g )
D n,m=0
m
g
n
n
Ŝ m Ŝ+
,
g = i!S/
✦ Expectations hÔi = Tr[Ô⇢ˆss ]. Truncate for some S.
!
✦ For continuous PT we demand continuous expectations of ‘local’
operators
X
!
ôi single qubit operator
Ô =
ôi
i
New type of phase transition!
Universality - quantum
✦ Magnetisation (local) critical exponent the same as for mean-field.
New type of phase transition!
Universality - quantum
✦ hŜz i/S ! 0, !/ ! 1. Hamiltonian part dominates.
!
✦ hŜz i/S ! 1, !/ ! 0. Pure state, fluctuations hŜx2 i/S 2 ! 0, !/ ! 0
!
✦ In general, quantum fluctuations relative system size O(S 1 ) .
!
✦ But S↵2 /S 2 ! 1/3, !/ ! 1 ! Reservoir-induced-fluctuations.
System size
green
…
blue
( S = 200)
New type of phase transition!
Universality - quantum
✦ Fluctuations reflected in the phase space distribution.
Q(z) = hz|ˆ
⇢ss |zi
Husimi Q-function
Magnetized phase
Incoherent phase
New type of phase transition!
Symmetries
✦ Symmetries.
!
• Closed case. [Â, Ĥ] = 0 $ Ĥ = Û Ĥ Û
⇢] = LÛ L̂Û 1 [ˆ
⇢]
• Open case. @ ⇢ˆ = LL̂ [ˆ
!
!
1
 conserved.
generally no conserved quantity.
✦ Guess Ŝx ! Ŝx , Ŝy ! Ŝy , Ŝz ! Ŝz
⇣
!
@ ⇢ˆ = i[ˆ
⇢, ! Ŝx ] +  2Ŝ ⇢ˆŜ+ Ŝ+ Ŝ ⇢ˆ
!
!
!
@ ⇢ˆ =
⇣
!
i[ˆ
⇢, ! Ŝx ] +  2Ŝ+ ⇢ˆŜ
Ŝ Ŝ+ ⇢ˆ
⇢ˆŜ+ Ŝ
⌘
⇢ˆŜ Ŝ+
⌘
Dual symmetry! Flip spectrum + “step up” instead of “step down”.
New type of phase transition
Second order phase transition with no
symmetry breaking!!
!
!
✦ Corresponding Hamiltonian system (drive + ‘internal’ energy)
!
!
!
!
Ĥ = ! Ŝx + Ŝz
clearly not critical.
!
✦ Interplay between ‘bath noise’ and unitary evolution.
New type of phase transition!
Quantum or classical
✦ Is it a quantum phase transition?
!
• Single degree of freedom.
Quantum fluctuations vanishingly
small in thermodynamic limit.
!
• Fluctuations due to coupling to
reservoir.
!
• Quantum critical region signatures of “quantum” survives
finite temperatures in the vicinity of
a critical point.
New type of phase transition!
Quantum or classical
✦ “Quantum”
!
• Coherence: Purity P ⌘ Tr[ˆ
⇢2 ] > 0 (thermodynamic limit).
⇢) between pair of qubits.
• Entanglement: Negativity N (ˆ
!Thermodynamic limit:
(a), full system purity.
Jump!! Extensive
operator, not local.
!
(b) Single qubit
purity. Pure state in
magnetised phase!
New type of phase transition!
Quantum or classical
✦ “Quantum”
!
• Coherence: Purity P ⌘ Tr[ˆ
⇢2 ] > 0 (thermodynamic limit).
⇢) between pair of qubits.
• Entanglement: Negativity N (ˆ
!Thermodynamic limit:
Scaled negativity
!
!
max(N ) = 1
Entanglement
vanishes ( ⇠ S
!
0.9
).
Peak at critical point.
New type of phase transition!
Quantum or classical
✦ “Quantum”
!
• Coherence: Purity P ⌘ Tr[ˆ
⇢2 ] > 0 (thermodynamic limit).
⇢) between pair of qubits.
• Entanglement: Negativity N (ˆ
!Thermodynamic limit:
Spin squeezing for
different system
sizes.
!
Entanglement
witness. Global
entanglement.
!
‘Entanglement
sharing’.
Outlook
1. (Equilibrium) Quantum phase transitions
• Symmetry breaking.
• Scale invariance and universality.
2. Open quantum systems
• Composite systems, density operators.
• Lindblad master equation.
3. Open quantum phase transitions
4. A new type of phase transition
5. Prospects
39
Phase transitions in open quantum systems!
✦ Found a model possessing an open PT, seemingly second order but
lacks any symmetry breaking ! New type.!
!
Open questions!
!
• Are these PT’s universal? Scale invariance? New classes?!
!
• Models driven more strongly by quantum fluctuations, length scale
(lattice models). !
!
• Gap closening of the Liouvillian, universal?!
!
• Is there a Mermin-Wagner theorem for open QPT’s?!
!
• Kibble-Zurek mechanism?
Thank you!