Fate of Kosterlitz-Thouless Physics in Driven Open Quantum Systems - - PowerPoint PPT Presentation

Functional Renormalization - from quantum gravity and dark energy to ultracold atoms and condensed matter March 07-10, 2017 IWH Heidelberg, Germany

Sebastian Diehl

Institute for Theoretical Physics, University of Cologne

collaboration:

Fate of Kosterlitz-Thouless Physics in Driven Open Quantum Systems

:

• L. Sieberer, E. Altman (Berkeley)
• G. Wachtel (Toronto)
• L. He (Cologne)

Universality in low dimensions: 2D

∼ e−r/ξ

• superfluidity

ρs 6= 0 ρs = 0

• KT transition: unbinding of vortex-antivortex pairs

hφ(r)φ∗(0)i ⇠ r−α

… also for out-of-equilibrium systems? … new universal phenomena tied to non-equilibrium?

• correlations

low temperature high temperature

• continuous phase rotations:

Kasprzak et al., Nature 2006

Experimental Platform: Exciton-Polariton Systems

k E excitons photons Imamoglu et al., PRA 1996 pump relaxation lower polaritons loss

• phenomenological description: stochastic driven-dissipative Gross-Pitaevskii-Eq

hζ⇤(t, x)ζ(t0, x0)i = γδ(t t0)δ(x x0)

microscopic derivation and linear fluctuation analysis: Szymanska, Keeling, Littlewood PRL (04, 06); PRB (07)); Wouters, Carusotto PRL (07,10) propagation elastic collisions two-body loss pump & loss rates

i∂tφ =  r2 2m µ + i(γp γl) + (λ iκ) |φ|2

• φ + ζ

u

• Bose condensation seen despite non-equilibrium conditions

Kasprzak et al., Nature 2006

• stochastic driven-dissipative Gross-Pitaevskii-Eq

Szymanska, Keeling, Littlewood PRL (04, 06); PRB (07)); Wouters, Carusotto PRL (07,10)

i∂tφ =  r2 2m µ + i(γp γl) + (λ iκ) |φ|2

• φ + ζ
• mean field
• neglect noise
• homogeneous solution φ(x, t) = φ0
• naively, just as Bose condensation in equilibrium!
• Q: What is “non-equilibrium” about it?

Experimental Platform: Exciton-Polariton Systems

stationary state!

u

• rewrite stochastic Gross-Pitaevski equation

“What is non-equilibrium about it?”

i∂tφc = δHc δφ∗

c

− iδHd δφ∗

c

+ ξ

Hα = Z ddx[rα|φc|2 + Kα|rφc|2 + λα|φ∗

cφc|4],

α = c, d

Im Re

coherent/ reversible dynamics incoherent/ irrev. dynamics example: two-body processes

Reλ Imλ

elastic two-body collisions inelastic two-body losses

• couplings located in the complex plane:

⇔ Hc ⇔ Hd

u u u

• Representation of stochastic Langevin dynamics as MSRJD functional integral

¯ S = Z

t,x

{φ∗

ci∂tφc − Hc + iHd}

S = Z

t,x

⇢ φ∗

q

δ ¯ S[φc] δφ∗

c

+ c.c. + i2γφ∗

qφq

• Z =

Z D[φc, φ∗

c, φq, φ∗ q] eiS[φc,φ∗

c,φq,φ∗ q]

• H. K. Janssen (1976); C. Aron et

al, J Stat. Mech (2011) generalisation to quantum systems (Keldysh functional integral)

• L. Sieberer, A. Chiochetta, U. Tauber,
• A. Gambassi, SD, PRB (2015)

Tβφc(t, x) = φ∗

c(−t, x),

Tβφq(t, x) = φ∗

q(−t, x) + i

2T ∂tφ∗

c(−t, x)

• Equilibrium conditions signalled by presence of symmetry under:

“What is non-equilibrium about it?”: Field theory

• Implication 1 [equivalence]: (classical) fluctuation-dissipation

hφc(ω, q)φ∗

c(ω, q)i = 2T

ω [hφc(ω, q)φ∗

q(ω, q) hφc(ω, q)φ∗ q(ω, q)i correlations responses (imaginary part)

➡ equilibrium conditions as a symmetry

⇔

i∂tφc = δHc δφ∗

c

− iδHd δφ∗

c

+ ξ

equilibrium dynamics

Im Re

non-equilibrium dynamics

Re Im

• coherent and dissipative dynamics may
• ccur simultaneously
• but they are not independent
• coherent and driven-dissipative dynamics do
• ccur simultaneously
• they result from different dynamical resources

symmetry protected no symmetry

➡ what are the physical consequences of the spread in the complex plane?

“What is non-equilibrium about it?”: Geometric interpretation

• Implication 2: geometric constraint

Review: L. Sieberer, M. Buchhold, SD, Keldysh Field Theory for Driven Open Quantum Systems, Reports on Progress in Physics (2016)

L∗ Lv

Outline

• mapping of the driven-dissipative GPE to KPZ-type equation
• fundamental difference to conventional context:

➡ weak non-equilibrium drive: two competing scales

• smooth non-equilibrium fluctuations -> emergent KPZ length scale
• non-equilibrium vortex physics -> emergent length scale
• result: different sequence in 2D and 1D

KPZ variable: condensate phase, compact ➡ strong non-equilibrium drive: new first order phase transition (one dimension)

Low frequency phase dynamics

• driven-dissipative stochastic GPE
• integrate out fast amplitude fluctuations:

∂tθ = Dr2θ + λ(rθ)2 + ξ

phase diffusion phase nonlinearity Markov noise Kardar, Parisi, Zhang, PRL (1986)

form of the KPZ equation

i∂tφ =  r2 2m µ + i(γp γl) + (λ iκ) |φ|2

• φ + ζ

φ(x, t) = (M0 + χ(x, t))eiθ(x,t)

see also: G. Grinstein et al., PRL 1993

gravitational field

x

particles deposited at rate

h(x, t) λ

surface roughening, fire spreading, bacterial colony growth..

• spin wave becomes non-linear
• nonlinearity: single-parameter measure of non-equilibrium strength

(ruled out in equilibrium by symmetry)

λ = 0

equilibrium Im

λ 6= 0

non-equilibrium Im Re Re

u

2 Dimensions

• E. Altman, L. Sieberer, L. Chen, SD, J. Toner, PRX (2015)
• G. Wachtel, L. Sieberer, SD, E. Altman, PRB (2016)
• L. Sieberer, G. Wachtel, E. Altman, SD, PRB (2016)

Lv

• RG flow of the effective dimensionless KPZ coupling parameter

strong coupling: disordered / rough non-equilibrium phase weak coupling: equilibrium phase

• general trend: non-equilibrium effects in systems with soft mode are
• enhanced in d = 1,2
• softened in d = 3 (below a threshold)

λ 6= 0

non-equilibrium Im Re

g2 = λ2∆ D3

g(L∗) = 1

Physical implication I: Smooth KPZ fluctuations

d

1 2 3

g 1

FRG analysis: Canet, Chate, Delamotte, Wschebor, PRL (2010), PRE (2012)

• RG flow of the effective dimensionless KPZ coupling parameter
• 2D: implication: a length scale is generated

microscopic (healing) length

λ 6= 0

non-equilibrium Im Re

g2 = λ2∆ D3

• exponentially large for
• weak nonequilibrium
• small noise level

λ

∆

Physical implication I: Smooth KPZ fluctuations

L∗ = a0e

16π g2 strong coupling: disordered / rough non-equilibrium phase weak coupling: equilibrium phase

g(L∗) = 1

d

1 2 3

g 1

FRG analysis: Canet, Chate, Delamotte, Wschebor, PRL (2010), PRE (2012)

Physical implications I: Absence of quasi-LRO

• generated length scale distinguishes two regimes:
• long-range behavior of two-point/ spatial coherence function:

r

hφ∗(r)φ(0)i

algebraic quasi-long range order (Kosterlitz-Thouless phase) sub-exponential non- equilibrium disordered (rough) phase

L∗

➡ algebraic order absent in any two-dimensional driven open system at the largest distances ➡ but crossover scale exponentially large for small deviations from equilibrium

hφ⇤(r)φ(0)i ⇡ n0eh[θ(x)θ(0)]2i

leading order cumulant expansion

L∗ = a0e

16π g2

∼ r−α

e−r2χ, χ ≈ 0.37 (d = 2)

Physical implications II: Non-equilibrium Kosterlitz-Thouless

• compact nature of phase allows for vortex defects in 2D!

vortex anti-vortex

• in 2D equilibrium: perfect analogy between vortices and electric charges
• log(r) interactions, forces

1/(✏r)

• dielectric constant = superfluid stiffness

✏−1

T<TKT$ T>TKT$ superfluid$=$dipole$gas$$ (“vortex$insulator”)$ Normal$=$plasma$ metallic$screening$

✏−1 → 0

✏−1 > 0

superfluid = dipole gas ➡ how is this scenario modified in the driven system? normal fluid = plasma metallic screening

∂tθ = Dr2θ + λ(rθ)2 + ξ

• KPZ equation for phase variable

P = (ε − 1) Eext = Z d2r rP(r)

Duality approach

∂tθ = Dr2θ + λ(rθ)2 + ξ

• KPZ equation for phase variable
• phase compactness = local discrete gauge invariance of

θt,x 7! θt,x + 2πnt,x

ψt,x = √ρt,xeiθt,x

θt,x ∈ [0, 2π), nt,x ∈ Z

• deterministic part: lattice regularization

unit lattice direction

=: L[θ]t,x

deterministic noise

∂tθx = − X

a

 D sin(θx − θx+a) + λ 2 (cos(θx − θx+a) − 1)

• + ηx

➡ needs to be taught to the KPZ equation:

θt,x ∈ [0, 2π)

Duality approach

∂tθ = Dr2θ + λ(rθ)2 + ξ

• KPZ equation for phase variable
• temporal part: stochastic update

✓t+✏,x = ✓t,x + ✏ (L[✓]t,x + ⌘t,x) + 2⇡nt,x

θt,x ∈ [0, 2π)

• NB: phase can jump, continuum limit eps -> 0 ill defined, derivatives discrete
• phase compactness = local discrete gauge invariance of

θt,x 7! θt,x + 2πnt,x

ψt,x = √ρt,xeiθt,x

θt,x ∈ [0, 2π), nt,x ∈ Z

θt,x ∈ [0,

➡ needs to be taught to the KPZ equation:

Duality approach: Comparison to non-compact case

∂tθ = Dr2θ + λ(rθ)2 + ξ

• KPZ equation for non-compact variable

Z = X

{˜ nt,x}

Z D[θ]eiS[θ,˜

n]

Z = Z D[˜ θ]D[θ]eiS[θ,˜

θ] manifestly gauge invariant! stochastic differential equation continuous noise MSRJD functional integral

⇔

stochastic difference equation discrete noise MSRJD functional integral

⇔

✓t+✏,x = ✓t,x + ✏ (L[✓]t,x + ⌘t,x) + 2⇡nt,x

lattice regularized deterministic term

• KPZ equation for compact variable

Duality approach

Z ∝ X

{nvX,˜ nvX, JvX,˜ JvX}

Z D[φ, ˜ φ, A, ˜ A]eiS[φ, ˜

φ,A, ˜ A,nv,˜ nv,Jv,˜ Jv]

• dual description:
• interpretation: study the associated Langevin equations

vortex density and current smooth spin wave fluctuations (equivalent KPZ equation)

• discrete gauge invariant dynamical functional integral

Z = X

{˜ nt,x}

Z D[θ]eiS[θ,˜

n]

S = X

t,x

˜ nt,x [−∆t✓t,x + ✏ (L[✓]t,x + i∆˜ nt,x)]

• introduce Fourier conjugate variables, use continuity equations to parameterise in terms of gauge fields,

Poisson transform

Electrodynamic Duality

KPZ non-linearity and noise

r · E = 2πnv r ⇥ E + 1 DB = 0 r ⇥ B ∂E ∂t = 2πJv ˆ z ⇥ r ✓λ 2 E2 + ¯ ζ ◆ r · B = 0

• Langevin equations = modified nonlinear noisy Maxwell equations

modified continuity eq phase dynamics irrotational flow

E = rφ A, B = Dr ⇥ A

fixed by modified gauge invariance

˜ E = rφ ∂t ˜ A, ˜ B = r ⇥ ˜ A

vortex density & current

dri dt = µniE(t, ri) + ξi

phenomenologically added vortex dynamics ∂t → 1/D

• formulated in electric and magnetic fields alone:
• reproducing KPZ: identify

E ⌘ ˆ z ⇥ rθ

& integrate out magnetic field, neglect vortices

∂tθ = Dr2θ + λ(rθ)2 + ξ

• next: integrate out gapless electric field degrees of freedom = phase fluctuations
• equilibrium : exactly
• non-equilibrium: perturbatively in

λ = 0 λ

A single vortex-antivortex pair

dr dt = µrV (r) + ξ r

• equation of motion for a single vortex-antivortex pair

r

equilibrium: Coulomb potential (2D)

➡ noise-activated unbinding for a single pair (at exp small rate) driven-dissipative system

V (r) ≈ 1 ε ln(r/a) − λ2 12ε3D2

• ln(r/a)3

V (r) = 1 ε ln(r/a)

Lv = a0e

2D λ length scale:

see also: I Aranson et al., PRB (1998) two-vortex problem

Many pairs: Modified Kosterlitz-Thouless RG flow

ε∆ y

equilibrium KT flow

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.02 0.04 0.06 0.08 0.10

KT transition

vortex unbinding

bound pairs

d" d` = 2⇡2y2 T dy d` =  2 − 1 2"T + 2 4"2D2 ✓1 4 + ` ◆ y

dT d` = 2T 2"2D2 ✓1 4 + ` ◆

✏ → ∞ ⇒ ρs → 0

dielectric constant superfluid stiffness

Many pairs: Modified Kosterlitz-Thouless RG flow

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.02 0.04 0.06 0.08 0.10

equilibrium KT flow modified non-equilibrium RG flow

KT transition

ε∆ y

dy d`

changes sign at a scale

Lv d" d` = 2⇡2y2 T

dT d` = 2T 2"2D2 ✓1 4 + ` ◆

dy d` =  2 − 1 2"T + 2 4"2D2 ✓1 4 + ` ◆ y

➡ vortex unbinding for any value of the noise strength

Summary: 2D

• two emergent length scales in complementary approaches:

Lv

KPZ length

Lv = a0e

2D λ

vortex length

• scaling for the relevant fixed points

hφ∗(r)φ(0)i ⇠ e−r2χ, χ = 0.4

KPZ fixed point

hφ∗(r)φ(0)i ⇠ e−r

free vortex/disordered fixed point

• for incoherently pumped exciton-polariton systems,

algebraic/equilibrium vortex/non-equilibrium

Lv ⌧ L∗

L∗ = a0e

16π g2

• E. Altman, L. Sieberer, L, Chen, SD, J. Toner, PRX (2015)
• L. Sieberer, G. Wachtel, E. Altman, SD, PRB (2016)
• G. Wachtel, L. Sieberer, SD, E. Altman, PRB (2016)
• caveats for observability:
• length scales exponentially large
• assumes stationary states (unknown

non-universal vortex dynamics)

1 Dimension

• L. He, L. Sieberer, E. Altman, SD, PRB (2015)
• L. He, L. Sieberer, SD, PRL (2017)

L∗ Lv

∼ e−a|t−t0|1/2

∼ e−b|t−t0|2/3

∼ e−c|t−t0|

equilibrium diffusive KPZ disordered

algebraic exponential

➡ Situation reversed compared to 2D! ➡ KPZ scaling should be observable in exciton-polariton experiments in 1D

(“bad cavity limit”, lifetime 1ps, system size 150 mu m)

Sequence of Scales

• L. He, L. Sieberer, E. Altman, SD, PRB (2015)

see also: K. Yi, V. Gladilin, M. Wouters, PRB (2015)

• L. He, L. Sieberer, SD, PRL (2017)
• direct numerical solution of driven-dissipative GPE in one dimension
• Study temporal instead of spatial coherence function
• Crossover scale

Tv ∼ eEc/σ

T∗ ∼ σ−2

noise level

subexponential KPZ scaling

exponential disordered scaling

numerical evidence

➡

what causes the emergent length scale beyond KPZ?

Space-time vortices in 1D XP condensate

• Physical origin: compactness of phase field

topologically nontrivial phase field configurations on (1+1)D space-time plane

• unbound at infinitesimal noise level (weak non-equilibrium)
• interaction potential:

(∂t + D∂2

x)−1 ∼ (Dt)−1/2e−x2/(4Dt)

• cf. 2D static equilibrium: r−2 ⇠ log(|x|)
• 1. temporal scaling (random uncorrelated charges)
• explains qualitative features
• 2. noise level dependence of crossover scale Tv ∼ eEc/σ

∼ e−c|t−t0|

vortex in space- time plane

spatial phase slip

⇔

+π +π

+0 +0

round trip

∆ϕ = 2π

(mapping to static 2D active smectic A liquid crystal)

Toner and Nelson, PRB (1984)

Strong non-equilibrium: Compact KPZ vortex turbulence

• In search of the phase diagram for XP condensates

Vortex turbulence (VT)

λ

noise level non-equilibrium strength

σ

Pv = O(1)

?

physics in strong non- equilibrium condition

Strong non-equilibrium: Compact KPZ vortex turbulence

• In search of the phase diagram for XP condensates

Noise activated vortices (TV) KPZ (conventional) dominated physics Vortex turbulence (VT)

λ

noise level non-equilibrium strength

σ

color code: vortex density on space-time plane

Pv = O(1)

• f filled triangles

i.e. Pv(˜ λ → ˜ λ∗−) and by changing r

with the same color and Pv(˜ λ → ˜ λ∗+), u from 1 4 to

first order non-equilibrium phase transition

Strong non-equilibrium: Compact KPZ vortex turbulence

• In search of the phase diagram for XP condensates

Noise activated vortices (TV) KPZ (conventional) dominated physics Vortex turbulence (VT)

λ

noise level non-equilibrium strength

σ

color code: vortex density on space-time plane

Pv = O(1)

amplification even by small phase fluctuations

• .
• reason: deterministic dynamical instability in compact KPZ: evolution of phase differences

decreases amplifies

➡ Transition to chaos?

chaotic solutions nonlinear dynamics: e.g. Aranson et al., RMP (2002)

• nset of vortex turbulence
• scaling of the momentum distribution at intermediate momenta (full stochastic GPE)

scaling due to thermally activated vortices:

Compact KPZ vortex turbulence: Signatures

nq = hψ∗(q)ψ(q)i ⇠ q−γ

γ ≈ 2 γ ≈ 5

scaling due to turbulent vortices:

• experiments: vortex turbulence favored in systems with strong diffusion, λ ∼ Kd/Kc

coherent propagation, inverse effective polariton mass diffusion constant

• F. Baboux et al. PRL (2016)
• flat band of 1D Lieb lattice realized with micropillar cavity arrays

Summary

Review: L. Sieberer, M. Buchhold, SD, Keldysh Field Theory for Driven Open Quantum Systems, Reports on Progress in Physics (2016)

Lv ⌧ L∗ Lv L∗

➡

2 dimensions:

➡

1 dimension:

➡ two intrinsic non-equilibrium length/time scales

Lv

L∗

Lv

➡

low dimensional driven open quantum systems: non-equilibrium always relevant at large distances

➡

phase dynamics: compact KPZ

➡

compactness crucial

d

2

λ

λ 6= 0

• E. Altman, L. Sieberer, L. Chen, SD, J. Toner, PRX (2015)
• L. Sieberer, G. Wachtel, E. Altman, SD, PRB (2016)
• G. Wachtel, L. Sieberer, SD, E. Altman, PRB (2016)
• L. He, L. Sieberer, E. Altman, SD, PRB (2015)
• L. He, L. Sieberer, SD, PRL (2017)
• weak non-equilibrium conditions
• strong non-equilibrium conditions

➡ phase transition to vortex turbulent regime ➡ challenge: analytical understanding via duality?

• L. He, L. Sieberer, SD, PRL

(2017) see also

• R. Lauter, A. Mitra, F.

Marquardt, arxiv (2016)