Non-Equilibrium Steady States BeyondIntegrabilityinteracting quantum systems Possibility of...
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Non-Equilibrium Steady StatesBeyond Integrability
Joe Bhaseen
TSCM Group
King’s College London
Beyond Integrability: The Mathematics and Physics of Integrability and
its Breaking in Low-Dimensional Strongly Correlated Systems
Centre De Recherches Mathematiques
University of Montreal
13-17 July 2015
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Strings, Cosmology & Condensed Matter
Benjamin Doyon Koenraad Schalm Andy Lucas
Ben Simons Julian Sonner
Jerome Gauntlett Toby Wiseman
King’s, Leiden, Harvard, Cambridge, Imperial
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Outline
• AdS/CFT and far from equilibrium dynamics
• Quenches and thermalization
• Energy flow between CFTs
• AdS/CFT offers new insights
• Higher dimensions
• Non-equilibrium fluctuations
• Current status and future developments
MJB, Benjamin Doyon, Andrew Lucas, Koenraad Schalm
“Energy flow in quantum critical systems far from equilibrium”
Nature Physics (2015)
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Progress in AdS/CMT
Transport Coefficients
Viscosity, Conductivity, Hydrodynamics, Bose–Hubbard, Graphene
Strange Metals
Non-Fermi liquids, instabilities, cuprates
Holographic Duals
Superfluids, Fermi liquid, O(N), Luttinger liquid
Equilibrium or close to equilibrium
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Gauge-Gravity DualityAdS/CFT correspondence (Anti-de Sitter/Conformal Field Theory)
Maldacena (1997); Gubser, Klebanov, Polyakov (1998); Witten (1998)
S. Hartnoll, Science 322, 1639 (2008)
For an overview see for example John McGreevy, Holographic duality
with a view toward many body physics, arXiv:0909.0518
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AdS/CFT Correspondence
d−1,1
z
R
AdSd+1minkowski
UVIR
...
Gubser–Klebanov–Polyakov–Witten
Z[φ0]CFT ≃ e−SAdS[φ]|φ∼φ0 at z=0
φ(z) ∼ zd−∆φ0(1 + . . . ) + z∆φ1(1 + . . . )
Fields in AdS ↔ operators in dual CFT φ ↔ O
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Utility of Gauge-Gravity Duality
Quantum dynamics Classical Einstein equations
Finite temperature Black holes
Real time approach to finite temperature dynamics in
interacting quantum systems
Possibility of anchoring to 1 + 1 integrable models
Beyond Integrability
Higher Dimensions
Non-Equilibrium Beyond linear response
Organising principles out of equilibrium
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Non-Equilibrium AdS/CMT
Quenches in Holographic Superfluids
MJB, Gauntlett, Simons, Sonner & Wiseman, “Holographic Superfluids
and the Dynamics of Symmetry Breaking” PRL (2013)
Quasi-Normal-Modes
Current Noise
Sonner and Green, “Hawking Radiation and Nonequilibrium Quantum
Critical Current Noise”, PRL 109, 091601 (2013)
Hawking Radiation
Superfluid Turbulence
Chesler, Liu and Adams, “Holographic Vortex Liquids and Superfluid
Turbulence, Science 341, 368 (2013); also arXiv:1307.7267
Fractal Horizons
Metric quenches, entanglement entropy, thermalization...
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Thermalization
Condensed matter and high energy physics
RTL T
Why not connect two strongly correlated systems together
and see what happens?
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AdS/CFT
Energy flow may be studied within pure Einstein gravity
S = 116πGN
∫
dd+2x√−g(R− 2Λ)
z
gµν ↔ Tµν
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Einstein Centenary
The Field Equations of General Relativity (1915)
Rµν − 12Rgµν + Λgµν = 8πG
c4 Tµν
http://en.wikipedia.org/wiki/Einstein field equations
gµν metric Rµν Ricci curvature R scalar curvature
Λ cosmological constant Tµν energy-momentum tensor
Coupled Nonlinear PDEs
Schwarzchild Solution (1916) RS = 2MGc2 Black Holes
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Non-Equilibrium CFTBernard & Doyon, Energy flow in non-equilibrium conformal field
theory, J. Phys. A: Math. Theor. 45, 362001 (2012)
Two critical 1D systems (central charge c)
at temperatures TL & TR
T TL R
Join the two systems together
TL TR
Alternatively, take one critical system and impose a step profile
Local Quench
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Steady State Energy FlowBernard & Doyon, Energy flow in non-equilibrium conformal field
theory, J. Phys. A: Math. Theor. 45 362001 (2012)
If systems are very large (L ≫ vt) they act like heat baths
For times t ≪ L/v a steady energy current flows
TL TR
Non-equilibrium steady state
J =cπ2k2
B
6h (T 2L − T 2
R)
Universal result out of equilibrium
Direct way to measure central charge; velocity doesn’t enter
Sotiriadis and Cardy. J. Stat. Mech. (2008) P11003.
Stefan–Boltzmann
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Energy Current FluctuationsBernard & Doyon, Energy flow in non-equilibrium conformal field
theory, J. Phys. A: Math. Theor. 45, 362001 (2012)
Generating function for all moments
F (λ) ≡ limt→∞
t−1 ln〈eiλ∆tQ〉
Exact Result
F (λ) = cπ2
6h
(
iλβl(βl−iλ) − iλ
βr(βr+iλ)
)
Denote z ≡ iλ
F (z) = cπ2
6h
[
z(
1β2l
− 1β2r
)
+ z2(
1β3l
+ 1β3r
)
+ . . .]
〈J〉 = cπ2
6h k2B(T2L − T 2
R) 〈δJ2〉 ∝ cπ2
6h k3B(T3L + T 3
R)
Poisson Process∫∞
0e−βǫ(eiλǫ − 1)dǫ = iλ
β(β−iλ)
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AdS/CFT
Steady State Region
Spatially Homogeneous
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Solutions of Einstein Equations
S = 116πG
∫
dd+2x√−g(R− 2Λ) Λ = −d(d+ 1)/2L2
Unique homogeneous solution = boosted black hole
ds2 =L2
z2
[
dz2
f(z)− f(z)(dt cosh θ − dx sinh θ)2+
(dx cosh θ − dt sinh θ)2 + dy2⊥]
f(z) = 1−(
zz0
)d+1
z0 = d+14πT
Fefferman–Graham Coordinates
〈Tµν〉s = Ld
16πG limZ→0
(
ddZ
)d+1 Z2
L2 gµν(z(Z))
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Boost SolutionLorentz boosted stress tensor of a finite temperature CFT
Perfect fluid
〈Tµν〉s = ad Td+1 (ηµν + (d+ 1)uµuν)
ηµν = diag(−1, 1, · · · , 1) uµ = (cosh θ, sinh θ, 0, . . . , 0)
One spatial dimension
a1 = Lπ4G c = 3L
2G
TL = Teθ TR = Te−θ 〈Ttx〉 = cπ2k2B
6h (T 2L − T 2
R)
Can also obtain complete steady state density matrix
Describes all the cumulants of the energy transfer process
The non-equilibrium steady state (NESS)
is a Lorentz boosted thermal state
Run past a thermal state with temperature T =√TLTR
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Steady State Density Matrix
〈O...〉 = Tr(ρsO...)
Tr(ρs)
ρs = e−βE cosh θ+βPx sinh θ
β =√βLβR e2θ = βR
βL
Lorentz boosted thermal density matrix
Generalized Gibbs Ensemble (GGE)
Describes all the cumulants of the energy transfer process
The non-equilibrium steady state (NESS)
is a Lorentz boosted thermal state
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Higher Dimensions
Shock Waves
Energy-Momentum conservation across the shocks
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Shock Solutions
Rankine–Hugoniot
Energy-Momentum conservation ∂µTµν = 0 across shock
〈T tx〉s = ad
(
T d+1L − T d+1
R
uL + uR
)
Invoking boosted steady state gives uL,R in terms of TL,R:
uL = 1d
√
χ+dχ+d−1 uR =
√
χ+d−1
χ+d χ ≡ (TL/TR)(d+1)/2
Steady state region is a boosted thermal state with T =√TLTR
Boost velocity (χ− 1)/√
(χ+ d)(χ+ d−1) Agrees with d = 1
Shock waves are non-linear generalisations of sound waves
EM conservation: uLuR = c2s , where cs = v/√d is speed of sound
cs < uR < v cs < uL < c2s/v reinstated microscopic velocity v
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Numerics I
Excellent agreement with predictions
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Numerics II
Excellent agreement far from equilibrium
Asymmetry in propagation speeds
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Conclusions
Average energy flow in arbitrary dimension
Lorentz boosted thermal state
Energy current fluctuations
Exact generating function of fluctuations
Generalizations
Other types of charge noise Non-Lorentz invariant situations
Bose–Hubbard and Unitary Fermi Gas
Different central charges Fluctuation theorems Numerical GR
Acknowledgements
B. Benenowski, D. Bernard, P. Chesler, A. Green
D. Haldane C. Herzog, D. Marolf, B. Najian, C.-A. Pillet
S. Sachdev, A. Starinets