From Critical Phenomena to Holographic Duality in Quantum ... · From Critical Phenomena to...
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From Critical Phenomena to HolographicDuality in Quantum Matter
Joe Bhaseen
TSCM Group
King’s College London
2013 Arnold Sommerfeld School
“Gauge-Gravity Duality and Condensed Matter Physics”
Arnold Sommerfeld Center for Theoretical Physics, Munich
5 - 9 August
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Lecture 3
“Holographic Approaches to Far From Equilibrium
Dynamics”
Accompanying Slides
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Outline
• Experimental motivation
• AdS/CMT and far from equilibrium dynamics
• Quenches and thermalization
• Holographic superfluids
• Heat flow between CFTs
• Potential for AdS/CFT to offer new insights
• Higher dimensions and non-equilibrium fluctuations
• Current status and future developments
Acknowledgements
Jerome Gauntlett Julian Sonner Toby Wiseman Ben Simons
Benjamin Doyon Koenraad Schalm
Bartosz Benenowski Bahman Najian
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ExperimentWeiss et al “A quantum Newton’s cradle”, Nature 440, 900 (2006)
Non-Equilibrium 1D Bose Gas
Integrability and Conservation Laws
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Experiment
Stamper-Kurn et al “Spontaneous symmetry breaking in a quenched
ferromagnetic spinor Bose–Einstein condensate”, Nature 443, 312 (2006)
Domain formation
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Experiment
Stamper-Kurn et al “Spontaneous symmetry breaking in a quenched
ferromagnetic spinor Bose–Einstein condensate”, Nature 443, 312 (2006)
Topological Defects
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Nature 464 1301 (2010)
Self-Organisation Domokos & Ritsch, PRL (2002)
Vuletic et al, PRL (2003) Nagy et al, Eur. Phys. J. D (2008)
⇓= |Zero Momentum〉 ⇑= |Finite Momentum〉
“Spins” Coupled to Light
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Observation of Dicke TransitionH = ωψ†ψ + ω0
∑
i
Szi + g
∑
i
(ψ†S−i + ψS+
i )
+ g′∑
i
(ψ†S+i + ψS−
i ) + U∑
i
Szi ψ
†ψ
Baumann, Guerlin, Brennecke & Esslinger, Nature 464, 1301 (2010)
Driven Open System
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Utility of Gauge-Gravity Duality
Quantum dynamics Classical Einstein equations
Finite temperature Black holes
Real time approach to finite temperature quantum
dynamics in interacting systems, with the possibility of
anchoring to 1 + 1 and generalizing to higher dimensions
Non-Equilibrium Beyond linear response
Temporal dynamics in strongly correlated systems
Combine analytics with numerics
Dynamical phase diagrams
Organizing principles out of equilibrium
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Progress on the CMT SideSimple protocals and integrability
Methods of integrability and CFT have been invaluable in
classifying equilibrium phases and phase transitions in 1+1
Do do these methods extend to non-equilibrium problems?
Quantum quench Parameter in H abruptly changed
H(g) → H(g′)
System prepared in state |Ψg〉 but time evolves under H(g′)
Quantum quench to a CFT
Calabrese & Cardy, PRL 96, 136801 (2006)
Quantum quenches in quantum spin chains
e.g. Calabrese, Essler & Fagotti, PRL 106, 227203 (2011)
Quantum quench in BCS pairing Hamiltonian
Andreev, Gurarie, Radzihovsky, Barankov, Levitov, Yuzbashyan, Caux...
Thermalization Integrability Generalized Gibbs
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Quantum Quench to a CFT
Calabrese and Cardy, “Time Dependence of Correlation Functions
Following a Quantum Quench”, PRL 96, 136801 (2006)
〈Φ(t)〉 = AΦb
[
π
4τ0
1
cosh[πt/2τ0]
]x
∼ e−πxt/2τ0
Scaling dimension x Non-universal decay τ0 ∼ m−1
Ratios of observables exhibit universality
〈Φ(r, t)Φ(0, t)〉 ∼ e−πxr/2τ0 t > r/2
Emergent length scale or temperature scale
AdS/CFT gives access to higher dimensional interacting
critical points and the possibility of universal results
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Quenches in Transverse Field IsingRecent work: Calabrese, Essler, Fagottini, PRL 106, 227203 (2011)
H(h) = − 12
∑∞l=−∞[σx
l σxl + hσz
l ]
QPT Ferromagnetic h < 1 Paramagnetic h > 1
Quantum quench h0 → h within the ordered phase h0, h ≤ 1
Exact results for long time asymptotics of order parameter
〈σxl (t)〉 ∝ exp
[
t∫ π
0dkπ ǫ
′h(k) ln(cos∆k)
]
ǫh(k) =√
h2 − 2h cos k + 1 cos∆k =hh0 − (h+ h0) cos k + 1
ǫh(k)ǫh0(k)
Exact results for asymptotics of two-point functions
〈σxl (t)σ
x1+l(t)〉 ∝ exp
[
l
∫ π
0
dk
πln(cos∆k)θ(2ǫ
′h(k)t− l)
]
× exp
[
2t
∫ π
0
dk
πǫ′h(k) ln(cos∆k)θ(l − 2ǫ′h(k)t)
]
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Quenches in Transverse Field Ising
Recent work: Calabrese, Essler, Fagottini, PRL 106, 227203 (2011)
Good agreement between numerics and analytics
Determinants Form Factors
Integrable quenches in interacting field theories
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BCS Quench DynamicsBarankov, Levitov and Spivak, “Collective Rabi Oscillations and Solitons
in a Time-Dependent BCS Pairing Problem”, PRL 93, 160401 (2004)
Time dependent BCS Hamiltonian
H =∑
p,σ
ǫpa†p,σap,σ − λ(t)
2
∑
p,q
a†p,↑a
†−p,↓a−q,↓aq,↑
Pairing interactions turned on abruptly
λ(t) = λθ(t)
Generalized time-dependent many-body BCS state
|Ψ(t)〉 =∏
p
[
up(t) + vp(t)a†p,↑a
†−p,↓
]
|0〉
Integrable
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Collective OscillationsBarankov, Levitov and Spivak, PRL 93, 160401 (2004)
Time-dependent pairing amplitude
∆(t) ≡ λ∑
pupv
∗p(t) = e−iωtΩ(t)
Equation of Motion
Ω2 + (Ω2 −∆2−)(Ω
2 −∆2+) = 0
Oscillations between ∆− and ∆+
0 10 20 300
0.5
1
Time × ∆0
|∆(t
)|/∆
0
∆−/∆
+=0.01
0 10 20 300
0.5
1
Time × ∆0
|∆(t
)|/∆
0
∆−/∆
+=0.5
0 10 20 300
0.5
1
Time × ∆0
|∆(t
)|/∆
0
∆−/∆
+=0.9
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Regimes of BCS Quench DynamicsBarankov and Levitov, “Synchronization in the BCS Pairing Dynamics
as a Critical Phenomenon”, PRL 96, 230403 (2006)
10−3
10−2
10−1
100
101
102
0
0.2
0.4
0.6
0.8
1
∆/∆ 0
∆s/∆
0
∆−
∆(T*)
∆+
∆a
Synchronization Dephasing
A B C
Initial pairing gap ∆s Final pairing gap ∆0
(A) Oscillations between ∆± (B) Underdamped approach to ∆a
(C) Overdamped approach to ∆ = 0
Emergent temperature T ∗ and gap ∆(T ∗)
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Bloch DynamicsAnderson Pseudo-Spins
H = −∑
p2ǫps
zp− λ(t)
∑
pqs−ps+q
0 50 100 150 2000
0.5
1
t ∆0
∆(t)
/∆0
30
0 1 2 3 4 50
1
2
t ∆0
∆(t)
/∆0
∆s=∆
AB
∆AB
< ∆s < ∆
BC
(a) ∆s=∆
BC(b)
∆(t) = ∆a +A(t) sin(2∆at+ α) A(t) ∝ t−1/2
∆(t) ∝ (∆st)−1/2e−2∆st
Non-Equilibrium Dynamical Phase Diagram
Yuzbashyan et al Andreev et al
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AdS QuenchesChesler & Yaffe, “Horizon formation and far-from-equilibrium
isotropization in supersymmetric Yang–Mills plasma”, PRL (2009)
ds2 = −dt2 + eB0(t)dx2⊥ + e−2B0(t)dx2
‖
The dependent shear of the geometry B0(t) =12c [1− tanh(t/τ)]
Jan de Boer & Esko Keski-Vakkuri et al, “Thermalization of
Strongly Coupled Field Theories”, PRL (2011)
ds2 =1
z2[
−(1−m(v))zddv2 − 2dzdv + dx2]
Vaidya metric quenches m(v) = 12M [1 + tanh(v/v0)]
Aparıcio & Lopez, “Evolution of Two-Point Functions from
Holography”, arXiv:1109.2571
Albash & Johnson, “Evolution of Holographic Entanglement
Entropy after Thermal and Electromagnetic Quenches”, NJP (2011)
Basu & Das, “Quantum Quench across a Holographic Critical
Point”, arXiv:1109.3309
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Holographic SuperconductorGubser, “Breaking an Abelian Gauge Symmetry Near a Black Hole
Horizon”, Phys. Rev. D. 78, 065034 (2008)
Hartnoll, Herzog, Horowitz, “Building a Holographic Superconductor”,
PRL 101, 031601 (2008)
Abelian Higgs Coupled to Einstein–Hilbert Gravity
S =
∫
dDx√−g
[
R− 2Λ− 1
4FabF
ab − |Daψ|2 −m2|ψ|2]
Fab = ∂aAb − ∂bAa Da = ∂a − iqAa Λ = −D(D−1)4L2
Finite temperature Black hole
Below critical temperature black hole with ψ = 0 unstable
Black hole with charged scalar hair ψ 6= 0 becomes stable
Spontaneous U(1) symmetry breaking
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Non-Equilibrium AdS SuperconductorsMurata, Kinoshita & Tanahashi, “Non-Equilibrium Condensation
Process in a Holographic Superconductor”, JHEP 1007:050 (2010)
Start in unstable normal state described by a
Reissner–Nordstrom black hole and perturb
Temporal dynamics from supercooled to superconducting
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AdS GeometryMurata, Kinoshita & Tanahashi, JHEP 1007:050 (2010)
AdS
boundary
event
horizon t = const.
z = const.
Excision of
the singularity
horizon
initial surface
Reissner-Nordström-AdS
+ perturbation
ψ(t, z) = ψ1(t)z + ψ2(t)z2 + ψ3(t)z
3 + · · · ≡ ψ1(t)z + ψ(t, z)z2
Gaussian perturbation of Reissner–Nordstrom–AdS
ψ(t = 0, z) = A√2πδ
exp[
− (z−zm)2
2δ2
]
A = 0.01, δ = 0.05, zm = 0.3
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Time Evolution of the Order ParameterMurata, Kinoshita & Tanahashi, JHEP 1007:050 (2010)
Start in normal state (Reissner–Nordstrom) & perturb
Numerical solution of Einstein equations
〈O2(t)〉 ≡√2ψ2(t)
1
10
0 1 2 3 4 5 6 7 8 9
|q<
O2>
|1/2 /
Tc
t Tc
0
2
4
6
8
10
0 2 4 6 8 10 12 14
|q<
O2>
|1/2 /
Tc
t Tc
T/Tc = 1.1, 1.2, 1.4 T/Tc = 0.2, 0.4, 0.6, 0.8
T > Tc Relaxation to N |〈O2(t)〉| = C exp(−t/trelax)
T < Tc Evolution to SC |〈O2(t)〉| = C1 exp(−t/trelax) + C2
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Questions
Where are the oscillations?
• Is there always damping or are there different regimes?
• Are the dynamics of holographic superconductors related to
condensed matter systems or intrinsically different?
• What is the role of coupling to a large number of critical
degrees of freedom?
• What is the role of large N and strong coupling?
• What is their influence on the emergent timescales?
• Do thermal fluctuations reduce the amplitude of oscillations?
• Beyond BCS and mean field dynamics using AdS/CFT?
• What is the dynamics at short, intermediate and long times?
• What happens if one quenches from a charged black hole?
AdS/CMT far from equilibrium?
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Setup
Homogeneous isotropic dynamics of CFT from Einstein Eqs
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Dynamical Phase DiagramConjugate field pulse
AdS4/CFT3 ψ(t, z) = ψ1(t)z + ψ2(t)z2 + . . .
ψ1(t) = δe−(t/τ)2
δ = δ/µi τ = τ /µi τ = 0.5 Ti = 0.5Tc
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Three Dynamical RegimesBhaseen, Gauntlett, Simons, Sonner & Wiseman, “Holographic
Superfluids and the Dynamics of Symmetry Breaking” PRL (2013)
(I) Damped Oscillatory to SC
〈O(t)〉 ∼ a+ be−kt cos[l(t− t0)]
(II) Over Damped to SC
〈O(t)〉 ∼ a+ be−kt
(III) Over Damped to N
〈O(t)〉 ∼ be−kt
Asymptotics described by black hole quasi normal modes
Far from equilibrium → close to equilibrium
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Approach to Thermal Equilibrium
Collapse on to Equilibrium Phase Diagram
Emergent temperature scale T∗ within superfluid phase
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Quasi Normal Modes
|〈O(t)〉| ≃ |〈O〉f +Ae−iωt|
Three regimes and an emergent T∗
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Recent Experiments
Endres et al, “The ‘Higgs’ Amplitude Mode at the Two-Dimensional
Superfluid-Mott Insulator Transition”, Nature 487, 454 (2012)
Time-dependent experiments can probe excitations
What is the pole structure in other correlated systems?
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Thermalization
RTL T
Why not connect two strongly correlated systems together
and see what happens?
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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 Heat 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 heat 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.
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Linear ResponseBernard & Doyon, Energy flow in non-equilibrium conformal field
theory, J. Phys. A: Math. Theor. 45, 362001 (2012)
J =cπ2k2
B
6h (T 2L − T 2
R)
TL = T +∆T/2 TR = T −∆T/2 ∆T ≡ TL − TR
J =cπ2k2
B
3h T∆T ≡ g∆T g = cg0 g0 =π2k2
BT3h
Quantum of Thermal Conductance
g0 =π2k2
BT3h ≈ (9.456× 10−13 WK−2)T
Free Fermions
Fazio, Hekking and Khmelnitskii, PRL 80, 5611 (1998)
Wiedemann–Franz κσT = π2
3e2 σ0 = e2
h κ0 =π2k2
BT3h
Conformal Anomaly
Cappelli, Huerta and Zemba, Nucl. Phys. B 636, 568 (2002)
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Experiment
Schwab, Henriksen, Worlock and Roukes, Measurement of the
quantum of thermal conductance, Nature 404, 974 (2000)
Quantum of Thermal Conductance
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Heuristic Interpretation of CFT Result
J =∑
m
∫
dk
2π~ωm(k)vm(k)[nm(TL)− nm(TR)]Tm(k)
vm(k) = ∂ωm/∂k nm(T ) = 1eβ~ωm−1
J = f(TL)− f(TR)
Consider just a single mode with ω = vk and T = 1
f(T ) =∫∞0
dk2π
~v2keβ~vk−1
=k2
BT 2
h
∫∞0dx x
ex−1 =k2
BT 2
hπ2
6 x ≡ ~vkkBT
Velocity cancels out
J =π2k2
B
6h (T 2L − T 2
R)
For a 1+1 critical theory with central charge c
J =cπ2k2
B
6h (T 2L − T 2
R)
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Stefan–BoltzmannCardy, The Ubiquitous ‘c’: from the Stefan-Boltzmann Law to
Quantum Information, arXiv:1008.2331
Black Body Radiation in 3 + 1 dimensions
dU = TdS − PdV
(
∂U∂V
)
T= T
(
∂S∂V
)
T− P = T
(
∂P∂T
)
V− P
u = T
(
∂P
∂T
)
V
− P
For black body radiation P = u/3
4u3 = T
3
(
∂u∂T
)
Vdu4u = dT
T14 lnu = lnT + const.
u ∝ T 4
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Stefan–Boltzmann and CFTCardy, The Ubiquitous ‘c’: from the Stefan-Boltzmann Law to
Quantum Information, arXiv:1008.2331
Energy-Momentum Tensor in d+ 1 Dimensions
Tµν =
u
P
P
. . .
Traceless P = u/d
Thermodynamics
u = T(
∂P∂T
)
V− P u ∝ T d+1
For 1 + 1 Dimensional CFT
u =πck2BT
2
6~v≡ AT 2 J =
Av2
(T 2L − T 2
R)
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Stefan–Boltzmann and AdS/CFTGubser, Klebanov and Peet, Entropy and temperature of black
3-branes, Phys. Rev. D 54, 3915 (1996).
Entropy of SU(N) SYM = Bekenstein–Hawking SBH of geometry
SBH =π2
2N2V3T
3
Entropy at Weak Coupling = 8N2 free massless bosons & fermions
S0 =2π2
3N2V3T
3
Relationship between strong and weak coupling
SBH = 34S0
Gubser, Klebanov, Tseytlin, Coupling constant dependence in the
thermodynamics of N = 4 supersymmetric Yang-Mills Theory,
Nucl. Phys. B 534 202 (1998)
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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(
1β2
l
− 1β2r
)
+ z2(
1β3
l
+ 1β3r
)
+ . . .]
〈J〉 = cπ2
6h k2B(T
2L − T 2
R) 〈δJ2〉 ∝ cπ2
6h k3B(T
3L + T 3
R)
Poisson Process∫∞0e−βǫ(eiλǫ − 1)dǫ = iλ
β(β−iλ)
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Full Counting StatisticsA large body of results in the mesoscopic literature
Free Fermions
Levitov & Lesovik, “Charge Distribution in Quantum Shot Noise”,
JETP Lett. 58, 230 (1993)
Levitov, Lee & Lesovik, “Electron Counting Statistics and Coherent
States of Electric Current”, J. Math. Phys. 37, 4845 (1996)
Luttinger Liquids and Quantum Hall Edge States
Kane & Fisher, “Non-Equilibrium Noise and Fractional Charge in the
Quantum Hall Effect”, PRL 72, 724 (1994)
Fendley, Ludwig & Saleur, “Exact Nonequilibrium dc Shot Noise in
Luttinger Liquids and Fractional Quantum Hall Devices”, PRL (1995)
Quantum Impurity Problems
Komnik & Saleur, “Quantum Fluctuation Theorem in an Interacting
Setup: Point Contacts in Fractional Quantum Hall Edge State Devices”,
PRL 107, 100601 (2011)
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Experimental Setup
Saminadayar et al, PRL 79, 2526 (1997)
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Shot Noise in the Quantum Hall EffectSaminadayar et al, “Observation of the e/3 Fractionally Charged
Laughlin Quasiparticle”, PRL 79, 2526 (1997)
R. de-Picciotto et al, “Direct observation of a fractional charge”,
Nature 389, 162 (1997)
SI = 2QIB Q = e/3
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Non-Equilibrium Fluctuation RelationBernard & Doyon, Energy flow in non-equilibrium conformal field theory,
J. Phys. A: Math. Theor. 45, 362001 (2012)
F (λ) ≡ limt→∞
t−1 ln〈eiλ∆tQ〉 = cπ2
6h
(
iλ
βl(βl − iλ)− iλ
βr(βr + iλ)
)
F (i(βr − βl)− λ) = F (λ)
Irreversible work fluctuations in isolated driven systems
Crooks relation P (W )
P (−W )= eβ(W−∆F )
Jarzynski relation 〈e−βW 〉 = e−β∆F
Entropy production in non-equilibrium steady states
P (S)P (−S) = eS
Esposito et al, “Nonequilibrium fluctuations, fluctuation theorems, and
counting statistics in quantum systems”, RMP 81, 1665 (2009)
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Lattice Models
RLT T
Quantum Ising Model
H = J∑
〈ij〉 Szi S
zj + Γ
∑
i Sxi
Γ = J/2 Critical c = 1/2
Anisotropic Heisenberg Model (XXZ)
H = J∑
〈ij〉(
Sxi S
xj + Sy
i Syj +∆Sz
i Szj
)
−1 < ∆ < 1 Critical c = 1
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Time-Dependent DMRG
Karrasch, Ilan and Moore, Non-equilibrium thermal transport and
its relation to linear response, arXiv:1211.2236
0
0.04
0.08
0 5 10 15
t
0
0.03
0.06
<J E
(n,t)
>-0.08
0
0.08
0.16
-20 0 20n
-0.2
0
h(n,
t)
λ=1
∆=−0.85, λ=1
TR=5
TR=0.67
TR=0.2TL=0.5
b=0
∆=0.8, λ=1, b=0
b=0.1b=0.2
b=0.4TL=∞TR=0.48
t=4
t=10
TL=∞TR=0.48
∆=0.5, b=0
λ=0.8λ=0.6
TL=0.5TR=5
n=2n=0
(a)
(b)
(c)
Dimerization Jn =
1 n odd
λ n even∆n = ∆ Staggered bn = (−1)nb
2
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Time-Dependent DMRG
Karrasch, Ilan and Moore, Non-equilibrium thermal transport and
its relation to linear response, arXiv:1211.2236
0 1 2 3 4 5
TR
-0.1
0
0.1
<J E
(t→
∞)>
0 1 2 3 4 5
-0.2
0
0.2
(a)
TL=∞
TL=1
TL=0.5
TL=0.2
∆=0.5
∆=2
TL=∞ TL=2 TL=1TL=0.5 TL=0.25curves collapse if
shifted vertically!
TL=0.125
0.1 1
TR
10-3
10-2
10-1
<J E
(t→
∞)> ∆=0, exact
∆=0, DMRG
0 0.1 0.2
0.105
0.11
0.115
0 0.4 0.8
0.56
0.6
0.64
(b) vertical shift ofcurves in (a)
c=1.01
fit toa+cπ/12 T2
~T2
∆=0.5
∆=2
~T−1
quantumIsing model
c=0.48
limt→∞〈JE(n, t)〉 = f(TL)− f(TR) f(T ) ∼
T 2 T ≪ 1
T−1 T ≫ 1
Beyond CFT to massive integrable models (Doyon)
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Energy Current Correlation FunctionKarrasch, Ilan and Moore, Non-equilibrium thermal transport and
its relation to linear response, arXiv:1211.2236
0
0.03
0.06
0 10 20
t
0
0.008
0.016
<J E
(t)J
E(0
)> /
N 0
0.008
0.016
λ=0.8
∆=−0.85, λ=1, T=0.5
λ=0.6λ=0.4
b=0
∆=0.5, b=0, T=1
λ=1 ∆=0.5, b=0, T=0.25
λ=1
λ=0.8
λ=0.6
b=0.1
λ=0.4
b=0.2
b=0.4
Bethe ansatz
(a)
(b)
(c)
Beyond Integrability
Importance of CFT for pushing numerics and analytics
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AdS/CFT
Heat flow may be studied within pure Einstein gravity
z
gµν ↔ Tµν
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Possible Setups
Local Quench Driven Steady State Spontaneous
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General Considerations
∂µTµν = 0 ∂0T
00 = −∂xT x0 ∂0T0x = −∂xT xx
Stationary heat flow =⇒ Constant pressure
∂0T0x = 0 =⇒ ∂xT
xx = 0
In a CFT
P = u/d =⇒ ∂xu = 0
No energy/temperature gradient
Stationary homogeneous solutions
without local equilibrium
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AdS/CFTPure Einstein Gravity S =
∫
d3x√−g(R− 2Λ)
Unique homogeneous solution in AdS3 is boosted BTZ
ds2 =r2
L2
[
−(
1− r20r2
cosh2 η
)
dt2 +r20r2
sinh(2η)dtdx
+L4
(1− r20
r2 )dr2 +
(
1− r20r2
sinh2 η
)
dx2
]
(neglect rotating black hole solutions)
r0 = 2πT unboosted temp L AdS radius η boost param
〈Tij〉 =L3
8πGNg(0)ij 〈Ttx〉 = L3
8πGN
r20
2L2 sinh(2η) =πL4GN
T 2 sinh(2η)
c = 3L/2GN
〈Ttx〉 = πc6 T
2 sinh(2η) = πc12 (T
2e2η − T 2e−2η)
TL = Teη TR = Te−η 〈Ttx〉 = cπ2k2
B
6h (T 2L − T 2
R)
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Work In Progress & Future Directions
Firmly establish 1D result for average energy flow
Uniqueness in AdS3 and identification of TL and TR
Energy current fluctuations
Exact results in 1 + 1 suggests simplifications in AdS3
Higher dimensions
Conjectures for average heat flow and fluctuations
Absence of left-right factorization at level of CFT
Free theories Poisson process
Generalizations
Other types of charge noise Non-Lorentz invariant situations
Different central charges Fluctuation theorems Numerical GR