emergent hydrodynamics in integrable systems out of …...a. bressan, hyperbolic conservation laws:...

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Emergent hydrodynamics in integrable systems out of equilibrium

Benjamin Doyon

Department of Mathematics, King’s College London

Bled, Slovenia, 8 June 2018

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Collaborators:

Alvise Bastianello, SISSA

Denis Bernard, École Normale Supérieure de Paris

M. Joe Bhaseen, KCL

Isabelle Bouchoule, Institue d’Optique, Paris

Olalla Castro Alvaredo, City, University of London

Jean-Sébastien Caux, University of Amsterdam

Jacopo De Nardis, École Normale Supérieure de Paris

Jérôme Dubail, Université de Lorraine

Rosemary Harris, Queen Mary University London

Robert Konik, Brookhaven National Laboratory

Maximilian Schemmer, Institut d’Optique, Paris

Herbert Spohn, Technische Universität München

Gerard Watts, KCL

Students:

Jason Myers, KCL (CANES)

Takato Yoshimura, CEA Saclay (on leave from KCL)

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Main papers

O. A. Castro-Alvaredo, B. Doyon, T. Yoshimura, Emergent hydrodynamics in integrable quantum systems

out of equilibrium, Phys. Rev. X 6, 041065 (2016)

B. Bertini, M. Collura, J. De Nardis, M. Fagotti, Transport in Out-of-Equilibrium XXZ Chains: Exact

Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201 (2016)

Both selected for a Viewpoint in Physics written by Jérôme Dubail

(http://physics.aps.org/articles/v9/153#c1)

Advertisements:

Conference on Non-Equilibrium Systems (CONES), King’s College London

25-27 June 2018, www.cones2018.com

Les Houches summer school on integrability in atomic physics and condensed matter

August 2018 (org.: J.-S. Caux, K. Kitanine, A. Kluemper, R. Konik)

Programme in Natal: Hydrodynamic of low-dimensional quantum systems

May 2019 (org.: BD, J. Dubail, G. Mussardo, M. Rajabpour, J. Viti)

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Plan

I. Basic theory of generalised hydrodynamics (GHD)

II. Numerical simulations and comparison with cold atom experiment

III. Further developments:

– steady states, geometry, method of characteristics

– zero temperature

– correlation functions, Drude weights

– large-deviation theory (a.k.a. full counting statistics)

– diffusion and viscosity

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I. Basic theory

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1. The problem: inhomogeneous dynamics of many-body integrable systems

Effects of integrability in the famous “Quantum Newton Cradle” experiment:

[Kinoshita, Wenger, Weiss 2006]

“Our results are probably explainable

by the well-known fact that a homoge-

neous 1D Bose gas with point-like colli-

sional interactions is integrable”

“Until now, however, the time evolution

of out-of-equilibrium 1D Bose gases

has been a theoretically unsettled is-

sue, as practical factors such as har-

monic trapping and imperfectly point-

like interactions may compromise inte-

grability”

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The Lieb-Liniger model describes point-like interactions of Galilean invariant Bose gases:

H =

∫dx h(x) =

∫dx

(1

2m∂xΨ

†∂xΨ +c

2Ψ†Ψ†ΨΨ

).

A simplification of the experimental problem is as follows. The initial state is an equilibrium

state with a inhomogeneous potential coupled to the density n(x) = Ψ†(x)Ψ(x):

〈A〉 = Tr (ρiniA)Trρini

, ρini = exp

[−β(H +

∫dxVini(x)n(x)

)].

Then the evolution occurs with the Hamiltonian in a different inhomogeneous potential

〈A(t)〉 = 〈eiHevotAe−iHevot〉, Hevo = H +∫

dxVevo(x)n(x).

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How can we compute the evolution of such a gas?

What are the general principles?

Can we reproduce the effects seen in the quantum Newton cradle experiment?

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2. From Gibbs ensembles to conventional hydrodynamics

For nice descriptions of standard concepts in hydrodynamics, see e.g.

A. Bressan, Hyperbolic conservation laws: An illustrated tutorial. In Modelling and Optimisation of Flows

on Networks, Cetraro, Italy 2009, Lecture Notes in Mathematics 2062, Springer, 2013

H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag, Heidelberg, 1991

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Hydrodynamics is the natural framework to describe inhomogeneous many-body phenomena.

The main idea behind hydrodynamics is “local thermodynamic equilibrium”.

It says that locally and on very short time scales (in fluid cells), the many-body system

“equilibrates” or relaxes. This means (naively) that locally we observe Gibbs states with

inverse temperature β(x, t) and chemical potential µ(x, t). Since things can be moving, then

in general these will be boosted by the local fluid velocity, with boost parameter ν(x, t):

ρGE(x, t) = e−β(x,t)(H−µ(x,t)N−ν(x,t)P )

and the hydrodynamic approximation is

〈O(x, t)〉ini ≈Tr (ρGE(x, t)O(0, 0))

TrρGE(x, t).

For instance (“local density approximation”):

ρini = e−β(H+

∫dxVini(x)n(x)) ⇒ ρGE(x, 0) = e−β(H+Vini(x)N).

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Hydrodynamics follows from local thermodynamic equilibrium + current conservation.

From microscopic conservation laws (for operators) to hydrodynamic equations (for Gibbs

averages)

∂th(x, t) + ∂xjh(x, t) = 0

∂tp(x, t) + ∂xjp(x, t) = 0

∂tn(x, t) + ∂xp(x, t) = 0

Take averages,hydro approximation⇒

∂th(x, t) + ∂xjh(x, t) = 0

∂tp(x, t) + ∂xjp(x, t) = 0

∂tn(x, t) + ∂xp(x, t) = 0

where

h(x, t) =Tr(ρGE(x, t) h)

TrρGE(x, t), jh(x, t) =

Tr(ρGE(x, t) jh)

TrρGE(x, t)

etc.

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Since there are only three parameters µ, ν, β to determine a boosted Gibbs state, there must

be two relations:

jh = F (h, p, n), jp = G(h, p, n).

These are the equations of state of the gas (which are highly model-dependent), and

combined with the above give the hydrodynamic equations for h, p, n.

• Note that h, p, n fix the potentials β, µ, ν. Thus hydrodynamics fixes the localspace-time dependent state.

• This is valid in the Euler limit: the limit of infinitesimal small variations in space and time.Beyond this scale, there are higher derivative corrections, such as viscosity terms.

• One usually write p = n v, giving the usual Euler equation

∂tv + v∂xv = −1

n∂xP

where the pressure is P = jp − nv2.

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3. From Gibbs ensembles to Generalized Gibbs ensembles (GGEs)

An old idea that got a lot of new interest recently starting with M. Rigol, V. Dunjko, V. Yurovsky, M.

Olshanii, Phys. Rev. Lett. 97, 050405 (2007).

Modern reviews:

A. Polkovnikov, K. Sengupta, A. Silva, M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011)

J. Eisert, M. Friesdorf, C. Gogolin, Nat. Phys. 11, 124 (2015)

C. Gogolin, J. Eisert, Rep. Prog. Phys. 79, 056001 (2016)

F. Essler, M. Fagotti, J. Stat. Mech. 2016, 064002 (2016)

L. Vidmar, M. Rigol, J. Stat. Mech. 2016, 064007 (2016)

E. Ilievski, M. Medenjak, T. Prosen, L. Zadnik, J. Stat. Mech. 2016, 064008 (2016)

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3. From Gibbs ensembles to Generalized Gibbs ensembles (GGEs)

But the Lieb-Liniger model is integrable. Integrable models possess an infinite number of

local conserved quantities

Qi =

∫dx qi(x), ∂tqi + ∂xji = 0, i = 0, 1, 2, 3, . . . unboundedly.

This gives rise to generalised Gibbs ensembles:

ρGGE = exp

[−∞∑

i=0

βiQi

].

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Combining: inhomogeneous dynamics of integrable systems

∂tqi + ∂xji = 0, qi, ji =Tr [ρGGE(x, t) qi, ji]

Tr ρGGE(x, t), i = 0, 1, 2, . . .

This is generalized hydrodynamics (GHD).

• The hydrodynamic principle is the emergence of local entropy maximization with respect toall available conserved charges, valid when variation lengths are large enough:

〈O(x, t)〉ini ≈Tr [ρGGE(x, t)O]

Tr ρGGE(x, t)

• There are equations of states: ji = Fi({qj}), and a bijective relation qi ↔ βi.

• Equations of conservations give dynamical equations determining the space-timedependent (generalized) Gibbs ensembles.

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Combining: inhomogeneous dynamics of integrable systems

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4. GHD in the quasi-particle language

for (generalized) thermodynamic Bethe ansatz:

C.N. Yang, C.P. Yang, J. Math. Phys. 10, 1115 (1969)

A. Zamolodchikov, Nucl. Phys. B 342, 695 (1990)

J. Mossel, J.-S. Caux, J. Phys. A 45, 255001 (2012)

For an understanding the relation GGE = (G)TBA in the XXZ chain:

E. Ilievski, E. Quinn, J.-S. Caux, Phys. Rev. B 95, 115128 (2017)

For GHD in the quasi-particle language:

O. A. Castro-Alvaredo, BD, T. Yoshimura, Phys. Rev. X 6, 041065 (2016)

B. Bertini, M. Collura, J. De Nardis, M. Fagotti, Phys. Rev. Lett. 117, 207201 (2016)

and the terms due to external force fields, temperature fields etc. were found in

BD, T. Yoshimura, SciPost Phys. 2, 014 (2017)

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Gibbs ensembles and generalized Gibbs ensembles can be described, in Bethe ansatz

integrable models, by using the (generalized) thermodynamic Bethe ansatz.

This is based on the fact that there emerge stable quasi-particles. The set of their momenta

and other quantum numbers is preserved under scattering, thus giving good quantum numbers

used to describe GGEs.

These quantum numbers are gathered into a spectral parameter θ characterizing the

quasi-particle. So, instead of the conserved densities qi, we use the conserved quasi-particle

densities ρp(θ):

{qi : i} ⇒ {ρp(θ) : θ}

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The quantity of charge Qi carried by the quasi-particle θ is a function hi(θ). So we have

qi =

∫dθ hi(θ)ρp(θ).

The GGE current averages ji are also given by linear functional on that space. Since ρp fully

determines the GGE, one can always write, for some veff(θ) = veff[ρp](θ),

ji =

∫dθ hi(θ)v

eff(θ)ρp(θ).

We now make GGEs space-time dependent. This means we promote ρp(θ) 7→ ρp(x, t; θ).We use completeness:

∂tqi + ∂xji = 0 ⇒∂

∂tρp(x, t; θ) +

∂

∂x

[veff(x, t; θ)ρp(x, t; θ)

]= 0

These are the GHD equations in the quasi-particle language.

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In order to get veff – the GGE equation of state – we need the model characteristics.

• spectral space: θ ∈ S , typically S = R× I where R represents the continuum ofvelocities and I a space of emerging particle types (including TBA strings, etc.).

• momentum function that defines relation between spectral space and real space, forinstance p(θ) = mθ or p(θ) = m sinh(θ).

• energy function that controls time evolution, for instance E(θ) = mθ2/2 orE(θ) = m cosh θ.

• differential scattering phase or TBA kernel that controls interactions, for instance ϕ(θ− θ′)=−id logS(θ − θ′)/dθ where S(θ − θ′) is the two-particle scattering amplitude.

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Using these model characteristics we can express the effective velocity. We find:

veff(θ) =E′(θ)

p′(θ)+

∫dα

ϕ(θ, α) ρp(α)

p′(θ)(veff(α)− veff(θ))

This is the “effectivization” of the group velocity E′(θ) / p′(θ).

This can be seen as the GGE equations of state (relating conserved currents to densities).

veff is in fact the velocity of excitations above finite-density states [Bonnes, Essler, Läuchli 2015].

The GGE equations of state can be proven using crossing symmetry in relativistic QFT, and in

the Lieb-Liniger model by taking the non-relativistic limit. It has been checked numerically in

the XXZ chain. There are now more complete proofs using form factor expansions [De Nardis,

Bernard, BD in progress; Yoshimura et al. in progress]

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We can also define the occupation function (or filling fraction)

n(θ) =ρp(θ)

ρs(θ)=

2πρp(θ)

p′(θ) +∫

dαϕ(θ, α)ρp(α)

These new hydrodynamic variables are the normal modes of GHD:

∂tn(x, t; θ) + veff(x, t; θ)∂xn(x, t; θ) = 0.

That is, the occupation function is the fluid coordinate that diagonalizes GHD. It is convectively

transported by the fluid, with propagation velocities veff(x, t; θ).

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All of this generalizes to the presence of force fields, etc.

It is the energy function that controls the time evolution. Assume that it is explicitly space

dependent E(θ) = E(x; θ). For instance in the repulsive Lieb-Liniger model in a force field,

Hevo = H +

∫dxVevo(x)n(x) ⇒ E(x; θ) = mθ2/2 + Vevo(x).

or evolved with inhomogeneous energy density,

Hevo =

∫dxJevo(x)h(x) ⇒ E(x; θ) = Jevo(x)mθ2/2.

Then the two following equivalent equations hold (here suppressing x, t; θ dependence):

∂tρp + ∂x[veffρp

]+ ∂θ

[aeffρp

]= 0

∂tn+ veff∂xn+ a

eff∂θn = 0

where the aeff is the “effectivization” of the acceleration−∂xE /∂θp.The local density approximation of thermal states e−βHevo are stationary solutions.

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5. Molecular dynamics and the effective velocity

BD, T. Yoshimura, J.-S. Caux, Phys. Rev. Lett. 120, 045301 (2018)

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We can simulate GHD using classical dynamics of particles in one dimension.

Consider a simple gas of hard rods, a gas of segments of lengths a lying on the line, and

propagating freely except for elastic collisions.

Upon collision, velocities are exchanged. A “quasi-particle” is a tracer of a given velocity. A

tracer jumps by a distance a at every collision.

By elastic collision, the number of tracers in a given velocity interval [v, v + dv] is preserved

by the dynamics, giving infinitely many conservation laws.

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Accumulating the jumps, an effective velocity develops at large scales:

It was shown rigorously that [Boldrighini, Dobrushin, Sukhov 1982]

veffcl (v) = v − a∫

dw ρp(w)(veffcl (w)− veffcl (v))

This is GHD if we identify v = θ and ϕ(θ, α) = −a !!!!

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We can generalize this to velocity-dependent jump lengths d(v, w) in any direction: the

“flea gas”.

The quasi-particles act like classical solitons of an integrable field theory, and indeed GHD

equations were found in soliton gases.

[Zakharov 1971; El 2003; El, Kamchatnov 2005; El, Kamchatnov, Pavlov, Zykov 2011]

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Simple counting of the additional distance due to the jumps give

veffcl (v) = v +

∫dw d(v, w) ρcl(w) (v

effcl (v)− veffcl (w))

One can check numerically that it works (here Lieb-Liniger model):

−2 −1 0 1 2

−1

0

1

v

veff

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II. Numerics and experiments

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6. An example: quantum Newton cradle

[J.-S. Caux, BD, J. Dubail, R. Konik, T. Yoshimura, 2017]

We are now ready to write the full dynamics for the original problem.

We represent the Bragg pulse used in the original experiment. We calculate ρharmop (x, θ)

associated to e−β(H+VharmoN), and then set the initial condition to

ρp(x, 0; θ) =1

2

[ρharmop (x, θ + θBragg) + ρ

harmop (x, θ − θBragg)

].

as the effect of the Bragg pulse followed by fast local entropy maximization.

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Remarks:

• No thermalization, no steady states: Euler equations are reversible, and possess infinitelymany conserved quantities (including the entropy),∫

dxdθ ρpf(n) any function f(n), e.g. entropy n log n+ (1− n) log(1− n)

• Both “horizontal” and “vertical” integrations are measurable:∫

dθ ρp(x, t; θ) is the

particle density at x, t, and∫

dx ρp(x, t; θ) is the self-similar particle density seen

on ray x/t = vgroup(θ) at large times after taking away the longitudinal trap.

• Eventually Euler equations are broken by viscosity / dissipation and stationarity isreached (coarse graining of fine structure, has been related to chaotic trajectories in

hard-rod gas [Cao, Bulchandani, Moore 2017]). Integrability breaking effects of potential still not

perceptible (no thermalization).

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7. Comparison with cold atom experiment: 1d expansion from a harmonic potential

[Bouchoule, BD, Dubail, Schemmer, in progress]

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III. Further developments

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8. Exact solutions

Consider the pratitioning protocol for generating non-

equilibrium steady states.

[Riemann 19th century; Spohn, Lebowitz 1977; Ruelle 2000; Tasaki

2000; Araki, Ho 2000; Aschbacher, Pillet 2003; Bernard, BD 2012]

The problem is invariant under x, t 7→ λx, λt: functions of the ray ξ = x/t:

(veff − ξ)∂ξn(ξ; θ) = 0.

In fluid dynamic terms, solution is a family (parametrised by θ) of contact singularities:

[Castro Alvaredo, BD, Yohimura & Bertini, Collura, De Nardis, Fagotti]

n(ξ; θ) = nL(θ)Θ(θ − θ?(ξ)) + nR(θ)Θ(θ? − θ(ξ)), veff(ξ; θ?(ξ)) = ξ.

Interpretations: Particles arrive either from the left

reservoir or the right reservoir depending on their

dressed velocity at the ray ξ.

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8. Exact solutions

There is a geometry behind GHD. Because of jumps, there is effectively additional space

available for propagation of quasi-particles, that depends on the local state.

[BD, Spohn, Yoshimura 2017].

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New coordinate q is related to real space x via density of states:

dq =2π ρs(x, t; θ)

p′(θ)dx

GHD equation becomes simple wave equation:

∂tn(q, t; θ) + vgroup(θ)∂qn(q, t; θ) = 0.

38

'

&

$

%

8. Exact solutions

Solution is immediate:

n(x, t; θ) = n(u(x, t; θ), 0; θ)

where the characteristics u(x, t; θ) solves∫ u

dy2π ρs(y, 0; θ)

p′(θ)=

∫ xdy

2π ρs(y, t; θ)

p′(θ)− vgroup(θ)t.

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x

t

u(x, t; ✓)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

x, 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

ve↵(✓)AAAB/HicbVDLSgNBEJz1GeNrNUcvg0GIl7Argh6DXjxGMA9IYpid9CZDZh/M9AaWJf6KFw+KePVDvPk3TpI9aGJBQ1HVTXeXF0uh0XG+rbX1jc2t7cJOcXdv/+DQPjpu6ihRHBo8kpFqe0yDFCE0UKCEdqyABZ6Elje+nfmtCSgtovAB0xh6ARuGwhecoZH6dmnymHVVQMH3p5UujgDZed8uO1VnDrpK3JyUSY563/7qDiKeBBAil0zrjuvE2MuYQsElTIvdREPM+JgNoWNoyALQvWx+/JSeGWVA/UiZCpHO1d8TGQu0TgPPdAYMR3rZm4n/eZ0E/eteJsI4QQj5YpGfSIoRnSVBB0IBR5kawrgS5lbKR0wxjiavognBXX55lTQvqq5Tde8vy7WbPI4COSGnpEJcckVq5I7USYNwkpJn8krerCfrxXq3Phata1Y+UyJ/YH3+AEnSlIQ=AAAB/HicbVDLSgNBEJz1GeNrNUcvg0GIl7Argh6DXjxGMA9IYpid9CZDZh/M9AaWJf6KFw+KePVDvPk3TpI9aGJBQ1HVTXeXF0uh0XG+rbX1jc2t7cJOcXdv/+DQPjpu6ihRHBo8kpFqe0yDFCE0UKCEdqyABZ6Elje+nfmtCSgtovAB0xh6ARuGwhecoZH6dmnymHVVQMH3p5UujgDZed8uO1VnDrpK3JyUSY563/7qDiKeBBAil0zrjuvE2MuYQsElTIvdREPM+JgNoWNoyALQvWx+/JSeGWVA/UiZCpHO1d8TGQu0TgPPdAYMR3rZm4n/eZ0E/eteJsI4QQj5YpGfSIoRnSVBB0IBR5kawrgS5lbKR0wxjiavognBXX55lTQvqq5Tde8vy7WbPI4COSGnpEJcckVq5I7USYNwkpJn8krerCfrxXq3Phata1Y+UyJ/YH3+AEnSlIQ=AAAB/HicbVDLSgNBEJz1GeNrNUcvg0GIl7Argh6DXjxGMA9IYpid9CZDZh/M9AaWJf6KFw+KePVDvPk3TpI9aGJBQ1HVTXeXF0uh0XG+rbX1jc2t7cJOcXdv/+DQPjpu6ihRHBo8kpFqe0yDFCE0UKCEdqyABZ6Elje+nfmtCSgtovAB0xh6ARuGwhecoZH6dmnymHVVQMH3p5UujgDZed8uO1VnDrpK3JyUSY563/7qDiKeBBAil0zrjuvE2MuYQsElTIvdREPM+JgNoWNoyALQvWx+/JSeGWVA/UiZCpHO1d8TGQu0TgPPdAYMR3rZm4n/eZ0E/eteJsI4QQj5YpGfSIoRnSVBB0IBR5kawrgS5lbKR0wxjiavognBXX55lTQvqq5Tde8vy7WbPI4COSGnpEJcckVq5I7USYNwkpJn8krerCfrxXq3Phata1Y+UyJ/YH3+AEnSlIQ=AAAB/HicbVDLSgNBEJz1GeNrNUcvg0GIl7Argh6DXjxGMA9IYpid9CZDZh/M9AaWJf6KFw+KePVDvPk3TpI9aGJBQ1HVTXeXF0uh0XG+rbX1jc2t7cJOcXdv/+DQPjpu6ihRHBo8kpFqe0yDFCE0UKCEdqyABZ6Elje+nfmtCSgtovAB0xh6ARuGwhecoZH6dmnymHVVQMH3p5UujgDZed8uO1VnDrpK3JyUSY563/7qDiKeBBAil0zrjuvE2MuYQsElTIvdREPM+JgNoWNoyALQvWx+/JSeGWVA/UiZCpHO1d8TGQu0TgPPdAYMR3rZm4n/eZ0E/eteJsI4QQj5YpGfSIoRnSVBB0IBR5kawrgS5lbKR0wxjiavognBXX55lTQvqq5Tde8vy7WbPI4COSGnpEJcckVq5I7USYNwkpJn8krerCfrxXq3Phata1Y+UyJ/YH3+AEnSlIQ=

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9. Zero-entropy GHD

[BD, J. Dubail, R. Konik, T. Yoshimura, Phys. Rev. Lett. 119, 195301 (2017)]

GHD simplifies drastically in the zero-entropy subspace

Let the occupation function n(θ) be formed of k disjoint, filled Fermi seas. Such occupation

functions have zero entropy. It turns out that the GHD equations simplify to a

finite-component hydrodynamics for the Fermi seas’ endpoints:

∂tθ±j + v

eff{θ}(θ

±j )∂xθ

±j = 0.

We refer to this as “2kHD”, for k = 1, 2, 3, . . .

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9. Zero-entropy GHD

Consider 2HD. A single Fermi sea: this is what we get in a zero temperature initial state.

This is a 2-component, Galilean invariant hydrodynamics. Hence the only possibility is

conventional hydrodynamics! That is,

2HD = conventional hydrodynamics (CHD)

∂tn + ∂x(vn) = 0, ∂tv + v∂xv = −1

mn∂xP, P = P(v, n).

Indeed a single Fermi sea is determined by fixing the density of particles and the boost

velocity. So the local density matrices have the form e−β(H−µN−νP )|β→∞.

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9. Zero-entropy GHD

Comparison with ABACUS quantum evolution:

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9. Zero-entropy GHD

In CHD, one usually encounters shocks that are sustained over time. At shocks, entropy is

continuously produced (the production of entropy at shocks is described by higher-derivative

terms neglected in Euler hydrodynamics).

However, in GHD, shocks are not sustained. Gradient catastrophes arise in 2HD, but

immediately “dissolve,” thanks to the availability of a large fluid space, into 4HD.

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9. Zero-entropy GHD

Remarks:

• CHD has been used in the past in order to describe the Lieb-Liniger model ininhomogeneous situations. It was partially successful. It has been observed that both at

collision of clouds and in propagation of density waves, shocks appear, beyond which CHD

is unable to reproduce tDMRG numerics. Here we have a full explanation of why CHD

works at zero temperature based on hydrodynamic principle only (local entropy

maximization), and exact hydrodynamic equations for describing what happens after the

gradient catastrophes.

• In real systems, exact gradient catastrophes do not ever occur: higher derivative terms(beyond Euler hydrodynamics) become important before. However, at very large scales

(Euler scales), gradients will look very sharp. GHD captures well what happens at such

scales.

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10. Correlation functions and linear fluctuating hydrodynamics

But we would like to go beyond averages of single observables. It turns out that Euler

hydrodynamics gives access to scaled, fluid-cell averaged dynamical correlation functions

〈O1(x1, t1) · · · ON (xN , tN )〉Eul

= limλ→∞

λN−1 〈Ō1(λx1, λt1) · · · ŌN (λxN , λtN )〉cλ.

The parameter λ characterize the fluid cell scale, and

〈O〉λ =Tr(e−

∫R dx

∑i βi(λ

−1x)qi(x)O)

Tr(e−

∫R dx

∑i βi(λ

−1x)qi(x))

The observables Ō(x, t) are appropriate fluid-cell averages

Ō(x, t) =∫

Nλ(x,t)

dyds

|Nλ|O(y, s).

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Linear fluctuating hydrodynamics and hydrodynamic projection operators

[BD, Spohn 2017]

The C operator.

A natural inner product is given by the bilinear form

〈O1|O2〉 =∫

dx 〈O1(x)O2(0)〉c =∫

dx 〈O1(x)O2(0)〉Eul

(after moding out the null space). A Hilbert space is obtained by completion.

The C operator of linear fluctuating hydrodynamics is defined by using conserved densities

Cij = 〈qi|qj〉

The qi span a sub-Hilbert space, that of pseudolocal charges. The C operator is simple to

calculate within TBA: we just have to differentiate with respect to βi the GGE average qj :

−∂qj∂βi

= − ∂∂βi

Tr(e−

∑k βkQkqj(0)

)

Tr(e−

∑k βkQk

) = 〈Qiqj(0)〉c = Cij

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For this, we use the pseudo-energy of TBA, which says how the state depends on βi:

n(θ) =1

1 + e�(θ), �(θ) =

∑

i

βihi(θ)−∫

dα

2πϕ(θ, α) log(1 + e−�(α))

After some calculations this gives

Cij =

∫dθ ρp(θ) (1− n(θ))hdri (θ)hdrj (θ)

where the “dressing” operation is defined by solving

hdr(θ) = h(θ) +

∫dα

2πϕ(θ, α)n(α)hdr(α)

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We can re-write this as a an integral operator on the spectral space

C = (1− nT )−1ρp(1− n)(1− Tn)−1

in the sense that

Cij =

∫dθ hi(θ)(Chj)(θ) =

∫dθdαC(θ, α)hi(θ)hj(α)

The integral operator T is

Th(θ) =

∫dθ

2πϕ(θ, α)h(α)

We see that hdr = (1− Tn)−1h.

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The B operator.

The B operator is defined using conserved currents

Bij = 〈ji|qj〉.

From general principles, it is symmetric, BT = B. Again it can be obtained by differentiating:

Bij = −∂jj∂βi

.

Using our known expression of the GGE average jj we get

Bij =

∫dθ ρp(θ) (1− n(θ)) veff(θ)hdri (θ)hdrj (θ)

That is

B = (1− nT )−1veffρp(1− n)(1− Tn)−1

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The Drude weight (Drude matrix).

The Drude weight is the matrix

Dij = limt→∞

∫dx 〈ji(x, t)jj(0, 0)〉c = 〈ji(t)|jj(0)〉

We realise that limt→∞〈ji(t)| projects onto the subspace of conserved quantities:

Dij =m∑

i′,j′=1

〈ji|qi′〉(C−1)i′j′〈qj′ |jj〉 that is D = BC−1B

This can be evaluated easily in the integral operator language

D = (1− nT )−1(veff)2ρp(1− n)(1− Tn)−1

giving

Dij =

∫dθ ρp(θ) (1− n(θ)) veff(θ)2hdri (θ)hdrj (θ)

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The A operator.

In order to obtain space-time dependent Euler scale correlations, we may use the evolution

operator A. This is defined as the derivative of the equations of state

Aij =∂ji∂qj

.

The quantity 〈ji(x, t)qj(0, 0)〉Eul equals

−δji(x, t)δβj(0)

= −∑

k

δqk(x, t)

δβj(0)

δji(x, t)

δqk(x, t)=∑

k

〈qk(x, t)qj(0, 0)〉EulAik

Taking t = 0 and integrating we find the (known) relation

B = AC

and so we obtain

A = (1− nT )−1veff(1− nT )

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The two-point functions of conserved densities.

Therefore we have

∂t〈qi(x, t)qj(0, 0)〉Eul + ∂x∑

k

Aik〈qk(x, t)qj(0, 0)〉Eul = 0

In Fourier space,

Sij(k, t) =

∫dx eikx〈qi(x, t)qj(0, 0)〉Eul = lim

k→0, t→∞kt fixed

∫dx eikx〈qi(x, t)qj(0, 0)〉c

Hence

∂tS = ikAS, S(k, 0) = C

The solution is

S(k, t) = eiktAC.

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That is,

Sij(k, t) =

∫dθ eiktv

eff (θ)ρp(θ) (1− n(θ))hdri (θ)hdrj (θ)

Back to real space,

〈qi(x, t)qj(0, 0)〉Eul =∫

dθ δ(x− veff(θ)t) ρp(θ) (1− n(θ))hdri (θ)hdrj (θ)

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More general two-point functions. [BD 2017]

Hydrodynamic projection principles allow us to generalize to generic fields:

〈O′(x, t)O(0, 0)〉Eul =∑

i,j,k,l

〈O′|qi〉C−1ij 〈qj(x, t)qk(0, 0)〉C−1kl 〈ql|O〉

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We obtain the necessary overlaps by differentiating GGE averages:

〈qi|O〉 = −∂〈O〉∂βi

=:

∫dθ ρp(θ) (1− n(θ))V O(θ)hdri (θ)

This gives

〈O′(x, t)O(0, 0)〉Eul =∫

dθ δ(x− veff(θ)t) ρp(θ) (1− n(θ))V O′(θ)V O(θ)

For instance,

V qi = hdri , Vji = veff hdri

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In the Lieb-Liniger model (with m = 1/2) we may take Pozsgay’s formulae [Pozsgay 2011] for

the Kth power of the particle density,

OK =1

(K!)2(Ψ†)K(Ψ)K

We find in a state defined by the occupation function n(p) (with p = θ)

V OK (p) =K∑

s=1

∫

RK−1

(K∏

r=1r 6=s

dpr2π

n(pr)hdrr (pr)

)hdrs (p) g

drs (p)

1dr(p)

where

hr(p) = pr−1, r = 1, 2, 3, . . .

and

gs(ps) =∏

j≥l

pj − pl(pj − pl)2 + (c/2)2

(defined as a function of ps with pr 6=s fixed parameters).

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All results also apply to classical integrable field theories. We look at sinh-Gordon model

[Bastianello, BD, Watts, Yoshimura 2017].

We evaluate numerically by Monte Carlo simulation the following two-point functions and

compare with our formulae (g is coupling):

1

t

∫ t

0

ds s 〈Tµµ (ζs, s)T νν (0, 0)〉 and1

t

∫ t

0

ds s 〈cosh(2gΦ(ζs, s)) cosh(2gΦ(0, 0))〉

Numerics

Analytic

Free

(a)

-1 0 10

0.05

0.1

0.15

ζ

〈Tμ μTν ν〉

Numerics

Analytic

(b)

-1 0 1

0

100

200

300

ζ

〈Ch(2)Ch(2)〉

Figure 6: The predictions (black curves) for the time average of two-point functions of(a) the trace of the stress-energy tensor, (46), and (b) vertex operators, (45) with (48),are compared to the numerical simulation (red curves). In the case (a) the free result ofAppendix B (dotted curve) is plotted for comparison. In order to improve the convergence,short times are avoided: the time average is taken in the interval [t0, t], rather than [0, t]as in eq. (46), with t0 = 5 and t = 20. The same parameters as those of Figure 5 areused.

NumericsAnalytic

(a)

0 4 8 12 16 200

0.2

0.4

Time

〈Tμ μTν ν〉

NumericsAnalytic

(b)

0 4 8 12 16 200

500

1000

Time

〈Ch(2)Ch(2)〉

Figure 7: The predictions (black curves) for the space integral of two-point functions of(a) the trace of the stress-energy tensor, (47), and (b) vertex operators, (28) with (48),are compared to the numerical simulation (red curves) as functions of time. At largetimes, the oscillating ripples are due to the Metropolis noise and are observed to reduceon increasing the number of samples. The same parameters m, �, g as those of Figure 5are used.

V k(✓) is given by

V k+1(✓) =g2

4⇡Vk+1

kX

l=0

V lV l+1 (2l + 1)n(✓)

pl(✓)dl(✓)

⇢p(✓)(48)

23

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11. Large deviation theory

[Myers, Bhaseen, Harris, BD in progress]

The statistics of the charge Qi being transported from the left to the right:

Prob(transfer of amount Q of charge Qi in time t ) ∼ e−tIi(Q)

where Ii(Q) is the large deviation function. Legendre transform is scaled cumulant

generating function for the integrated current at 0:

Fi(λ) =∞∑

n=1

cnn!λn = lim

t→∞t−1 log 〈 exp

[λ

∫ t

0

ji(0, s) ds

]〉

[free fermions: Levitov-Lesovik; quantum impurities: Saleur et al.; CFT: Bernard, BD]

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Main idea: current integral gives λ-dependent homogeneous, stationary states (GGE)

〈exp[λ∫∞−∞ ji(0, s)ds

]O〉

〈exp[λ∫∞−∞ ji(0, s)ds

]〉

= 〈O〉GGEλ

We used

dλ〈O〉GGEλ =

∫ ∞

−∞ds 〈ji(0, s)O〉cGGEλ

and our formulae for one- and two-point functions to get

∂nλ∂λ

= sign(veff)nλ(1− nλ)hdri,λ

Get an expression for Fi(λ), which can be evaluated numerically, and expressions for the

cumulants cn.

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Numerical checks in the hard rod gas:

Then by defining

n±(1 � a⇢±)�1f±(✓) (3.49)

we can show that this hydro equation with the initial conditions 3.46 is solved by:

n(⇠; ✓) = n�(✓)⇥(✓ � ✓?(⇠)) + n+(✓)⇥(�✓ + ✓?(⇠)) (3.50)

where,

veff (⇠; ✓?) = ⇠ (3.51)

The key thing to notice is that it seems there is ambiguity coming from the ambiguity inherent in⇥(0). However consider the physical situation and ensure we are consistent in the choice of conventionof the ⇥ function. If ⇥(0) = 0 then somehow, in the steady state, the particle density between thetwo connections must drop to zero. If ⇥(0) = 1 then we somehow have an extra dense point in thesteady state of the system. However with the choice ⇥(0) = 12 we get an average of the left and rightparticle densities which seems to physically be the most reasonable.

3.7 Results

Figure 1: Plot of moments scaled in time. These are for 1000000 samplings. There are 45000 particlesin this simulation.

4 The Full SCGF

Consider the first moment:

c1 =

Zd✓

2⇡E0(✓)n(✓)H(✓) (4.1)

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Then by defining

n±(1 � a⇢±)�1f±(✓) (3.49)

we can show that this hydro equation with the initial conditions 3.46 is solved by:

n(⇠; ✓) = n�(✓)⇥(✓ � ✓?(⇠)) + n+(✓)⇥(�✓ + ✓?(⇠)) (3.50)

where,

veff (⇠; ✓?) = ⇠ (3.51)

The key thing to notice is that it seems there is ambiguity coming from the ambiguity inherent in⇥(0). However consider the physical situation and ensure we are consistent in the choice of conventionof the ⇥ function. If ⇥(0) = 0 then somehow, in the steady state, the particle density between thetwo connections must drop to zero. If ⇥(0) = 1 then we somehow have an extra dense point in thesteady state of the system. However with the choice ⇥(0) = 12 we get an average of the left and rightparticle densities which seems to physically be the most reasonable.

3.7 Results

Figure 1: Plot of moments scaled in time. These are for 1000000 samplings. There are 45000 particlesin this simulation.

4 The Full SCGF

Consider the first moment:

c1 =

Zd✓

2⇡E0(✓)n(✓)H(✓) (4.1)

16

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12. Diffusion and viscosity

[De Nardis, Bernard, BD in progress]

The diffusion matrix Dij is defined via the Kubo formula (which involves subtracting the Drude

matrix Dij )

(DC)ij =

∫

Rdt

(∫

Rdx 〈ji(x, t)jj(0, 0)〉c −Dij

)

This changes the evolution of two-point functions as

∂tSij(x, t) +∑

k

(Aik∂x ∓

1

2Dik∂

2x

)Skj(x, t) = 0 (t ≷ 0)

or, in Fourier transform,

S(k, t) = exp

[−ikAt− 1

2k2D|t|

]C

It also adds a “Navier-Stokes” term to the GHD equation

∂tρp(θ) + ∂x(veff(θ)ρp(θ)) =

1

2∂x

∫dαD(θ, α)∂xρp(α)

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We find an exact expression for the diffusion operator

(DC)ij =

∫dθdαK(θ, α)hdri (θ)h

drj (α)

with

K(θ, α) = g(θ)w(θ)δ(θ − α)− g(θ)U(θ, α)g(α), g(θ) = ρp(θ)(1− n(θ))

where

U(θ, α) =ϕdr(θ, α)ϕdr(α, θ)

(2πρs(θ))(2πρs(α))|veff(θ)− veff(α)|

w(θ) =

∫dα g(α)

ϕdr(θ, α)2

(2πρs(θ))2|veff(θ)− veff(α)|

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Remarks:

• The operator DC is symmetric and positive, as it should.

• There is positive Yang-Yang entropy production:

S = −∫

dθdx ρs (n log n+ (1− n) log(1− n))

satisfies

∂tS = appropriate diagonal matrix element (DC)ii > 0

• Corresponding formulae for quantum chains, classical gases and field theories, andcorrectly specialises to the known diffusion operator in the hard rod gas [Boldrighini, Suhov 97].

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Main steps of calculation:

• Expansion into intermediate excitations of particles and holes in the GGE background:

〈ji(x, t)jj(0, 0)〉c

=

∫dθpdθhe

i(kp−kh)x−i(�p−�h)t〈GGE|ji|θpθh〉〈θpθh|jj |GGE〉

+

∫dθ1,2p dθ

1,2h e

i∑2a=1(k

ap−k

ah)x−i(�

ap−�

ah)t〈GGE|ji|θ1,2p θ1,2h 〉〈θ1,2p θ

1,2h |jj |GGE〉

+ · · ·

• Intergal over x gives δ(∑(kp − kh)). First line (one particle-hole pair) gives Drude weight,subtracted.

• Use of conservation equations:

〈GGE|ji|θ1,2,...p θ1,2,...h 〉 =∑

a

(�ap − �ah) fi(θ1,2,...p θ1,2,...h )

where fi has simple structure, with simple “kinematical poles” at θap = θ

bh.

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• Intergal over t gives δ(∑(�p − �h)).• At two particle-hole pairs: the conditions of conservation of energy and momentum impliesconservation of the set of momenta:

2∑

a=1

kap =2∑

a=1

kah,2∑

a=1

�2p =2∑

a=1

�ah ⇒ {θap} = {θah}

This specialises to the pole of fi, cancelling the factor (�ap − �ah)2.

• At higher particle-hole pairs: the conditions don’t fix the set, hence the factor (�ap − �ah)2dominates and makes the contribution vanish.

• General properties of integrable systems fix the kinematical poles.

Main message, interpreting holes as outgoing particles: two-body processes give rise to

diffusion, with amplitude given by two particle-hole pair matrix element of conserved currents.

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Conclusions

We have obtained a set of hydrodynamic equations describing the dynamics at large scales

(Euler scale) in integrable systems of all types, and the diffusive (Navier-Stokes) correction.

This is a theoretical framework “solving” the famous quantum Newton cradle experiment, and

giving exact results for a wealth of problems in non-equilibrium dynamics of integrable systems.

• GHD equations in the quasi-particle language have wide applicability:– quantum chains and quantum field theory;

– classical gases such as the hard rod gas and soliton gases;

– classical integrable field theory and most likely classical integrable chains as well.

• We also found formulae for Euler-scale two-point functions in inhomogeneous,non-stationary states [BD 2017].

Still to do: superdiffusion, subdiffusion; higher-derivative corrections; integrability breaking and

emergence of conventional hydrodynamics from GHD, ...

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...

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emergent hydrodynamics in integrable systems out of …...a. bressan, hyperbolic conservation laws:...

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