On the Generalised Hydrodynamics of Integrable QFT
with non-diagonal scattering
Francesco Ravanini
Alma Mater Studiorum - Università di Bologna
and
INFN - Sezione di Bologna

Workshop New Trends in Integrable Systems
Osaka City University - Sep 19, 2019

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 20191 / 24

Non equilibrium quantum physics

Experimental results and theoretical progresses
Advent of cold atoms, optical lattices andmolecular electronic devices⇓
new experimental insight in phenomena ofquantum non-equilibrium statistical mechanics
Kinoshita, Wenger, Weiss: A quantum
Newton’s cradle, Nature 440 (2006) 900

There has been much theoretical progress in the last decade. Studies focused on:quantum quench: responses to excitations or pulses
emergent hydrodynamics: steady properties (do not vary in time)
Paradigms:
effective reservoirs −→ open, non-unitary systems
hamiltonian reservoirs −→ close, unitary systems
F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 20192 / 24

Non equilibrium quantum physics

Partition protocol
123

Prepare two semi-infinite halves of a homogeneous 1D quantum system
thermalized independently at temperatures TL and TR
At time t = 0 connect the two halves so that they can exchange energy andparticles
The initial state |inii evolves for t > 0 with Hamiltonian H = HL + HR + δH.
At large times it reaches a steady regime

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 20193 / 24

Non equilibrium quantum physics

Partition protocol
123

Prepare two semi-infinite halves of a homogeneous 1D quantum system
thermalized independently at temperatures TL and TR
At time t = 0 connect the two halves so that they can exchange energy andparticles
The initial state |inii evolves for t > 0 with Hamiltonian H = HL + HR + δH.
At large times it reaches a steady regime

For an observable O the steady state limit means
Ost := lim lim he iHt Oe −iHt iinit→∞ L→∞

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 20193 / 24

Non equilibrium quantum physics

Local thermodynamic equilibrium
Assumption: local thermodynamic equilibrium
We observe the system on the scale of clouds of particles (10−6 m) rather
than at the scale of particles (10−10 m).
After some local relaxation time, physical properties vary only on space-timescales much larger than microscopic ones.
The system decomposes in fluid cells, each one in thermal equilibrium.
Potentials β(x, t) vary slowly in adjacent cells.

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 20194 / 24

Non equilibrium quantum physics

Local thermodynamic equilibrium
Assumption: local thermodynamic equilibrium
We observe the system on the scale of clouds of particles (10−6 m) rather
than at the scale of particles (10−10 m).
After some local relaxation time, physical properties vary only on space-timescales much larger than microscopic ones.
The system decomposes in fluid cells, each one in thermal equilibrium.
Potentials β(x, t) vary slowly in adjacent cells.
As a consequence:
Averages of local observables tend at large times, to averages evaluated inlocal Gibbs ensembles with space-time dependent potentialshO(x, t)i = Tr[ρ(x, t)O]

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

hOiβ(x,t)

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Non equilibrium quantum physics

Conservation laws and normal modes
Conserved quantities Qi =R

dxqi (x, t) in involution =⇒ conservation laws

∂t qi (x, t) + ∂x ji (x, t) = 0 ,i = 1, ..., NAverage of densities qi (x, t) = hqi i and currents ji (x, t) = hji i also satisfy∂t qi (x, t) + ∂x ji (x, t) = 0

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Non equilibrium quantum physics

Conservation laws and normal modes
Conserved quantities Qi =R

dxqi (x, t) in involution =⇒ conservation laws

∂t qi (x, t) + ∂x ji (x, t) = 0 ,i = 1, ..., NAverage of densities qi (x, t) = hqi i and currents ji (x, t) = hji i also satisfy∂t qi (x, t) + ∂x ji (x, t) = 0
Equation of State links ji = Fi (q)JacobianJij =∂Fi (q)∂qj

F. Ravanini (Unibo)=⇒∂t qi (x, t) +X

Jij ∂x qj (x, t)) = 0

j

GHD in non-diagonal S-matrix IQFT

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Non equilibrium quantum physics

Conservation laws and normal modes
Conserved quantities Qi =R

dxqi (x, t) in involution =⇒ conservation laws

∂t qi (x, t) + ∂x ji (x, t) = 0 ,i = 1, ..., NAverage of densities qi (x, t) = hqi i and currents ji (x, t) = hji i also satisfy∂t qi (x, t) + ∂x ji (x, t) = 0
Equation of State links ji = Fi (q)JacobianJij =∂Fi (q)∂qj=⇒∂t qi (x, t) +X

Jij ∂x qj (x, t)) = 0

j

J can be diagonalized with a change of coordinates q → n (normal modes)∂t ni (x, t) + vieff ∂x ni (x, t)) = 0
vieff can be interpreted as velocity of propagation of the i-th normal mode ni .
Invariance of this equation under rescaling (x, t) 7→ (ax, at) shows that thesolutions should depend only on ξ = x/t.
F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Generalized Hydrodynamic Description

Generalized Gibbs Ensemble (GGE)
Goal of this approach:
Compute the profile of functions ni (ξ).
Integrable systems −→ infinity of conserved charges = constraints on the ensemble
Density matrix has to take them into account to describe the dynamics of thesystemP

ρGGE =

e − i βi Qi
P
Tr[e − i βi Qi ]Complete set of charges Qi = {Ii , Xs,i }, local and quasi-local
Local charges P
Ij can be expressed in terms ofdensities Ij = ` ij (`) which have support on a
finite number of sites, e.g. in XXZ


dj
iπ
Ij = −i
log
T
θ
+
1
dθj
2
θ=0
F. Ravanini (Unibo)

Quasi-local
P charges Xs,j also have densities
Xs,j = ` xs,j (`) but their support is on an
extended region with exponentially decaying
norm [Ilievski et al. 2015]


dj
iπ
Xs,j = −i j log Ts θ +
dθ
2
θ=0

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Factorised S-matrix and TBA
In QFT2 integrability implies factorised S-matrix, no particle production andconservation of the set of momenta. [Parke, 1979]
Study the steady states after local relaxation time by use of Thermodynamic
Bethe Ansatz [Yang, Yang 1966 - Al. Zamolodchikov, 1990]
For IQFT2 with diagonal factorized S-matrix =⇒ Doyon, Castro-Alvaredo,
Yoshimura, Phys. Rev. X6, 041065 (2016) (DCY)

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Factorised S-matrix and TBA
In QFT2 integrability implies factorised S-matrix, no particle production andconservation of the set of momenta. [Parke, 1979]
Study the steady states after local relaxation time by use of Thermodynamic
Bethe Ansatz [Yang, Yang 1966 - Al. Zamolodchikov, 1990]
For IQFT2 with diagonal factorized S-matrix =⇒ Doyon, Castro-Alvaredo,
Yoshimura, Phys. Rev. X6, 041065 (2016) (DCY)
However, many theories have internal degrees of freedom and symmetries
organising particles in multiplets. The S-matrix is non diagonal(a,b) kl
(Sab )kl)mn (θij )mn (θij ) = Uab (θij )(RState with N-particles |θ1 , a1 ; ...; θN , aN i on a periodic box of length L. Add
to this state a probe particle with rapidity θ and impose periodic boundary
conditions on the wave function.

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Diagonalisation of color transfer matrix
This leads to Bethe-Yang condition of quantization of momenta
e ipa (θ)L =

N
Y
j=1

Sabj (θ − θj ) =NY

Uab (θ − θj )Tra

j=1

N
Y
km
(R (a,b) )mj j nj+1
(θ − θj )
j
j=1

|

{z

}

T (θ|{θj })=color transfer matrix

R (a,b) -matrix acts on the a, b multiplets of particles, indices k, l, m, n run in a
multiplet. Between different multiplets a, b the S-matrix is block-diagonal.

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Diagonalisation of color transfer matrix
This leads to Bethe-Yang condition of quantization of momenta
e ipa (θ)L =

N
Y
j=1

Sabj (θ − θj ) =NY

Uab (θ − θj )Tra

j=1

N
Y
km
(R (a,b) )mj j nj+1
(θ − θj )
j
j=1

|

{z

}

T (θ|{θj })=color transfer matrix

R (a,b) -matrix acts on the a, b multiplets of particles, indices k, l, m, n run in a
multiplet. Between different multiplets a, b the S-matrix is block-diagonal.
Diagonalisation of T by Bethe ansatz. The eigenvalues of T enter the
Bethe-Yang equation (BYE).
They depend on the rapidities θn of the particles but also on someparameters un characterising the states (the Bethe roots), that are
determined by a set of Bethe Ansatz Equations (BAE).

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Thermodynamic limit
Thermodynamic limit N → ∞. The set of rapidities θ1 , ..., θN and of Betheroots u1 , ..., uM tend to continuum.
solutions uk organize in n-strings
(n)

(n)

uk,α = rk +iπ(n + 1 − 2α)2,

α = 1, ..., nintroduce density of possible string centres σn (θ)
density of occupied string centres ρn (θ)
density of holes (unoccupied string centres) ρ̄n (θ) = σn (θ) − ρn (θ)
densityof occupied
quasi-particle states ρp (θ) such thatR
PN
dθρp (θ)...
j=1 ... 7−→

Split the product on all Bethe roots as
M
Y
k=1

··· =

Mn Y
n
YY

···

n∈U k=1 α=1

U = set of all possible types of strings (depends on the model)
F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Perturbed coset CFT’s and Dynkin TBA
Here we study the case where R(θ) is the R-matrix of Uq (G), restricted at q =root of 1, for some algebra G = A, D, E . In the RSOS basis, these S-matrices
describe the perturbed CFT coset models
Gk × G `+ φid,idadj
Gk+`
S(θ) = X (θ)Sk (θ) ⊗ S` (θ)
Spectrum of kinks of mass m(a) (Perron-Frobenius eigenvector of G) separatingcolored vacua with RSOS(k) × RSOS(`) strucutres.

TBA is encoded on product of Dynkin
diagrams G  Ak+`−1
FR, Tateo, Valleriani 1992 — Quattrini, FR,
Tateo 1993
see also: Kuniba, Nakanishi, Suzuki, 1994 and2011

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Thermodynamic Bethe Ansatz (TBA)
Log-derivatives of BYE+BAE lead toσn(a) (θ) = δn` p 0(a) (θ) +

rankG
X

ϕ(a,b) ∗ ρ(b)n (θ) +k+`−1
X

In,m ϕ ∗ ρ(a)
m (θ)

m=1

b=1

Kernels (g = dual Coxeter G)1ϕ̃(a,b) (κ) = 2π  δ ab −G ab
2 cosh πκ
g!−1− δ ab ,ϕ(θ) =

g
2 cosh g2θ

G ab is the incidence matrix of the G Dynkin diagram, In,m is the incidence matrixof Ak+`−1 Dynkin diagram

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Thermodynamic Bethe Ansatz (TBA)
Log-derivatives of BYE+BAE lead toσn(a) (θ) = δn` p 0(a) (θ) +

rankG
X

ϕ(a,b) ∗ ρ(b)n (θ) +k+`−1
X

In,m ϕ ∗ ρ(a)
m (θ)

m=1

b=1

Kernels (g = dual Coxeter G)1ϕ̃(a,b) (κ) = 2π  δ ab −G ab
2 cosh πκ
g!−1− δ ab ,ϕ(θ) =

g
2 cosh g2θ

G ab is the incidence matrix of the G Dynkin diagram, In,m is the incidence matrixof Ak+`−1 Dynkin diagram
Minimizaton of free energy with the constraints of BYE and BAE leads to theG  Ak+`−1 TBA equations
log yn(a) (θ) = δn` p 0(a) − ϕ(a,b) ∗ log Yn(b) (θ) + Inm ϕ ∗ log Ym(a) (θ)
F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Integrabilty and TBA

Universal form and Y-system(a)

Functions yn

defined as(a)ρn (θ)
density of "string centres"yn(a) (θ) = (a)=density of holesρ̄n (θ)(a)Yn(a) (θ) = 1 + yn(a) (θ) =σn (θ)(a)ρ̄n (θ)(a)Ŷn(a) (θ) = (1 + yn(a) (θ)−1 )−1 =ρn (θ)(a)σn (θ)= nn(a) (θ) = occupation numbers

This can be recast in a set of functional equations (Y-system of G  Ak+`−1 type) rankG
k+`−1
Y
Y
iπ
iπ
Ŷn(b) (θ)Gab
Ym(a) (θ)Inm
yn(a) θ +yn(a) θ −
=
g
g
m=1
b=1

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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TBA for GHD

Compact notation
Introduce the matrix Φ of kernel functions with entries
(a,b)
Φn,m
= ϕ(a,b) (θ)δn,m − In,m δ a,b ϕ(θ)

Also introduce a vector notation for the currents
q = {qi , i = 1, 2, 3, ...}
TBA can be compactly written as(b)
log yn(a) = νn(a) − Φ(a,b)
n,m ∗ log Ym(a)where νn = m(a) δn` cosh θ
Averages of densities can be written as
q (a) =

k+`−1
X Z

(a)

dθh(a) (θ) log Y` (θ)

n=1
F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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TBA for GHD

Formulation of TBA with GGE
Conserved charges act asQ|θ1 , a1 ; ...; θN , aN i =NXh(ak ) (θk )|θ1 , a1 ; ..., θN , aN i

k=1h(a) (θ) one particle eigenvalue of Q. If relativistic (θ = rapidity):(a)h1 (θ) = m(a) cosh θ,(a)h2 (θ) = m(a) sinh θ

If Galilean (θ = velocity):(a)h1 (θ) = e (a) (θ) = m(a) θ2 /2 ,(a)h2 (θ) = p (a) (θ) = m(a) θ

Q0 = N, Q1 = H, Q2 = P... and [Qi, Qj ] = 0 (also quasi-local charges)

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 201914 / 24

TBA for GHD

Formulation of TBA with GGE
Conserved charges act asQ|θ1 , a1 ; ...; θN , aN i =NXh(ak ) (θk )|θ1 , a1 ; ..., θN , aN i

k=1h(a) (θ) one particle eigenvalue of Q. If relativistic (θ = rapidity):(a)h1 (θ) = m(a) cosh θ,(a)h2 (θ) = m(a) sinh θ

If Galilean (θ = velocity):(a)h1 (θ) = e (a) (θ) = m(a) θ2 /2 ,(a)h2 (θ) = p (a) (θ) = m(a) θ

Q0 = N, Q1 = H, Q2 = P... and [Qi, Qj ] = 0 (also quasi-local charges)
In the formulation of TBA we assume the generalized hamiltonian:XZ(a)
HGGE =
ρ(a)n (θ)wn (θ)dθa,n(a)
New driving term wn (θ, x, t) = β(x, t) · δn,` h(a) (θ, x, t)
F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 201914 / 24

TBA for GHD

Formulation of TBA with GGE IIThis allows to:Extend the phase space of the Hamiltonian considering the higher charges asinteractions
Calculate the averages of quantitites using GGE density matrix

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 201915 / 24

TBA for GHD

Formulation of TBA with GGE IIThis allows to:Extend the phase space of the Hamiltonian considering the higher charges asinteractions
Calculate the averages of quantitites using GGE density matrix
Minimization of generalised free energy FGGE leads to generalised TBA(b)log yn(a) = wn(a) − Φ(a,b)
n,m ∗ log Ym

Z
FGGE =

F. Ravanini (Unibo)dθwn(a) (θ) log Yn(a) (θ)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 201915 / 24

TBA for GHD

Formulation of TBA with GGE IIThis allows to:Extend the phase space of the Hamiltonian considering the higher charges asinteractions
Calculate the averages of quantitites using GGE density matrix
Minimization of generalised free energy FGGE leads to generalised TBA(b)log yn(a) = wn(a) − Φ(a,b)
n,m ∗ log Ym

Z
FGGE =dθwn(a) (θ) log Yn(a) (θ)

For any density h(θ) define the dressing operation as(a)dr(b) (b) dr[h(a)− Φ(a,b)n,m ∗ nm [hm ]n ] = δn,` h

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

Osaka - Sep 19, 201915 / 24

TBA for GHD

Spectral current and effective velocity
(a)

(a)

q (a) = {qi } and ρ(a) = {ρn } are alternative complete sets to characterize the
GGE ensemble.
Z
Z
(a)
(a)
(a)0
(a)
(a)
q = dθρ` (θ)h (θ) = dθn` (θ)[p` (θ)]dr h(a) (θ)
Z
(a)
(a)
= dθn` (θ)p (a)0 (θ)[h` (θ)]dr
Currents can be obtained by a Double Wick rotation (crossing operation C)(x, t) 7→ (it, −ix) ,

j

(a)

θ 7→
Z

=
(a)

iπ
−θ
2

,

(a)

(a)0

(e (a) , p (a) ) 7→ (ip (a) , −ie (a) )dθn` (θ)[e` (θ)]dr h(a) (θ)
(a)

(a)0

Define the spectral currents ρ̂n (θ) = nn (θ)[en (θ)]drZ(a)j (a) = dθρ̂` (θ)h(a) (θ)
F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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TBA for GHD

Effective velocity

j (a) =

Z

(a)

(a)

dθv` (θ)eff ρ` (θ)h(a) (θ)

Effective velocity (Group velocity is v (a) (θ)gr = e (a)0 (θ)/p (a)0 (θ))(a)0[en (θ)]dr
vn(a) (θ)eff = (a)0[pn (θ)]dr(a)=ρ̂n (θ)(a)=ρn (θ)(b)(a,b)
b,m Φn,m ∗ ρ̂m (θ)P(b)
p (a)0 (θ)δn` − b,m Φ(a,b)
n,m ∗ ρm (θ)

e (a)0 (θ)δn` −(a)vn(a) (θ)eff = v (a) (θ)gr δn` +

(a)

P

(a)

ρn ∗ vn (θ)eff − vn (θ)eff
p (a)0 (θ)(a)

This is the velocity of propagation of quasi-particles. It is determined by ρn thatcompletely characterize the steady state.

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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TBA for GHD

Bethe-Boltzmann equation(a)Euler equation ∂t h(a) + ∂x j (a) = 0 can be written in terms of ρn(a)(a)(a) eff∂t ρ(a)∂x ρ(a)n + ∂x ρ̂n = 0 = ∂t ρn + vnnor after some manipulation, in terms of the occupation number n∂t nn(a) (θ) + vn(a) eff (θ)∂x nn(a) (θ) = 0
which is the diagonal form with normal modes propagating with velocity veff .
Similar Euler equation holds for σ and ρ̄ and so entropy is conserved∂t s + v eff ∂x s = 0
as it should be in a fluid without viscosity.

F. Ravanini (Unibo)

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Measurable quantities

Partition protocol problem
Solving the two-reservoir system corresponds to solving the initial value problem(∂t n(θ, x, t) + v eff (θ)∂x n(θ, x, t) = 0n(θ, x, 0) = nin (θ, x) = nL (θ)Θ(−x) + nR (θ)Θ(x)Ansatzn(θ, x, t) = nin (θ, x − v eff (θ)t)
givingn(θ, x, t) = nL (θ)Θ(−x + v eff (θ)t) + nR (θ)Θ(x − v eff (θ)t)
If v eff (θ) is monotonic in θ, the equation v eff (θ)t − x = 0 has a unique solution
θ? −→ Solution to initial value problem(n(θ, x, t) = nL (θ)Θ(θ − θ? (x, t)) + nR (θ)Θ(θ? (x, t) − θ)v eff (θ? (x, t)) = xt

F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Measurable quantities

Numerical implementation I
Functions θ? (ξ) are also solutions ofp(θ? (ξ), ξ)dr = 0
Numerical procedure
1

Solve TBA for right and left using w (θ) = βL,R cosh θ −→ L,R (θ)

2ComputenL,R (θ) =3

1
1 + e L,R (θ)

Fix ξ. Choose inital value θ? (ξ)0 = 0. Solve p(θ? (ξ), ξ)dr = 0 iteratively
nn (θ) = nL (θ)Θ(θ − θn? ) + nR (θ)Θ(θn? − θ)Zdα?
p(θn+1)dr = p(θn? ) +ϕ(θn? − α)nn (α)p(θn? )dr
2π?
?
p(θn+1) = 0 −→ θn+1

F. Ravanini (Unibo)

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Osaka - Sep 19, 201920 / 24

Measurable quantities

Numerical implementation II
1

Use the stable θ? (ξ) to compute the total occupation numbern(θ, ξ) = nL (θ)Θ(θ − θ? (ξ)) + nR (θ)Θ(θ? (ξ) − θ)23

4

We can now compute all dressed quantities h(θ, ξ)dr as we need
Finally we can use n(θ, ξ) and hi (θ, ξ)dr to compute all average densities andcurrents qi (ξ), ji (ξ)
The procedure can be repeated at different rays ξ

In the CFT limit one can compare this results with the expected prediction(Bernard, Doyon 2016)
2
2
JCFT = πc
12 (TL − TR )F. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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Measurable quantities

Preliminary numerical results for TIM
We have performed some numerical checks for the simplest non-diagonal A1  A2
model: the Tricritial Ising Model perturbed by its least relevant operator φ13 .
Initial solutions with discontinuityσ=

F. Ravanini (Unibo)

TL
βR
=
βL
TR

,

K (θ) = n` (θ)p` (θ)

GHD in non-diagonal S-matrix IQFT

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Measurable quantities

Conclusions
TBA methods allow to access exact information about non-equilibriumfeatures of integrable systems.
In particular, in emergent hydrodynamic paradigma, one can describe exactly
and non-perturbatively the stationary currents corresponding to steady states
between two thermal reservoirs.
The TBA has been generalized to cases with non-diagonal S-matrix and put
in relation with Y-systems. Modifications in the Euler equations have been
pointed out.
Many issues have still to be developed within this technique. For example:
Where possible, an NLIE should be used instead. This would give access tostudies in very important models, like e.g. sine-Gordon theory (next step)
Numerical simulations are to be performed and checked against these
theoretical results (TEBD method)

F. Ravanini (Unibo)

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Measurable quantities

Thank youF. Ravanini (Unibo)

GHD in non-diagonal S-matrix IQFT

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