Thermodynamic Bethe ansatz for non-equilibrium...

Thermodynamic Bethe ansatz for non-equilibrium steadystates in integrable QFT

Olalla A. Castro-Alvaredo

School of Mathematics, Computer Science and EngineeringDepartment of Mathematics

City University London

EPSRC NetworkPlus Workshop on Non-equilibrium QuantumSystems, University of Nottingham

29-30 May 2014

This talk is based mainly on the work

Olalla A. Castro-Alvaredo, City University London TBA for non-equilibrium steady states in IQFT

O. C.-A., Yixiong Chen, Benjamin Doyon and MarianneHoogeveen, Thermodynamic Bethe ansatz for non-equilibrium

steady states: exact energy current and fluctuations in

integrable QFT, J. Stat. Mech. (2014) P03011; arXiv:1310.4779

O. C.-A., Yixiong Chen, Benjamin Doyon and MarianneHoogeveen, Thermodynamic Bethe ansatz for non-equilibrium

steady states: exact energy current and fluctuations in

integrable QFT, J. Stat. Mech. (2014) P03011; arXiv:1310.4779

It builds on previous results for CFT which have been discussedby Benjamin Doyon in a previous talk.

Overview of the talk

Introduction to the TBA approach

Adapting the TBA approach to the study ofnon-equilibrium steady states (NESS)

The NESS energy current: examples

NESS c-functions: examples

Energy current as as Poisson process

Non-additivity of the NESS current: examples

Conclusions

Introduction

Integrable Quantum Field Theories in 1+1 dimensions possesvery desirable properties: they can be “solved” exactly,meaning that their full scattering matrix and matrix elementsof local fields can be obtained non-perturbatively.

Introduction

Nowadays quasi-one-dimensional integrable systems can berealized in the laboratory and they exhibit unique properties.

Introduction

The Thermodynamic Bethe Ansatz (TBA) approach is aclassical approach to the study of IQFTs.

Introduction

It was introduced by Zamolodchikov (1990) as a method for thecomputation of the ground state energy of IQFT on an infinitecylinder whose circumference is identified as compactified time.

Introduction

It was introduced by Zamolodchikov (1990) as a method for thecomputation of the ground state energy of IQFT on an infinitecylinder whose circumference is identified as compactified time.Alternatively, we may regard this as a formulation of QFT atfinite temperature T .

TBA: Particles on a Trip Around the World

The BA equations arise from the requirement of periodicity ofthe wave function

��

eiLMA sinh θA

SAB(θA − θB) = 1

LMA sinh θA +∑

δAB(θA − θB) = 2πnA, A = 1, · · · , n

Review of the TBA approach

Taking the thermodynamic limit NA, L → ∞ with NA/Lfinite (NA is the number of particles of species A) andrequiring thermodynamic equilibrium the TBA equationsemerge:

ǫA(θ) = MAβ cosh θ −∑

ϕAB ∗ LB(θ)

Here ǫA(θ) are the pseudo-energies, β = 1/T ,

ϕAB = −id ln(SAB(θ))dθ

, LB(θ) = ln(1 + e−ǫB(θ)) and *indicates convolution f ∗ g(θ) := 1

f(θ − θ′)g(θ′)dθ′.

ǫA(θ) = MAβ cosh θ −∑

ϕAB ∗ LB(θ)

Here ǫA(θ) are the pseudo-energies, β = 1/T ,

ϕAB = −id ln(SAB(θ))dθ

, LB(θ) = ln(1 + e−ǫB(θ)) and *indicates convolution f ∗ g(θ) := 1

f(θ − θ′)g(θ′)dθ′.

The free energy is

f(β) = −β

∫ ∞

−∞

dθLA(θ) cosh θ := −πβ2ceff(β)

TBA and CFT

The function ceff(β) is a scaling function which can beinterpreted as an “off-critical” Casimir energy [Blote,Cardy & Nightingale’86; Affleck’86].

TBA and CFT

In the UV limit limβ→0 ceff(β) = ceff := c− 24∆ where ceffis the effective central charge, c is the central charge and ∆is the lowest conformal dimension in the Kac’s table of theunderlying CFT.

TBA and CFT

In the UV limit limβ→0 ceff(β) = ceff := c− 24∆ where ceffis the effective central charge, c is the central charge and ∆is the lowest conformal dimension in the Kac’s table of theunderlying CFT.

Other CFT properties may be recovered from the TBAequations. For example, when expressed in terms ofY -systems the latter exhibit periodicities which have beenrelated to the dimension of the perturbing field[Zamolodchikov’91].

TBA for non-equilibrium steady states

The TBA contains all of the key elements we need for aNESS formulation.

It has temperature dependence naturally built in and itallows for the computation of generalized energies“pseudo-energies”

Probably, the simplest modification of the TBA equationsis the following:

ǫA(θ) = WA(θ)−∑

ϕAB ∗ LB(θ)

WA(θ) = MAβl cosh θ for θ > 0 and WA(θ) = MAβr cosh θfor θ < 0.

ϕAB ∗ LB(θ)

WA(θ) = MAβl cosh θ for θ > 0 and WA(θ) = MAβr cosh θfor θ < 0.

This generalization follows from fundamental properties ofintegrable systems, especially that in the infinite past andfuture, states become well-separated, well-definedcollections of wave packets behaving like free particles[Doyon’12]

The NESS energy current

The current is a measure of energy transfer from the hot tothe cold reservoir. In the TBA context this naturallycorresponds to a derivative of the free energy:

J(βl, βr) =dfa(βl, βr)

dθMA cosh θxA(θ)

1 + e−ǫA(θ),

dθMA cosh θxA(θ)

1 + e−ǫA(θ),

where a is an auxiliary parameter which we introduce in

ǫA(θ) = WA(θ) + a pA(θ)−∑

(ϕAB ∗ log(1 + e−ǫB ))(θ)

dθMA cosh θxA(θ)

1 + e−ǫA(θ),

where a is an auxiliary parameter which we introduce in

ǫA(θ) = WA(θ) + a pA(θ)−∑

(ϕAB ∗ log(1 + e−ǫB ))(θ)

fa(βl, βr) is the free energy, as defined earlier. The

functions xA(θ) =dǫA(θ)da

a=0can be obtained by solving

xA(θ) = MA sinh θ +∑

ϕAB ∗xB

1 + eǫB

Current and L-functions: sinh-Gordon model

c-functions: roaming trajectories model

We have seen in Benjamin’s talk that in CFTJ(βl, βr) =

cπ12 (β

−2l − β−2

cπ12 (β

−2l − β−2

The quantity c1(Tl, Tr) :=πJ(βl,βr)12(T 2

l−T 2

r )is a function of

temperature which is zero for low temperatures andapproaches the central charge c for high temperatures.

cπ12 (β

−2l − β−2

l−T 2

r )is a function of

From examples, it behaves as a new c-function!

cπ12 (β

−2l − β−2

l−T 2

r )is a function of

From examples, it behaves as a new c-function!

We can define a whole family of such functions if we alsoexamine the current’s cumulants.

Energy current as a Poisson process

It has been shown [Bernard & Doyon’13] that the scaledcumulants generating function F (z) in a system with pureenergy transmission is given by

F (z) =

0dsJ(βl − s, βr + s)

F (z) =

The energy current scaled cumulants for such theories aresimply Cn+1 =

dznJ(βl − z, βr + z)|z=0.

F (z) =

In CFT the function F (z) =∫

dqw(q)(ezq − 1) withw(q) = cπ

12 e−βlq for q > 0 and w(q) = cπ

12eβrq for q < 0.

F (z) =

12eβrq for q < 0.

It is a superposition of Poisson processes for every energyE representing jumps towards the right or left withMaxwell-Boltzman factors e∓βl,rE.

F (z) =

12eβrq for q < 0.

It is a superposition of Poisson processes for every energyE representing jumps towards the right or left withMaxwell-Boltzman factors e∓βl,rE.

This interpretation also holds for massive free theories.Numerical analysis suggests that it may also hold for othernon-trivial QFTs.

Additivity/Non-additivity of the current: examples

In CFT the current has the form J(βl, βr) = f(βl)− f(βr).Therefore it satisfies the additivity property

P (β, σ) :=J(β, σβ) + J(σβ, σ2β) + J(σ2β, β)

J(β, σ2β)= 0

Question: does this still hold for massive integrable QFT?Answer: It seems from our numerics and some perturbativecalculations that the answer is NO.

J(β, σ2β)= 0

Question: does this still hold for massive integrable QFT?Answer: It seems from our numerics and some perturbativecalculations that the answer is NO. This can be seen quitestrikingly in the roaming trajectories model

Conclusions

We have proposed a new approach to the computation ofthe energy current in NESS of integrable 1+1 dimensionalQFTs.

Conclusions

We have proposed a new approach to the computation ofthe energy current in NESS of integrable 1+1 dimensionalQFTs.Within this approach we have numerically obtained thecurrent for several models and found a relationshipbetween the current and its cumulants and a new family ofc-functions.

Conclusions

We have proposed a new approach to the computation ofthe energy current in NESS of integrable 1+1 dimensionalQFTs.Within this approach we have numerically obtained thecurrent for several models and found a relationshipbetween the current and its cumulants and a new family ofc-functions.The monotonicity properties of these c-function imposeconstraints on the growth of the cumulants with a changein mass scale.

Conclusions

We have proposed a new approach to the computation ofthe energy current in NESS of integrable 1+1 dimensionalQFTs.Within this approach we have numerically obtained thecurrent for several models and found a relationshipbetween the current and its cumulants and a new family ofc-functions.The monotonicity properties of these c-function imposeconstraints on the growth of the cumulants with a changein mass scale.We have found numerical evidence for an interpretation ofthe current flow as a Poisson process both in CFT andQFT.

Conclusions

We have proposed a new approach to the computation ofthe energy current in NESS of integrable 1+1 dimensionalQFTs.Within this approach we have numerically obtained thecurrent for several models and found a relationshipbetween the current and its cumulants and a new family ofc-functions.The monotonicity properties of these c-function imposeconstraints on the growth of the cumulants with a changein mass scale.We have found numerical evidence for an interpretation ofthe current flow as a Poisson process both in CFT andQFT.We have found evidence that the additivity of the currentin not preserved in QFT. This is unlike results [Karrasch,Ilan & Moore’12] although they consider the non-universal,higher temperature regime of gapless chains.

Future directions

Investigate the non-additivity property further, for exampleby comparing our results to DMRG simulations on gappedspin chains.

Future directions

Prove that the functions cn(Tl, Tr) are c-functions.

Future directions

Prove that the functions cn(Tl, Tr) are c-functions.

Carry out TBA numerics for non-diagonal theories(e.g. sine-Gordon model).

Thermodynamic Bethe ansatz for non-equilibrium...

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