Correlation Functions of In- and Out-of-Equilibrium Integrable Models J. De Nardis

Summary

Since the birth of the scienti�c method it became widespread in the physical sci-ences the idea that natural phenomena can always be understood by studyingthe physics of each of their elementary constituents. While this attitude led toimmense discoveries on the sub-atomic particles that brought to a uni�ed the-ory of electromagnetism, weak and strong interactions inside the nuclei (the socalled standard model), on the other hand it left many big questions with noproper answer. For example, one of the most fascinating concepts of modern-day physics is emergence, the process whereby complex macroscopic propertiesarise from the interaction of many simple-behaved constituents. For example,when interactions among the single constituents of a system becomes large, theproperties of the whole are in general completely di�erent from the propertiesof the single element. New undiscovered physics therefore emerges from thepresence of such non-trivial interactions which cannot usually be treated per-turbatevely, namely they cannot be addressed assuming that the interactionsare small. When the constituents are con�ned to a one-dimensional geometryfor example interactions becomemore andmore e�ective. The tra�c on a singleline road can switch from �uid to jammed by slightly increasing the density ofcars above a certain critical value. In the same way the movement of many in-teracting quantum particles in one dimension is collective and they behave as a�uid. When prepared in an out-of-equilibrium situation they reorganize them-selves in a unusual, hard-to-predict manner that possibly leads to new exoticstates of matter.

Quantum systems with a large number of strongly interacting constituents areusually very hard to address. To solve a quantum mechanical problem one in-deed has to diagonalize the Hamiltonian of the system which is a matrix of thesame size as the Hilbert space, whose dimension grows exponentially with thenumber N of elementary particles in the system. The study of quantum systemsat equilibrium is based on a number of tools that either rely on the weakness ofthe interaction, like perturbation theory, or on the assumption that the �uctu-ations of physical observables around their equilibrium values are assumed tobe small, as it is done in mean-�eld approaches. These approaches are based onassumptions that can rarely be checked with an exact solution for any numberof particles N . However if we restrict ourselves to one spacial dimension it turnsout that it is possible to �nd exact solutions for models which are truly interact-ing andphysically relevant. Thesemodels are called integrable and the two exam-ples addressed in this thesis are the XXZ Heisenberg spin chain 1 and the Lieb-

1With Hamiltonian H � J∑L

j�1

(σx

j σxj+1 + σ

yj σ

yj+1 + ∆σ

zj σ

zj+1

)with σαj the Pauli matrices associ-

ated to each spin in the lattice.

Liniger model for point-interacting bosons 2. Like non-interacting theories arereducible to single particle physics, in these models the many-body scatteringsamong the particles are reducible to sequences of two-body processes 3. This re-duction is enforced by a certain number of internal symmetries which also playthe role of constrains for the non-equilibrium time evolution of the density ma-trix associated to the system. As a remarkable consequence the time evolutionfrom a generic initial state keeps much more memory of the initial conditionsthan in a generic interacting system, where the density matrix simply relaxes tothe usual Gibbs distribution.

Figure 1: Experimental data for thetime evolution (from top to bottom) ofphase-phase correlations 〈Ψ+(z)Ψ(z′)〉(with Ψ+(z),Ψ(z′) the bosonic cre-ation/annhilation operator) as a functionof the relative distance z � z − z′ of a one-dimensional Bose gas after having beingcoherently split into two parts. The greenline corresponds to the expectation valueof the operator Ψ+(z)Ψ(z′) on the steadystate, which can be described by the Gener-alized Gibbs Ensemble. The experiment isrealized with contact interacting cold atomsin an e�ectively one-dimensional trap suchthat the quantum gas can be treated as aLieb-Liniger gas of bosons.

If the initial state is a pure state whichis not an eigenstate of the Hamiltonian(global quantum quench) then the asymp-totic steady state, which e�ectively de-scribes the equilibrium expectation valuesof all the local observables, is not triv-ially characterized by a single tempera-ture, as it is the case in the Gibbs ensem-ble, but by many more additional dataon the initial state. A typical problemis to determine the thermodynamicallyrelevant information on the initial statethat one needs to know in order to re-produce the equilibrium expectation val-ues of all the local observables (repre-sented by operators that act non-triviallyonly on a �nite sub-region of the system),which are indeed the ones that we can di-rectly observe in a laboratory 4 (see �gure1).

The aim of this thesis is to develop amethod to study the non-equilibrium dy-namics of integrable systems in their ther-

2With Hamiltonian H �∑N

j�1∂2

∂x2j+ 2c

∑1≤i< j≤N δ(xi − x j ) acting on bosonic wave functions

Ψ(x1 , . . . , xN )3Namely there is no creation of newmomenta in the scattering. If N particles scatter with initial

momenta (k1 , . . . , kN ) then the �nal state is described by a new set of momenta which is simply apermutation P of the initial one (k′1 , . . . , k

′

N ) � P(k1 , . . . , kN ). On the other hand each particle getsa non-trivial phase shift which forces the momenta (k1 , . . . , kN ) to satisfy a set of non-linear coupledequations, the so called Bethe equations.

4Wede�ne a local observable as represented by an operator O(x) acting on a sub-region [x , x+δx]of the entire volume L, such that limth δx/L � 0, and with a negligible intensity, compared to thetotal energy of the system limth

〈O〉〈H〉 � 0. We denote with limth the thermodynamic limit with a

number N →∞ of particles in a volume L →∞with �xed density D � N/L.

modynamic limit. This is the regimewhere we can expect to obtain some explicit results and where we can describethe system in terms of its thermodinamically emerging degrees of freedom. Thisis usually a much easier description than the one in terms of the positions ormomenta of all the constituents. Moreover it is only in the thermodynamic limitwhere we can formulate general statements on the thermalization of an isolatedsystem, where we observe true quantum relaxation of local observables whenwe restrict to a �nite part of the whole system. In this case indeed the comple-mentary part acts as an e�ective in�nite bathwhich induces relaxation processeson the local observables.

In the following part we list the main results of this thesis.

Overlaps in integrable models

In order to have an exact description of the non-equilibrium time evolution givenby a quantumquench one needs to know the overlaps between the basis of eigen-states of themodel, which is given by the Bethe states, and the initial state, whichin general is very di�erent from any of the eigenstates 5. A physically interestingclass of initial states is constituted by the eigenstates of the same model with adi�erent coupling constant. While the algebraic approach to the exact solutionof integrable models, the algebraic Bethe ansatz, provides handy expressionsfor the overlaps between Bethe states with the same coupling, an analogous re-sult valid for states with two arbitrarily di�erent couplings has not been foundyet. In this thesis we provide the exact overlaps for two speci�c cases where theinitial state is the ground state of the free theory. For the Lieb-Liniger modelthis is achieved by setting the coupling constant of the interactions between theparticles in the gas to zero c → 0, leading to a many-body free bosonic theory.Equivalentlywe obtain the overlaps of the Bethe state of the XXZ spin chainwiththe ground state of the model in the limit of in�nite anisotropy ∆ → ∞, whenthe the XXZ model reduces to the classical longitudinal Ising spin chain. Thesetwo expressions for the overlaps are remarkably elegant in terms of the quasi-momenta which parametrize the Bethe states and they are very similar to theformula giving the norm of the Bethe states. This suggests that we may be ablein the future to �nd explicit expressions for more generic overlaps.

5To compute the time evolution after a quench from the initial state |Ψ0〉 of a generic operator Oone typically inserts two resolutions of the identity 1 �

∑λ |λ〉〈λ | using the basis of the eigenstates

of the model such that 〈Ψ0 |O(t) |Ψ0〉 �∑µ∑λ 〈µ|O |λ〉e

it(Eµ−Eλ )〈Ψ0 |µ〉〈λ |Ψ0〉. It is then necessary

to know the overlaps between any eigenstate and the initial state 〈Ψ0 |µ〉 ∀µ. We say that the initialstate |Ψ0〉 is far from any of the eigenstates of the system if the overlaps decay exponentially withthe number of constituents N , 〈Ψ0 |µ〉 ∼ e−Nc[µ]

∀µ with<c ≥ 0.

Post-quench steady state in the Lieb-Liniger gas and in the XXZspin chainThis thesis provides the �rst non-Gibbs steady states after a non-equilibriumtime evolution of truly interactingmodels. The non-equilibrium evolution is im-plemented by letting an initial state unitarily evolve under an integrable Hamil-tonian as the Lieb-Liniger gas for point-interacting bosons and the XXZ spinchain, a model for interacting spins on a lattice. Due to the presence of non-trivial conserved quantities in the models, the expectation values of local opera-tors, as the density-density correlation or the nearest-neighbors spin-spin corre-lation, do not relax to expectation values which are reproducible by a Gibbs-likeensemble.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

x

t = 10−1

t = 10−2

t = 10−3

t = 5× 10−2

t = 5× 10−3

t = 0

Figure 2: Time evolution (black linesfrom top to bottom) of phase corre-lations 〈BEC(t) |Ψ+(x)Ψ(0) |BEC(t)〉as a function of the relative distancex of a one-dimensional Bose gas afterthe quench from the ground state ofthe free theory (Bose-Einstein conden-sate |BEC〉) to the in�nite repulsiveregime (hard-core bosons). The reddotted line shows the steady stateexpectation values which is not givenby an e�ective thermal Gibbs ensemblelimt→∞〈BEC(t) |Ψ+(x)Ψ(0) |BEC(t)〉 ,Tr

(e−βHΨ+(x)Ψ(0)

)but by the

saddle point state |ρsp〉 givenby quench action approachlimt→∞〈BEC(t) |Ψ+(x)Ψ(0) |BEC(t)〉 �

〈ρsp |Ψ+(x)Ψ(0) |ρsp〉. This approach

also reproduces the whole time evolu-tion towards the equilibrium with theminimal amount of information on theinitial state.

Even a generalization of it, the Gen-eralized Gibbs Ensemble 6 whichaims to include all the other lo-cal constraints in the system, failsto give correct predictions. Inthe case of the Lieb-Liniger modelthe conserved quantities are indeedplagued with creepy divergencesthat do not allow to properly eval-uate them on the initial state. Inthe XXZ spin chain the set of lo-cal conserved quantities turns tonot be complete and more unknownadditional constraints need to betaken into account. We then ad-dress the problem with an alterna-tive approach, the so called quenchaction approach, which does not relyon any assumption on the internalsymmetries of the system but onlyon the emerging form of the over-laps between the eigenstates and theinitial state in the thermodynamiclimit. This gives a set of e�ectiveconstraints that are used to deter-mine the steady state, which turnsto be the exact one. The question of

6The Gibbs ensemble is expected to reproduce the equilibrium values of the physical operators inmost interacting systems limt→∞〈Ψ0 |O(t) |Ψ0〉 � Tr

(e−βH O

)/Tr

(e−βH

)where β−1 is the e�ective

temperature of the initial state. If there are more local conserved quantities than just the singleHamiltonian then this is expected to generalize to the GeneralizedGibbs Ensemble e−

∑n βn Qn where

{Qn } is the maximal set of local operators that commute with the Hamiltonian H.

how to reproduce such exact steadystates with the alternative approach of the Generalized Gibbs Ensemble is notanswered yet.

Time evolutionWhile some facts on the steady state after a non-equilibrium time evolution arein general established, on the contrarymuch less is known on the time evolutiontowards equilibrium. This is in general a much more complicated problem. Aquite common statement for example is that in order to reproduce its whole timeevolution a larger amount of information on the initial state is needed, comparedto the one necessary to reconstruct the steady state at in�nite times. However weshow in this thesis that this is not the case in a quenched integrable model (see�gure 2). The thermodynamic limit indeed selects only a restricted amount ofdata on the initial state, which are responsible for the whole time evolution upto in�nite times, when the system has relaxed to its steady state. This is a directconsequence of the quench action approach, which also provides a general toolto access the e�ective time scales of the non-equilibrium dynamics which areencoded in the post-quench e�ective velocity of propagation of the excitations.This is also a functional of the initial state and does not depend on all its detailsbut only on its macroscopic parameters (like its energy density for example).Di�erent initial states therefore will lead to e�ective �eld theories with di�erentlight velocities.Besides its theoretical relevance the whole time evolution is also much morerelated to experimental realizations. Cold atoms experiments are indeed usuallylimited in the range of time, making usually impossible to observe the true post-quench steady state in a real setting (See �gure 1).