Thermalization/Relaxation in integrable and free field theories

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Thermalization/Relaxation in

integrable and free field theories

THESIS

submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

PHYSICS ANDASTRONOMY

Author : Tereza Vakhtel

Student ID : s2228637

Supervisor : Koenraad Schalm

2ndcorrector : Jan Zaanen

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Thermalization/Relaxation in

integrable and free field theories

Tereza Vakhtel

Instituut-Lorentz, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

August 16, 2020

Abstract

Classical integrable theories fail to thermalize. This situation can be at-tributed to the presence of the extensive number of integrals of motion that preclude exploration of the phase space. The further question to ask is what happens in the quantum integrable case. The answer turns to be more sophisticated than in the classical case and may, in particular, de-pend on a) what one means by thermalization and b) which local observ-ables are considered. The no-go condition presented in this thesis will help to clarify these conditions when thermalization is defined in a particular way. This no-go condition will be applied to several examples of quantum integrable models.

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Chapter

1

Introduction

Prepare a physical system (in the thermodynamic limit) in a non-equilibrium state and let it evolve. Generically, the observables like the particle den-sity or the average velocity will equilibrate and will be given by the Gibbs distribution [1], [2]. This is an experimental fact and that is why statistical mechanics is so successful. The construction typically goes as follows: as-sume there’s an ensemble capable of describing the expectation values of physical observables in the system. Ensemble is a probability distribution were certain probability is assigned to each microstate that satisfies some constraints present in the system. Suppose there are no constraints: a sys-tem is connected to a thermostate it’s in a thermal equilibrium with and only the average energy is fixed (macrocanonical ensemble). Maximiza-tion of the entropy and the conservaMaximiza-tion of energy being additive for two independent systems yield the Gibbs distribution[4]. The equivalence of the microcanonical and the macrocanonical ensembles makes the descrip-tion applicable to both isolated and open thermodynamic systems. Never-theless, there’s some lack of understanding of the underlying microscopic reasons for why this approach works so well [3].

For example, the assumption of existence of a probabilistic ensemble that describes the properties of the system is not obvious. One of the fa-mous ways to explain it in the classical case is to invoke the ergodicity hypothesis [5] (assume an isolated system and no conservation laws ex-cept the total energy E): an ergodic system will visit each microstate with the energy E with equal probability throughout the long-time evolution, thus the time average of an observable is the same as the average of the observable over all these states. In short, the ensemble that we made up is needed to capture the long-time average [3].

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2 Introduction

of chaotic systems [6–9] and it’s believed to hold for any chaotic system [3]. A typical definition of classical chaos is that small difference in ini-tial condition cause the two trajectories to be exponenini-tially far from each other [3]. A generic system is chaotic, although there’re exceptions like in-tegrable systems [10]. A hamiltonian system is called inin-tegrable when the number of integrals of motion is the same as the number of degrees of free-dom∗. In this case, the Liouville’s integrability theorem is applicable[11]. What it says is that for the coordinates and momenta of the system(p, q) there’s a canonical transformation (p, q) → (I,Θ), where I denotes the set of the integrals of motion. The solutions of the equations of motion become (N is the number of the integrals of motion):

Ij(t) =const

Θj(t) =Ωjt+Θj(0), j =1· · ·N (1.1)

Thus the evolution happens on an N−dimensional torus and is obviously not chaotic. Also, the full exploration of the phase subspace given by the constant energy is not possible because now the dimensionality of the mo-tion is significantly reduced. The way in which this situamo-tion is often un-derstood is that the integrals of motion prevent the system from the explo-ration of the phase space [3]. The number of these integrals of motion is crucial: a classical system that has less integrals of motion than the number of degrees of freedom is generically chaotic[3].

A picture comparing a single-particle motion in a 2D integrable and a 2D chaotic system (Bunimovich stadium [8]) helps to gain some intuition about that. The figure on the left features the trajectory of a particle bounc-ing from a circle-shaped cavity. It’s visible that some of the points inside the cavity are never visited despite the fact that the potential energy is the same everywhere inside it. At the same time, the system shown on the right is chaotic and the trajectory seems to cover the space inside evenly. This system was proven to be ergodic[8].

We can ask how isolated quantum systems thermalize. There’re two important facts related to thermalization in quantum systems. The first fact is that the notion of trajectory is absent in quantum mechanics, so formulating anything similar to the ergodicity hypothesis becomes prob-lematic. The second fact is that an isolated system prepared in a pure state cannot evolve into a thermal mixed state given by the Gibbs density ma-trix ρ = e−β ˆ_ZH, where ˆH is the hamiltonian and Z is the partition function. Otherwise, the unitarity of evolution would’ve been violated. Because of

∗_{For instance, for M particles in 3 dimensions one has 3M pairs}₍_q

i, pi)of the

coordi-nates and momenta, so the number of integrals of motion has to be 3M.

2

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3

Figure 1.1: a)Trajectory of a particle in a circular cavity (integrable case) and b)trajectory of a particle inside the Bunimovich stadium (chaotic case). Source: H.J. Stockmann, Scholarpedia 5 (2010), p. 10243 [12]

that, it’s not possible that the expectation values of all operators will look thermal and typically only local operators are considered.

One of the central ideas in the field of quantum thermalization is the Eigenstate Thermalization Hypothesis[13–15]. Roughly speaking, the as-sumption is that the eigenstates of generic hamiltonians are such that the matrix elements of certain class of observables are given by an ansatz, in-spired by the Random Matrix Theory[16]. If this holds, the time dependent expectation values of such observables (in a state being a superposition of the energy eigenstates with close energies) will relax to the expectation values predicted by the corresponding microcanonical ensemble[13–15]. This assumption was proven to be true in certain systems [3].

Now, what happens in the quantum integrable case? First of all, there’s no generally accepted definition of what a quantum integrable system is[17]. Typically what is meant by it is a system with an extensive num-ber of commuting integrals of motion. Also, it’s important that the inte-grals of motion are local and additive[3]. For example, any hamiltonian has the commuting set of the energy projection operators but they don’t make any hamiltonian integrable. Secondly, the question needs to be bet-ter specified. The common way to study thermalization in integrable sys-tems is by performing a quench and letting the system evolve[18]. Quench is an abrupt change of the parameters of the hamiltonian H(p1,· · ·pn) →

H(p0₁,· · ·p0n). Typically one starts with an eigenstate of the initial

hamilto-nian and asks what the expectation values of some local observables will look like after the quench is performed.

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4 Introduction

it can be justified is as follows. The pure state|ψiof the system evolves as:

|ψ(t)i =∑ncne−

i

¯hEnt|E_ni, (1.2)

where Enand|Eniare the eigenenergies and the eigenvectors of the

corre-sponding hamiltonian and cn are the coordinates of the state in the energy

eigenbasis at t=0+(after the quench has been performed). Suppose we’re interested in the time evolution of the observableO. The time-dependent expectation value will be given by:

hψ(t)|Oˆ|ψ(t)i =∑_n,me i ¯h(Em−En)tc∗_mc_nhE_m| O |E_ni =_∑_mhEm| O |Emi |cm|2+∑m6=ne i ¯h(Em−En)tc_m∗c_nhE_m| O |E_ni. (1.3) The first sum in the second line is time independent. The second sum is expected to decay with time due to dephasing, even if the system is integrable. This was verified for 1D lattice hard-core bosons for example [52].

What is the value that the exectation value will equilibrate to? Turns out that quantum integrable systems exhibit lack of thermalization simi-larly to the classical integrable case. There was even an experiment that demonstrated it (a quantum Newton’s cradle)[53]. As in the classical case, the lack of thermalization is associated with the existence of the integrals of motion constraining the evolution. It was conjectured [52] that the ex-pectation values will be given by the following density matrix:

ˆρGGE =

exp −_∑_kλkˆIk

Trexp −_∑_kλkˆIk

, (1.4)

where Ikare the integrals of motion and λk are fixed by the condition that

Tr(ρGGEIk) gives the expectation value of the integral of motion for the

initial state. Such a mixed state is referred to as the Generalized Gibbs Ensemble. This type of a density matrix (GGE) was shown to give the correct expectation values of few body observables for a large number of integrable systems[52][19–41]. Nevertheless, sometimes, for some initial states, the expectation values in integrable systems can be given by the Gibbs ensemble[42–49].

What we conclude from this chapter is that quantum integrable sys-tems exhibit lack of thermalization and the question about when an ob-servable looks thermal has to have a detailed answer. In this thesis, I’ll depart from the typical way in which thermalization in integrable systems is studied. The initial state will be a thermal state instead of an energy 4

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5

eigenstate, and there will be no quench but a small deformation of the hamiltonian. More focus will be put on the observable being studied. It will turn out that the expectation value can be thermal or not depending on the perturbation and the observable.

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Chapter

2

The question and the main result of

the thesis

2.1 Definition of thermalization in this thesis

In this thesis, we will consider an integrable quantum many-body system coupled to a heat bath, so that at times t<0 its density matrix is given by the Gibbs ensemble:

ˆρ(t <0) = e

−β ˆH

Z (2.1)

where ˆH is the unperturbed hamiltonian of the system and β is the inverse temperature and Z is the partition function.

At the moment t=0 a perturbation to the hamiltonian is switched on, which persists forever:

δH =

Z

dd−1xφ(x, t)O(ˆ x, t). (2.2)

Here, ˆO(x, t) is some hermitian operator and φ(x, t)is the source profile. We will always assume the perturbation to be small and that the linear re-sponse approach is valid. Another assumption that we make is that the un-perturbed hamiltonian H0 is invariant under translations in space. Then,

the perturbation of the expectation value of the observable ˆO(x, t)is given by: δhO(x, t)i = Z dd−1x0dt0GR x−x0, t−t0 φ x0, t0 , (2.3)

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8 The question and the main result of the thesis

where GR(x−x0, t−t0) is the retarded Green’s function, defined

specifi-cally for the operator ˆO(x, t): GR x−x0, t−t0

= −iΘ t−t0 _ˆ

O(x, t), ˆO x0, t0_β. (2.4) h i_βstands for the thermal average and[, ]for (anti)commutator, depend-ing on whether ˆO is a fermionic or a bosonic operator. If GR decays

ex-ponentially with time (as t → ∞), then δhO(x, t)i goes to 0 as t → ∞ and we say that ˆO thermalizes. This means that despite perturbing the system, the expectation value returns to that of the Gibbs ensemble. Note that this is different from quenching a system in a pure state and saying that the expectation values of some local operators are given by the Gibbs statistics.

To give an example of an application of the definition, we will look at the case where thermalization doesn’t happen. Let’s consider a free bosonic field theory in 1+1D, with the following lagrangian:

L= 1 2 Z dxh(∂tφ(x, t))2− (∂xφ(x, t))2 i . (2.5)

Then its Euclidean Green’s function will be:

GE(ωn, k) = 1

ω2n+k2+m2

, (2.6)

where ωn = 2πnT, n ∈ Z are Matsubara frequencies and T -

tempera-ture. The retarded and Euclidean Green’s functions are connected in the following way:

GR(ω, k) = −GE(−iω+η, k). (2.7)

The infinitesimal and positive number η here is to ensure that the poles are located slightly below the real ω axes. Now, to obtain the time behavior of the Green’s function, we calculate the inverse Fourier transform:

GR(t, k) = 1 2π Z ∞ −∞dω e−iωt (ω+iη)2−k2−m2 = −√ 1 k2+m2 sin p k2₊_m2tΘ₍_t₎ . (2.8)

We see, however, that there’s no time decay, so the operator ˆφ(x, t)doesn’t

thermalize. There’s another example, where thermalization in an inte-grable model DOES happen. This example will involve the transverse 8

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field Ising model in 1+1 which can be mapped to a free fermion theory. The hamiltonian is:

H = −J

N

∑

i=1

gσ_ix+σ_izσ_iz+1 , (2.9)

with the spin operators σ_ia that obey the conventional commutation rela-tions hσ_ia, σ_jb

i

= 2ieabcδij . If one perturbs the system with the

magneti-zation operator σ_iz and looks at its response, the corresponding retarded

Green’s function in the continuum limit at g = 1 and at high enough T

will be [50, 54]: G(_Rσz)(ω, k) ∼ 1 T7/4 Γ(7/8) Γ(1/8) Γ1 16 −iω +k 4πT Γ 161 −iω −k 4πT Γ15 16 −i ω+k 4πT Γ 15 16 −i ω+k 4πT . (2.10)

This function has poles at the following frequencies:

ω = −4πiT n+ 1 16 ±k, n =0, 1, 2,· · · . (2.11)

These frequencies have a negative imaginary part which implies that the Fourier transformed retarded Green’s function will decay with time. Thus, the magnetization operator thermalizes.

2.2 The no-go condition and its proof

Now that we’ve seen both examples of an operator that thermalizes and an operator that doesn’t, it’s a natural question to ask what properties of an operator define whether this happens or not. In this chapter, I will present a condition which, if satisfied by an operator, implies that the operator does not thermalize. This condition will be followed by a proof which is quite straightforward. But first, let’s represent the retarded Green’s func-tion in a more convenient form:

GR(x, t) = −iΘ(t)h[O(x, t), ˆˆ O(0, 0)]iβ

= −iΘ(t) Z(β)

∑

_n e

−βEn

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10 The question and the main result of the thesis

hamiltonian.O(x, tˆ )can be represented as follows: ˆ

O(x, t) =eiHte−iP·xO(ˆ 0, 0)e−iHteiP·x, (2.13) where H is the unperturbed hamiltonian of the quantum system and P is the momentum operator. By inserting the unit operator ˆI=∑_n|ni hn|, we end up with the following:

GR(x, t) = −iΘ(t) Z(β) _m,n

∑

e −βEn e−i(Em−En)t+i(Pm−Pn)·x₋ _(2.14) −(−1)2sei(Em−En)t−i(Pm−Pn)·x |hm|O|ˆ ni|2. (2.15) The Fourier transform of the retarded Green’s function is:

GR(k, ω) = − i Z(β) _n,m

∑

e −βEn δ(d−1)(k− (Pn−Pm)) ω+ (En −Em) +iη − (2.16) −(−1)2sδ (d−1)₍_k_{− (}_P m−Pn)) ω+ (Em−En) +iη |hm|O|ˆ ni|2. (2.17) As in the previous section, η is an infinitesimal real number to ensure the correct position of the poles. This way to represent the Green’s function is called Lehmann spectral representation. Now we’re ready to state and prove the theorem:

Theorem. If there exists a finite set of single valued functions FO

i (Pn−Pm)

N

i=1

such thathm|O|ˆ ni|2 ₌ _{0 implies E}

n −Em = F

(O)

i (Pn−Pm), then the

pertur-bation ˆOwill not thermalize.

F_iO do not have to be continuous and N must be finite even for the infinite system size.

Proof. If the assumption in the theorem holds, there will be N groups of terms in the sum 2.16 with a common denominator En−Em:

GR(k, ω) = (2.18) = N

∑

i=1 H_i(O)(β, k) ω+F_i(O)(k) +iη − (−1)2s H (O) i (β,−k) ω−F_i(O)(−k) +iη ! , (2.19) where H_i(O)(β, k) = − i Z(β)

∑

m,n|En−Em=Fi(O)(Pn−Pm)

e−βEn_δ(d−1)₍_k_{− (}_P

n−Pm)) |hm|O|ni|2

(2.20)

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are the functions independent of ω. Since all F_iOare real functions, all the poles of the Green’s function are located on the real axes and therefore its inverse Fourier image will not decay with time. Therefore, the operatorO won’t thermalize.

Note that we didn’t prove the converse: the theorem doesn’t imply that all non-thermalizing operators satisfy the no-go condition. In the next sec-tions, we will consider thermalizing operators (that also turn to violate the no-go condition) as well as those satisfiying the no-go condition and there-fore not thermalizing. The examples of the integrable theories will include the free boson theory in (1+1)D and the transverse field Ising model. In these models, we will see an easy recipe to construct an operator violating the no-go condition and thermalizing.

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Chapter

3

Examples

3.1 2D Conformal field theories

The structure of this chapter is as follows: I’ll first define what a confor-mal field theory is and show that conforconfor-mal theories in 1+1 dimensions are special because of their infinitely large symmetry group. This makes those models integrable and therefore relevant for the study. In (1+1)D, I’ll discuss a certain subset of operators and prove that some of them sat-isfy the no-go condition and fail to thermalize while the rest violate it and thermalize. This implies that the no-go theorem works both ways on this subset. After that, I’ll focus on a specific case of a (1+1)D CFT: free mass-less boson theory and illustrate how the no-go theorem works on some examples.

3.1.1 Basic background

Conformal field theories are of relevance for a physicist because they de-scribe the critical behaviour of systems at second order phase transitions [55]. To define what a conformal field theory is, I’ll introduce the notion of the conformal symmetry group in d dimensions first. Assume we have a flat(!) metric gµν = ηµν of signature (p, q) where p, q are some

pos-itive integer numbers. The line element ds2 in this metric is given by ds2 = gµνdxµdxν. Because gµν is a tensor, its transformation properties

under a coordinate change x →x0 can be written in the following way:

gµν →g 0 µν x 0 = ∂x α ∂x0µ ∂xβ ∂x0νgαβ(x). (3.1)

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14 Examples

Conformal transformations are such coordinate transformations that the transformed metric differs from the old metric only by a multiplication by a function: gµν(x) → g 0 µν x 0 =Ω(x)gµν(x). (3.2)

For d>2, all the conformal transformations turn to be generated by: 1. Translations:

x →x0 = x+a (3.3)

2. Linear transformations given by SO(p, q):

x →x0 =Λx, Λµ_ν ∈ SO(p, q) (3.4)

3. Dilatations:

x→ x0 =λx, λ ∈R (3.5)

4. Special conformal transformations:

x→ x0 = x+bx

2

1+2b·x+b2_x2, (3.6)

where b is just a real number.

In d = 2 however the situation turns to be more interesting. Since we

have just two coordinates x1, x2, we can define z, ¯z = x1±ix2. The line

element ds2 can be rewritten as ds2 = dzd¯z. Note that z and ¯z are treated as independent variables. If one examines the question whether a given coordinate transformation is conformal or not, one quickly comes to the conclusion that all the conformal trasnformations are generated by the holomorphic functions:

z → f(z), ¯z → ¯f(¯z), (3.7)

so that the line element stays positive real and transforms as: ds2 =dzd¯z→ d f dz 2 dzd¯z. (3.8)

The generators of all such transformations can be written as:

`n = −zn+1∂z `¯n = −¯zn+1∂¯z (n ∈Z). (3.9)

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One may consider the so called global conformal group which consits of conformal transformations that are well defined and invertible every-where on the Riemann sphere. This group is generated by globally de-fined infinitesimal generators{`−1,`0,`1} ∪`¯−1, ¯`0, ¯`1 . We can identify `−1, ¯`−1 as generators of translations, `0+¯`0 as generators of dilatations

and i(`₀−`¯₀) – rotations. `₁, ¯`₁ generate special conformal transforma-tions. In the number of dimensions higher than 2 there’s no distinction between the global and local conformal groups because they are identical.

It happens that the global conformal algebra{`−1,`0,`1} ∪

_¯

`−1, ¯`0, ¯`1

turns to be useful for characterising the physical states. If one works in the basis of eigenstates of `0, ¯`0 (note that `0 and ¯`0 commute), one can

la-bel these eigenstates by their eignevalues h, ¯h which are called conformal weights. Because`0+`¯0 and i(`0−`¯0)generate dilatations and rotations

respectively, the scaling dimension ∆ is h+ ¯h and the spin s is h−¯h. ∆ characterizes how the state transforms under dilatations and s - under rotations. Now that we introduced basic notions, we can define what a conformal field theory is. Let’s stay more general and assume an arbitrary

p+q = d number of dimensions again. A conformal field theory is the

field theory that satisfies the following criteria [55]:

1. There’s a set of fields Ai(x). This set must (in particular) contain the

derivatives of all the fields Ai(x)and it’s therefore infinite;

2. There’s a subset of fields φi ∈ Aiwhich is called ’primary fields’ that

transform in the covariant way under global conformal transforma-tions x→ x0: φj(x) → ∂x0 ∂x ∆j/d φj x0 , (3.10)

where ∆j is called the dimension of φj and is one of the

character-istics of the operator. If this is satisfied, the theory will be covari-ant under the global conformal transformations in the sense that any correlators of the quasi-primary operators will satisfy the following transformation property when a global transformation is performed:

hφ1(x1). . . φν(xn)i = ∂x0 ∂x ∆1/d x=x1 · · · ∂x0 ∂x ∆n/d x=xn hφ1(x10). . . φn(x0n)i . (3.11)

3. The rest of the operators Ai’s can be expressed as linear combinations

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16 Examples

4. The vacuum state|0iis invariant under the global conformal group. Similarly, if an operator in a 2-dimensional CFT satisfies the transforma-tion rules as in 3.10 under the local conformal group, it’s called primary. The transformation property 3.11 restricts the form of correlators of the quasi-primary fields. For example, for two point functions it implies that they must have the form:

hφ1(x1)φ2(x2)i = ( _c 12 r2∆₁₂ ∆1 =∆2 =∆ 0 ∆1 6=∆2 (3.12) Here, r12 =|x1−x2|and ∆1,∆2 are dimensions of the fields φ1, φ2

respec-tively. Similarly, in 2D, the primary operators transform as: Φ(z, ¯z) → ∂ f ∂z h ∂ ¯f ∂ ¯z ¯h Φ(f(z), ¯f(¯z)) (3.13) h, ¯h are called the conformal weights of the operators. Note that h and ¯h are always real numbers so bar here doesn’t mean complex conjugation. If a primary operator is a function of z or ¯z only, so that either h=0 or ¯h=0, it’s called chiral primary. Because of such transformation properties, the two-point functionsΦ1,Φ2of primary operators are constrained to be (by

analogous reasonings to the previously discussed case): hΦ1(z1, ¯z1)Φ2(z2, ¯z2)i =

C12

z2h₁₂¯z2¯h₁₂, z12 =z1−z2, ¯z12 = ¯z1−¯z2. (3.14) The last element that we need is the procedure called radial quantization[56]. Let’s fully turn into 2 dimensions since now. Suppose our system is of the finite size L with periodic boundary conditions. This means that our hamiltonian lives on an infinite flat cylinder with circumference L (the in-finite direction corresponds to time t∈ [−∞, ∞]). The time coordinate is t and the space coordinate is x∗. In principle, we’re free to parametrize the space as we want, so one can perform the following conformal coordinate transformation:

z=exp 2π(t+ix) L

. (3.15)

This maps the cylinder to a complex plane. Equal time circles on the cylin-der get mapped onto the concentric circles on the plane. It’s easy to see from 3.15 that time and space translations on the cylinder correspond to 16

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Figure 3.1:Mapping from the cylinder to the plane. Source: [51]

the dilatations and rotations on the plane correspondingly. It happens that it’s more convenient to deal with a QFT on the plane than on the cylinder, due to the opportunity to use the properties of the contour integrals it gives. A non-trivial fact is that when a critical theory on the cylinder is mapped in this way to the complex plane, the resulting theory is a CFT with the local generators as defined in 3.9. Because the map 3.15 is confor-mal, the correlators of the primary fields on the plane get mapped back to the cylinder in the known way 3.11. If one is interested in the correlators at finite temperature for the infinite system size, than the finite L direc-tion becomes the finite inverse temperature β = 1/T direction. Thus, the coordinate transformation 3.15 is replaced by z = exp{2πT(t+ix)}. The correlators calculated on the plane have to be mapped back on the cylinder with this transformation.

3.1.2 General results for 2D CFTs

The transformation properties of 2D CFTs with respect to the generators 3.9 result in the infinite number of mutually commuting conserved charges that can be constructed using the KdV hierarchy [57]. Thus, it’s a relevant class of theories to consider in this thesis. In this subsection, I’ll give an outline of the main results of our paper [58]. The operators in the focus are the primary operators.

Because the two-point function of two primary fields (calculated in the flat metrics) is fixed to have the form as in 3.12, one can easily calculate the time ordered two point function at finite temperature by employing the radial mapping trick that was briefly discussed in the previous subsection. IfOis a primary operator with the conformal weights h, ¯h, β is the inverse temperature and τ is the Euclidean time, then the time ordered correlation ∗_{We take the units of time to be the same as units of distance here and in many other}

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18 Examples

function is:

hT_E{O(x, τ)O(0, 0)}i_β =  2π 2 β2 h+¯h 1 sinh2hπ β(x+iτ) sinh2¯hπ β(x−iτ) . (3.16) However, what we need is the retarded Green’s function, which is given by:

hTE{O(x, τ)O(0, 0)}i_β = hO(x, τ)O(0, 0)iβΘ(τ)+

+e2πi(h−¯h)hO(0, 0)O(x, τ)i_βΘ(−τ) (3.17)

We can obtain real time correlators by analyitically continuing:

τ →it±e, (3.18)

e is an infinitesimal positive number to ensure the right ordering. The

resulting retarded Green’s function is: GR(x, t) = −iΘ(t) 2π2 β2 h+¯h " 1 sinh2hπ β(x−t+ie) sinh2¯hπ β(x+t−ie) − 1 sinh2hπ β(x−t−ie) sinh2¯hπ β(x+t+ie) # (3.19)

It’s easy to see that this function is 0 everywhere except at the lightcone x = ±t. Let’s pick x=t and see how the Green’s function behaves:

GR(x =t, t) ∼ 1

sinh2¯h2πt_β .

(3.20)

It’s easy to see that for non-zero ¯h the function decays which means that the operator thermalizes according to our definition. Thus, a generic pri-mary operator (i.e. the one with non-zero both h and ¯h) doesn’t thermalize. I’ll show later that such operators also violate the no-go condition. Now, let’s consider a chiral primary operator. Assume that ¯h = 0. Its retarded Green’s function is given by:

GR(x, t) = −iΘ(t) 2π 2 β2 h   1 sinh2hπ β(x−t+ie) − 1 sinh2hπ β(x−t−ie)  . (3.21) 18

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Because e is infinitesimal, this vanishes for any x 6= t. For e |x−t| 1, one can write:

GR(x, t) = −iΘ(t)2h 1 (x−t−ie)2h − 1 (x−t+ie)2h . (3.22)

This of course doesn’t decay in time, because it’s either infinite (on the light cone) or zero (anywhere else). For example, if 2h is one, then the following is true: 1 2πielim→0 1 u−ie− 1 u+ie =δ(u). (3.23)

And if 2h is an integer larger than 1, then one has: lim e→0 1 (u−ie)n − 1 (u+ie)n = (−1)n 2πi (n−1)! dn−1 dun−1δ(u). (3.24)

In the following paragraphs, I’ll show that such (chiral) operators also sat-isfy the no-go condition.

If one considers a hamiltonian H describing some physical system of size L, then it’s initially defined on an infinite cylinder with circumference L, as explained in the subsection 3.1.1. By looking at 3.15, one can check that the hamiltonian of the system gets mapped to 2π_L (`0+`¯0) and the

momentum operator is mapped to 2π_L (`0−`¯0). Here,`0and ¯`0are defined

in the coordinates of the plane, as in 3.9. The eigenstates of the hamilto-nian and the momentum are the eigenstates of`0 and ¯`0 as well, so they

can be labeled by their eigenvalues h and ¯h: h_i, ¯h_i, in agreement with the conventions introduced in the subsection 3.1.1. Now, let’s consider an operator O and check whether the no-go condition is satisfied or not. From the previous paragraph it follows that all the matrix elements in the eigenbasis of the hamiltonian will look like h1, ¯h1

O

h₂, ¯h₂. The corre-sponding∆E and ∆P are given by ∆E = 2π_L (∆h+∆¯h),∆P= 2π_L (∆h−∆¯h). For a generic primary operator (i.e. with non-zero h and ¯h), the non-zero matrix elements are such that neither ∆h nor ∆¯h are 0, so ∆E will not be a function of ∆P. However, if the operator O is chiral, it means that it’s either z or ¯z independent so it commutes with`0 or ¯`0. The consequence

is that either ∆h or ∆¯h must be 0 in order for the matrix element to be non-vanishing. In this case, it’s clear that for non-zero matrix elements, ∆E = ±∆P. Thus, the no-go condition is clearly satisfied. This explains why the retarded Green’s function for a chiral (not necessarily primary)

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20 Examples

operator doesn’t decay†. The consequence of the above arguments and

the results for the retarded Green’s functions given in this subsection is that the no-go theorem works both ways when applied to the subset of the operators called primary operators. It’s not clear whether the condition is necessary for all the other operators that are neither chiral nor primaries too. Still, the no-go condition is satisfied for any (not only primary) chiral operator. I’ll also consider an example of a non-primary operator which isn’t chiral in the next subsection.

3.1.3 Free massless boson theory in 2D

One of the special cases of a 2D CFT is the free massless boson theory. In this subsection, I’ll show how the no-go condition works in this case. Let’s start with the classical Lagrangian:

L= 1 2 Z dxh(∂tφ(x, t))2− (∂xφ(x, t))2 i . (3.25)

The corresponding hamiltonian is:

H = 1 2 Z dxhπ2+ (∂xφ)2 i , π(x, t) = ∂tφ(x, t). (3.26)

To canonically quantize, one typically rewrites [59] the field in terms of the creation annihilation operators in the following way:

φ(x) = Z _dp 2π 1 p2ω(p) a(p) +a(−p)†eipx π(x) = Z _dp 2π(−i) r ω(p) 2 a(p) −a(−p)†eipx. (3.27)

a(p)and a†(p)obey: [a(p), a†(p0)] =2πδ(p−p0). In terms of these oper-ators, the hamiltonian becomes:

H = Z _dp 2πω(p) a(p)†a(p) +1 2 h a(p), a(p)†i , (3.28)

with ω=|p|. From the commutators: h

H, a(p)†i =ω(p)a(p)† , [H, a(p)] = −ω(p)a(p), (3.29)

†_{We calculated the Green’s function in the thermodynamic limit where L} _→ _{∞ and}

here the size of the system appears to be finite. Still, the argument should work for the L→∞ too.

20

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we can write how the creation and annihilation operators evolve: a(p)(t) = eiHta(p)e−iHt =a(p)e−iω(p)t,

a(p)†(t) =eiHta(p)†e−iHt =a(p)†eiω(p)t, (3.30) so that the field φ evolves as:

φ(x, t) = Z _dp 2π 1 p2ω(p)

a(p)eipx−iω(p)t+a(−p)†eipx+iω(p)t. (3.31)

It’s technically more convenient to put the theory on the cylinder of size L. The momenta and frequencies are then quantized:

ω(p) = 2π_L |k|, p= 2π_L k, k∈ Z

ak = √1_La(p), a†k = √1_La†(p),

(3.32) so that now the ladder operators ak and a†_k are normalized to obey the

commutation relationship[a†_k, ak] =δk,k0. The energy eigenstates then take

the form: |Ei = ⊗_k|nki, E =

∑

k nkωk = 2π L

∑

_k |k|nk. (3.33)

It will be more convenient for the future to have the left- and right-moving (or, equivalently, holomorphic and antiholomorphic) parts separated in the field φ: φ(x, t) =∑∞_k=04πkL αkak+α ∗ ka†k +¯αka−k+¯α ∗ ka†−k αk = √ 4πk L e− 2πi L k(t−x), ¯α_k = √ 4πk L e− 2πi L k(t+x). (3.34)

Special care should be taken about the zero modes k=0, however, I’ll skip this part since it won’t have any drastic effect on the result. The momen-tum operator P(x, t)in the continuum limit is P(x, t) = R dp p a†(p)a(p). The momentum is conserved for this hamiltonian so every eigenstate has a well defined momentum:

P=

∑

k 2πk L a † kak. (3.35)

Unlike for the energy 3.33, we have just k here instead of|k|. This will be important for the examples I’ll consider.

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22 Examples

A chiral primary operator

The first example I will look at is a chiral primary operator. Although

φ(x, t)isn’t a primary operator, the left- (or right-) moving derivative ∂φ(x, t) =

(∂x−∂t)φis. The calculation in the massless bosonic theory is

straightfor-ward: h∂φ(x, t)∂φ(0, 0)iβ = ∑∞k=1αk(x, t)α∗k(0, 0)aka†k β+α ∗ k(x, t)αk(0, 0)a†kak β =_∑∞_k₌₁hn_ki_β α_k(x, t)α∗_k(0, 0) +α_k∗(x, t)α_k(0, 0)+∑_k∞₌₁α_k(x, t)α∗_k(0, 0) = 4π L2 ∑∞k=1k h hn_ki_βe−2πikL (t−x)+e2πikL (t−x) +e−2πikL (t−x) i . (3.36)

The bosonic k−modes are non-interacting and the occupation numbers

nk obey the Bose-Einstein distribution: hnkiβ = 1 e2π|k|βL −1 = ∞

∑

m=1 e−2π|k|βL m. (3.37)

Interchanging the sums over m and k and defining d=t−x, the first term in 3.36 gives:

2π L2 ∑∞m=1

cos2π(d−_Limβ)+cos2π(d+_Limβ)−2 cos(2πd L )−cosh _2πmβ L 2 = − 2 π ∑ ∞ m=1 (d−mβ)(d+mβ) (d2₊_m2_β2₎2 +O 1/L 2 = 1 πd2 − π β2 1 sinh2dπ β +O 1/L2 . (3.38) For the second term (which is also the zero temperature two-point func-tion), we have: h0|∂φ(x, t)∂φ(0, 0)|0i = ∞

∑

k=1 4πk L2 e −2πik L d _(3.39) = − π L2_sin2πd L = − 1 πd2 +O 1/L2. (3.40)

For the sum to converge,−2πik

L d has to have a negative definite real part.

Therefore, we will take d = t−x−ie with e ≥ 0 so that the final answer is well-defined. The answer is:

h∂φ(x, t)∂φ(0, 0)iβ = − π β2 1 sinh2π(t−_βx−ie) . (3.41) 22

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This isn’t the final answer though because this is just the time ordered part. The full retarded Green’s function we need is:

GR(x, t) = −iΘ(t) hO(x, t)O(0, 0)iβ− hO(0, 0)O(x, t)iβ . (3.42)

It’s easy to see from the last line of 3.36 that the only contribution to this difference will come from the second term (the the first term is invariant

under the transformation t−x → x−t ). At the same time, the second

term is just the zero-temperature value, so the term that we need to sub-stract in order to get the retarded Green’s function is:

h0|∂φ(0, 0)∂φ(x, t)|0i = ∞

∑

k=1 4πk L2 e 2πik L (t−x)_. _(3.43)

This time, t−x must have a positive-definite imaginary part for the sum to converge, so the result of the summation will be:

h0|∂φ(0, 0)∂φ(x, t)|0i = − 1

π(t−x+ie)2 +O

1/L2. (3.44)

The retarded Green’s function is: GR(x, t) = i πΘ(t) 1 (t−x−ie)2 − 1 (t−x+ie)2 (3.45) = −2Θ(t)δ0(t−x). (3.46)

The function shows no exponential decay and the operator doesn’t ther-malize. It’s expected because, as was mentioned in the beginning, ∂φ is a chiral operator with h =1 and ¯h = 0. Because the theory is free, it’s easy to check which matrix elements don’t vanish directly (and convince our-selves that the no-go condition is satisfied). Consider the matrix element:

⊗_kn0_k|∂φ| ⊗knk = ∞

∑

l=1 D ⊗_kn0_k αlal+α ∗ la†l ⊗knk E . (3.47)

If one looks at a particular summand labeled by l, it’s clear that in order for the matrix element to be non-zero, all the occupation numbers for k 6=l

have to match except for k = l whose occupation number must differ

ex-actly by one. The difference in momentum between the states|⊗_knkiand

⊗_k₆₌_ln_k; n_l =n_k±1 is 2π_L ∑_kk(n_k− (n_k±δ_k,l)) = ∓2π_L l . At the same time, the difference in energy is 2π_L ∑k|k| (nk− (nk±δk,l)) = ∓2πL l. Thus

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24 Examples

A composite operator

To check a non-chiral and non-primary case, let’s consider the following operator:

C(x, t) =: ∂φ(x, t)¯∂φ(x, t) : . (3.48) Where :: denotes normal ordering. C(x, t)can be represented as

C(x, t) =: O₁(x, t)O₂(x, t) :, where O₁ = ∂φ and O2 = ¯∂φ. To

calcu-late the retarded Green’s function for this operator, it’s useful to keep the following identity in mind:

GC_R(x, t) = −iΘ(t)h[C(x, t), C(0, 0)]i_β = (3.49) = −iΘ(t)[hO₁(x+ζ, t)O1(ζ, 0)[O2(x, t),O2(0, 0)] (3.50) +O1(x+ζ, t)[O2(x, t),O1(ζ, 0)]O2(0, 0)iβ (3.51)

+hO1(ζ, 0)[O1(x+ζ, t),O2(0, 0)]O2(x, t) (3.52) +[O1(x+ζ, t),O1(ζ, 0)]O2(0, 0)O2(x, t)iβ]. (3.53)

ζ is an infinitesimal real number to prevent any potential problems with

contact divergencies. Another result that we need is: h¯∂φ(x, t)¯∂φ(0, 0)i_β = −π β2 1 sinh2π(t+_βx−ie) (3.54) so that GR(x, t) = − i πΘ(t) 1 (t+x−ie)2 − 1 (t+x+ie)2 . (3.55)

Then the retarded Green’s function becomes: G:∂φ ¯∂φ:_R (x, t) = Θ(t) 2 β2 1 sinh2π β(t−x−ie) δ 0₍_t₊_x₎₊ + 1 sinh2π β(t+x+ie) δ 0₍_t₋_x₎ . (3.56)

The Green’s function is zero away from the light cone, but on the light cone it decays exponentially with time. Let’s check whether the operator satisfies the no-go condition or not. By substituting 3.34 into the definition of the operator 3.48, we obtain:

: ¯∂φ∂φ := ∞

∑

k,k0₌₁ h αk¯αk0a_ka₋_k0+_α_k¯α∗_k0a†₋_k0a_k+α∗_k¯αk0a†_ka₋_k0+_α∗_k¯α∗_k0a†_ka†₋_k0 i (3.57) 24

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I’ll consider each of the four terms separately. For the term with annihila-tion operators only, one has:

⊗_lnl|aka−k0| ⊗_ln0_l =

∏

l6=k,k0 δnln0_l ! δnk,n0_k+1 δn_−k0,n0_−k0+1 q n0_kn0₋_k. (3.58) The matrix element is non-zero if and only if the momentum difference between the two states is P0−P = 2π_L (k−k0), while the energy difference is given by E0−E = 2π_L (k+k0). Thus∆E can be adjusted independently of∆P for the matrix elements that are non-zero. The same is true for all the other terms and they cannot cancel each other such that for the remaining non-zero matrix elements ∆E = Fi(∆P). This is because the sets of the

pairs of the eigenstates for which the matrix elements don’t vanish don’t overlap for any of the two terms.

Vertex operators

There’s another class of operators that are easy to check which are also primary. Those are called vertex operators. For example, one of such op-erators is:

Vξ(x, t) =: e

iξφ(x,t)

: (3.59)

ξ is a real number. This operator isn’t hermitian, so let’s consider two

hermitian linear combinations that make sense as physical operators: cos(ξφ) :≡

Vξ +V_ξ†

2 , : sin(ξφ) :≡

Vξ−V_ξ†

2i . (3.60)

It’s interesting to study how such complicated operators (that interact with many conserved charges in some sense) will change the expectation val-ues. Let’s also define the chiral versions of these operators:

V_ξ(±)(x, t) =: eiξφ(±)(x,t) : , Vξ(x, t) =V (−) ξ (x, t)V (0) ξ V (+) ξ (x, t). (3.61)

φ/V(±,0)(x, t) contain only positive-, negative- or zero-modes. Based on

the arguments from the subsection 3.1.2, such chiral operators will obey the no-go condition and won’t thermalize. Because we’re dealing with a free theory, we can calculate the retarded Green’s functions and check whether the condition applies and see the theorem at work in this partic-ular case.

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26 Examples

Let’s compute the two-point correlation functions for Vξ and V

(±)

ξ and

their hermitian conjugates first. The following identity will be helpful (which can be viewed as a special case of the Baker-Campbell-Hausdorff formula):

hψ|: eA1 :: eA2 : |ψi = hψ|: eA1+A2 : |ψieh0|A1A2|0i. (3.62)

By substituting A1 = −iξφ(x, t)and A2 =iξφ(0, 0)we obtain:

D : V_ξ†(x, t) :: Vξ(0, 0) : E β =D: eA1+A2 _:E β eh0|A1A2|0i _(3.63) A1+A2 = ∞

∑

k=1 γkak−γ∗_ka†_k+γ¯ka−k−γ¯∗_ka†−k , (3.64)

where γkand ¯γk are:

γk = iξ √ 4πk h 1−e−2πiL k(t−x) i , ¯γk = iξ √ 4πk h 1−e−2πiL k(t+x) i . (3.65)

We computeh0|A1A2|0ifirst. The non-vanishing terms are: h0|A1A2|0i = C+ξ2 ∞

∑

k=1 e−2πiL k(t−x)+e−2πiL k(t+x) 4πk (3.66) =C− ξ 2 4π h log1−e−2πiL (t−x−ie) +log1−e−2πiL (t+x−ie) i . (3.67)

Here, C is a real constant that accounts for the zero-mode contribution and ie is inserted to ensure the convergence of the sums. Exponentiation gives:

eh0|A1A2|0i ₌_C0 h 2ie−iπL(t−x−ie)sin π(t−x−ie) L i−ξ2_4π × ×h2ie−iπL(t+x−ie)sin π(t+x−ie) L i−ξ2_4π ∼ 1 h_{2iπ(t−x−ie)} L iξ2 4πh_{2iπ(t+x−ie)} L iξ2 4π . (3.68) The terms of the formVξ(x, t)Vξ(0, 0) are subleading in

1

L. This is why

they are ignored in the calculation for : cos(ξφ) : and : sin(ξφ) :. For the

finite temperature response we also need: eA1+A2 _:

β. The details of the

calculation can be found in the Appendix 5. The result of these calculations 26

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3.1 2D Conformal field theories 27 is: D : eA1+A2 _:E β =K   π(t+x−ie) βsinh π(t+x−ie) β   ξ2 4π   π(t−x−ie) βsinh π(t−x−ie) β   ξ2 4π , (3.69) where K is a real constant. With this, we can write down the finite temper-ature correlation function:

D V_ξ†(x, t)Vξ(0, 0) E β =   π2L2 4β2_sinhπ(t−x−ie) β sinhπ(t+_βx−ie)   ξ2 4π . (3.70) The vertex operator is a primary operator [55] as well as : cos(ξφ) : and :

sin(ξφ) :. From the above expression, one can read off the conformal

weights of these operators:

(h, ¯h) = ξ2 8π, ξ2 8π . (3.71)

It’s enough information to apply the result in 3.1.2 and say that the cor-responding Green’s functions will decay ‡. At the same time, it follows from the same calculation as in 3.1.2 that the retarded Green’s function of the chiral operator V_ξ(+)+h.c. won’t decay. Now, let’s check whether the no-go condition is satisfied or not for these operators. Consider two energy eigenstates, labelled by the occupation number of each mode k: |⊗_knki and ⊗_kn0_k. We can write: : eiξφ :=

∏

k eiξµ∗ka†keiξµkak _(3.72) =

∏

k e−ξ2|µk|2_eiξµkak_eiξµ∗ka†k, (3.73) where µ∗_k = √i 4π|k|e 2πi L (|k|t−kx). Thus D ⊗kn0k : e iξφ : ⊗knk E =

∏

k e−ξ2|µk|2 D n0_k e iξµ_ka_k eiξµ∗ka†k nk E . (3.74)

‡_{It’s assumed here that ξ}2₆₌_{2πn where n is an integer number. However, even when}

ξ2=2πn, the retarded Green’s function will still decay, although the calculation will be

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28 Examples

One can recognize the following expression for the coherent state that obeys the following relation:

eαa†_|₀_{i =} _e|α|2₂ _| αi = ∞

∑

l=0 αl √ l!|li. (3.75)

By making use of it, we have: e−ξ2|µk|2n0 k eiξµkakeiξµ ∗ ka†k |n_ki =∑_l k,l0_kl 0 k (iξµ_k)l0k √ l0_k! (_√ak)n0k n0_k! (a† k)nk √ n_k! (iξµ∗_k)lk √ l_k! |lki =_∑_l k,lk0 (iξµ_k)l0k √ l0_k! (iξµ_√∗_k)lk l_k! 1 √ nk!n0k! r (l_k0+n0_k)!(lk+nk)! l0_k!lk! δlk+nk,l 0 k+n0k =      (iξµk)nk−n 0 k q_n k! n0_k! 1F (R) 1 1+nk, 1+nk−n0k;−ξ2|µk|2 , if nk ≥n0k iξµ∗_kn0k−nk r n0_k! nk! 1F (R) 1 1+n0_k, 1+n0_k−nk;−ξ2|µk|2 , otherwise ≡c nk, n0_k; ξ (3.76) where 1F (R) 1 (a, b; z) ≡ 1F1(a, b; z)/Γ(b) is a regularized hypergeometric

function. This leads to ⊗n0_kV_ξ ⊗n_k =_∏_kc nk, n0_k; ξ ⊗n0_k|: cos(ξφ): | ⊗nk = 1₂ _∏_kc nk, n0_k; ξ +_∏_kc nk, n0_k;−ξ ⊗n0_k| : sin(ξφ) :| ⊗nk = _2i1 _∏_kc nk, n0_k; ξ −_∏_kc nk, n0_k;−ξ . (3.77) Note that c nk, n0_k; ξ = (−1)(nk−n0k)c n_k_{, n}0

k; ξ so that the matrix

ele-ments for : cos(ξφ) : (: sin(ξφ) :) vanish only for those states where

∑knk−n0k is odd (even). This is certainly not sufficient to have∆E given

by a finite number of functions of∆P, so the no-go condition is clearly vi-olated for these operators. At the same time, for the chiral vertex operator V_ξ(+)the matrix elements will be zero whenever n0₋_¯k 6= n₋_˜kfor at least one ˜k > 0, since the operator doesn’t contain any creation/annihilation oper-ators for the negative modes. As a consequence,∆E = _∑_k_>₀(nk−n0_k)k =

∆P and so ∆E is fixed by ∆P for the non-zero elements.

3.2 Transverse field Ising model

The next example we will consider is the transverse field Ising model. We will show that the hamiltonian can always be diagonalized in terms of 28

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fermionic creation annihilation operators and therefore the theory is inte-grable. The hamiltonian of the system is as follows:

H = −J

N

∑

i=1

gσ_ix+σ_izσ_iz+1 , (3.78)

where σ_ia are spin operators with the commutation relations: hσ_ia, σ_jb

i = 2ieabcδijσc. It describes spins living on a 1D chain where the neighbouring

σ_iz, σ_iz₊₁ couple to each other and the magnetic field is applied in the x

direction.

One of the reasons why we consider the model is that the correlators of σz are known to be described by the Gibbs ensemble after a quench is performed [64]. Therefore, it’s interesting to check whether this operator thermalizes in our picture or not§.

In the thermodynamic limit, the system has a critical point at g = 1. This point separates an ordered g > 1 and disordered g < 1 phase. The theory is described by a CFT at this critical point and therefore some of our previous calculations will be applicable. This is the limit in which we want to calculate the retarded Green’s function for σz and check whether the no-go condition is violated or not. Let’s show that the model is also a free fermionic model at any coupling g. To do this, we will need the following tranformation: σ_ix =1−2c_i†ci , σ_iz = −

∏

i<i 1−2c†_jcj ci+c†i , (3.79)

which is called Jordan-Wigner transformation. One can check that the re-sulting creation/annihilation operators obey fermionic (anti-)commutation relations: n ci, c†j o =δij ci, cj =nc†_i, c†_jo=0. (3.80)

By plugging in the operators in the initial hamiltonian and Fourier trans-forming them ck = √1_N ∑jcje−ikrj (rj denotes the cites), we obtain:

H = J

∑

k h 2(g−cos(ka))c†_kck+i sin(ka) c†−kc†k+c−kck −gi, (3.81) §_{Although it’s not entirely clear how our set up is related to the the conventional set}

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30 Examples

here, a is the lattice constant. Finally, we perform a Bogoliubov tranforma-tion:

ck =cos(θk/2)γk+i sin(θk/2)γ−†k (3.82)

with θk defined by:

tan θk = sin (ka)

g−cos(ka). (3.83)

The result is the following diagonalized hamiltonian:

H =

∑

k ek γ_k†γk− 1 2 (3.84) with dispersion relation

ek ≡2J(g cos θk−cos(θk+ka)) =2J

q

1+g2₋_{2g cos}₍_ka₎_. _(3.85)

By looking at the spectrum, one can see that the gap of the theory is:

ek=0 =2J|g−1|, (3.86)

so that it becomes gapless at g = 1. At this point the dispersion relation becomes: ek =4J sin ka 2 =2Ja|k| + O(ka)2. (3.87) For ka<<1, this is the dispersion relation of relativistic massless particles with the ”speed of light” equal to 2Ja.

In the continuum limit, we can ignore O((ka)2)terms. By defining Ψ(xi) =

1 √

aci (3.88)

and looking at 3.81, we get [60]

H = E0+ Z dx v 2 Ψ†∂Ψ† ∂x −Ψ ∂Ψ ∂x +mΨ†Ψ + · · · (3.89) with m=2J(1−g), v=2Ja (3.90) 30

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and E0denoting some constant. If one defines a pair of Majorana fermions

as ψ1 = Ψ +Ψ+ √ 2 , ψ2 =i Ψ−Ψ+ √ 2 , {ψ1, ψ2} = 0, ψ 2 1 =ψ22 = 1 2a, (3.91) and then performs a rotation in the space of ψ1and ψ2:

ψ+ = ψ1 +ψ2 √ 2 , ψ− = ψ2_√−ψ1 2 , (3.92)

The Euclidean action in terms ψ+and ψ−can be rewritten as follows:

SE = 1

2 Z

dxdτ(ψ+(∂τ+iv∂x)ψ++ψ−(∂τ−iv∂x)ψ−+2imψ−ψ+).

(3.93) At the critical point, the mass is 0 and we’re left with a two-dimensional conformal field theory of two (non-interacting) Majorana fermions. Here,

ψ−and ψ+ are holomorphic and antiholomorphic, respectively.

From the Jordan-Wigner transformation 3.79 and 3.92,3.91,3.88 it can be derived that σzis mapped to an operator that looks quite complicated:

σz(x) ∼ eπ

Rx

−_∞ψ−(y)ψ+(y)dy₍_ψ₊₍_x_{) −}_ψ₋₍_x₎₎_. _(3.94)

Therefore, if one wants to calculate a correlator of this operator and the re-tarded Green’s function at the critical point, it appears to be not a straight-forward task to do. However, it turns that σzcorresponds to a primary

op-erator with conformal weights (h, ¯h) = (1/16, 1/16). This can be shown in multiple ways. For example, on of the ways is to use the spin-disorder duality of the model [61]. We will take a different route, however, and use the technique called bosonization. It turns that by applying this technique, the correlation function of σz can be obtained from the square root of the correlation function of a vertex operator in a free bosonic field theory.

3.2.1 Bosonization: basic rules and the intuition behind

them

In this subsection, I’ll state a recipe of how to map a fermionic theory to a bosonic one and explain the basic ideas behind some of the rules. In the following subsection, this recipe will be applied to the transverse field Ising model, with some minor modifications that are necessary in this case.

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32 Examples

This recipe works only if the number of spatial dimensions is 1. This sub-section isn’t going to explain how the bosonization rules actually work and why, for a more detailed review see [67]. These bosonization rules will be discussed in a rather trivial context which is a mapping of a free fermion theory into a free boson theory. Still, the knowledge of this ex-ample will be sufficient to bosonize the transverse field Ising model at the critical point.

The idea of this mapping was suggested by many people. Two of them were Coleman [65] and Mandelstam [66]. Suppose we have a free massless fermion theory in (1+1)D. We can write the Euclidean action (with τ being Euclidean time) as

S=2

Z

dτdx R+∂¯zR+L+∂zL , (3.95)

where z = τ+ix_v, ∂z = 1₂(∂τ−iv∂x). R and L are right- and left-moving

(holomorphic and antiholomorphic ) fermionic fields correspondingly. From now on, we will forget about v since it doesn’t change the result signfi-cantly. Now, the statement is that this fermionic theory is equivalent to the bosonic theory with the action:

S = 1 2 Z dτdx(∂τΦ)2+ (∂xΦ)2 , (3.96)

whereΦ(z, ¯z) = φ¯(¯z) +φ(z). φ(z)and ¯φ(¯z) are the holomorphic and

an-tiholomorphic parts of the bosonic field. The correspondence between the bosonic and fermionic operators is as follows:

R= √1 2πaexp n i√4π ¯φ(¯z) o (3.97) L= √1 2πaexp n −i√4πφ(z)o, (3.98)

where a is the lattice constant again. The ”=” shouldn’t be taken too seri-ously here because the expressions on the right hand side aren’t complete and so-called Klein factors [67] are needed to make the equalities into op-erator identities. Still, these factors can often be safely omitted if one is dealing with bilinears of R, L, for example. Another important bosoniza-tion rule is for the currents of the right- and left-moving fermions

R+R= −√i π∂¯z ¯ φ(¯z) (3.99) L+L = √i π∂zφ (z). (3.100) 32

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relating the currents to the derivatives of the right- and left-moving bosonic fields. The commutation rules for the bosonic fields are¶ (remember z =

τ+ix):

[∂xφ(x), φ(y)] = i

2δ(x−y) (3.101)

[∂xφ¯(x), ¯φ(y)] = −i

2δ(x−y). (3.102)

The first way to see that the mapping actually makes sense is to notice that the spectra of excitations of both theories are the same. Another good sign is that the equalities 3.99 relate the conserved charges in the fermionic theory to the conserved charges in the bosonic theoryk. How can we un-derstand 3.97? Let’s consider the field L. Suppose we start from a vacuum state and create a left-moving fermion at the point x0. This means that the

charge density L†L is δ(x−x0). By integrating 3.99, we can see that the

value of φ should jump at x =x0. One can produce such a domain wall by

acting by an operator canonically conjugated to φ in the following way: L†(x0) ∼e−i

√

πR−x0∞dxπφ(x). (3.103)

Now, what is πφ? By looking at 3.101 and comparing it to the definition

of a canonically conjugated operator[πφ(x), φ(y)] =iδ(x−y), we can see

that: πφ(x) = 2∂xφ(x). (3.104) So that: L†(x0) ∼ e−i √ 4πRx0 −∞dx∂xφ(x) _∼_e−i √ 4πφ(x0) _(3.105)

as we have in 3.97. Another property that can be checked straightfor-wardly is that the fermionic fields anticommute. By integrating 3.101, we have:

[φ(x), φ(y)] = i

4sign(x−y) (3.106)

[φ¯(x), ¯φ(y)] = −i

4sign(x−y). (3.107)

¶_{They can be derived from the bosonization procedure, if it’s done consequentially,}

like in [67] for example.

k_{The conservation laws can be derived from the continuous symmetries φ} _→

φ+

a, ¯φ → φ¯+b for the bosonic action and the U(1) gauge symmetry for the right- and

left-moving fields in the fermionic action. One can easily establish the correspondence between the transformations of the fields by looking at the bosonization rules 3.97

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34 Examples

The constant of integration is fixed by the requirement that the expres-sions on the right hand side are antisymmetric, as commutators have to be. On the other hand, the Baker-Campbell-Hausdorff formula applied to two operators A, B that satisfy[A, B] ∼ 1 writes as:

eAeB =e−[A,B]eBeA. (3.108)

Applied to the field φ, it yields: L(x)L(y) = _2πa1 e−i √ 4πφ(x) e−i √ 4πφ(y) = _2πa1 e4π[φ(x),φ(y)]_e−i √ 4πφ(y)_e−i√4πφ(x)

= _2πa1 eiπ sign(x−y)e−i

√

4πφ(y)_e−i√4πφ(x) _{= −}_L₍_y₎_L₍_x₎_,

(3.109)

so that L indeed anticommutes with itself, as a fermion field should. The same can be verified for the field R. The treatment of anticommutation relations of L, L†is a bit more laborious because one has to deal with some divergencies here (the proof can be found in [67]). The bosonization rules outlined here will be applied to the transverse field Ising model in the next subsection.

3.2.2 Bosonization technique applied to the transverse field

Ising model

The transverse field Ising model cannot be mapped to a free bosonic the-ory in a straightforward way. This is so because the transverse field Ising model at the critical point isn’t a theory of two chiral fermions but it’s a theory of two chiral Majoranas instead 3.93. The second theory yields dif-ferent physics predictions (heat capacity, for example) from those of the first one and has different conservation laws. Because the theory of two chiral fermions is equivalent to the free massless boson theory, there’s no way the Majorana theory can be mapped to this bosonic theory. Neverthe-less, there’s a way to overcome this problem so that the correlators that we want to know can be computed as correlators in a free massless bosonic theory, of some relatively simple-looking operators.

We will follow the bosonization procedure of the transverse field Ising model described in the paper by Zuber and Itsykson [62]. The main trick is to double the theory by introducing one more pair of chiral Majorana fermions and defining new fermionic fields R, L in the following way:

R= ψ 1 ++iψ2+ √ 2 , L= ψ1−_√+iψ2− 2 . (3.110) 34

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The indices 1 and 2 denote the first and the second copy of the Ising model. One can check that R, L will satisfy the canonical anticommutation rela-tions for fermions. Both ψ1± and ψ2±have the same action as 3.93 but with

m = 0 because we’re at the critical point. As a result, the action for R, L is the same as in the previous subsection 3.95, with v = 2Ja that will be taken to be 1 for simplicity. This means that the outlined bosonization recipe is applicable. Still, the theory we’re bosonizing is a bit different and what we hope for is that the correlators in this theory will be related to the correlators in the transverse field Ising model in some simple way.

Let’s consider the spin-spin correlator of the single copy squared: hσz1(n)σz1(0)i 2 = hσz1(n)σz1(0)σz2(n)σz2(0)i def = hσz(n)σz(0)i. (3.111)

The latter equality holds because the two copies are non-interacting. It’s convenient to represent σ_za(n)σ_za(0)(a =1, 2) as follows :

σza(n)σza(0) = n−1

∏

m=0 σza(m+1)σza(m) = n−1

∏

m=0 2iaψ₂a(m)ψ₁a(m+1). (3.112) ψa₁, ψ₂a are just the copies of the Majorana fields ψ1, ψ2defined in 3.91. The

second equality follows from 3.79, 3.91 and 3.88. By making use of the anticommutation relations 3.91, one gets:

hσz1(n)σz1(0)i 2 =hF1(n)F(n−1)F2(0)i, (3.113) Fj(m) =2aψ1j(m)ψ2j(m) =exp h πaψ1_j(m)ψ2_j(m) i (3.114) F(n−1) = n−1

∏

m=1 F1(m)F2(m) =exp " πa n−1

∑

m=1j=

∑

1,2 ψ1_j(m)ψ2_j(m) # . (3.115) Substitution of 3.92 into 3.114 gives (in the continuum limit a →0):

F1,2(x) =2aψ11,2(x)ψ1,22 (x) = =a ψ1+(x)ψ2+(x) +ψ1−(x)ψ2−(x) ∓a ψ1+(x)ψ2−(x) +ψ1−(x)ψ2+(x) . (3.116) Eq. 3.110 and the bosonization rules 3.97, 3.99 lead to the result:

F1,2(x) = − ia √ π∂xΦ (x) ∓ i π sin √ 4πΦ(x). (3.117)

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36 Examples Finally: F(n−1) =exp " πa n−1

∑

m=1j=

∑

1,2 ψ1_j(m)ψ2_j(m) # →exp −i√π Z x−a a ∂yΦ dy , (3.118) so that: hσz(x)σz(0)i = −_√ia π∂xΦ(x) − i π sin √ 4πΦ(x)exp−i√πΦ(x−a) × ×expi√πΦ(a) −_√ia π∂xΦ(0) + i πsin √ 4πΦ(0) . (3.119)

In the limit a → 0, some of the terms will blow up. The regularization

procedure is described in [63]. I quote the answer: hσ_z(n)σz(0)i = hσ_z1(x)σ_z1(0)i2 = 1 π2: sin √ πΦ(x):: sin√πΦ(0): . (3.120) As a result, one can represent σz(x) = σ_z1(x)σ_z2(x) in the doubled Ising

model as

σz ∼: sin √

πΦ(x) : (3.121)

and compute the correlation functionhσz(x)σz(0)i. To obtain its value for

the original transverse field Ising model, one has to take the square root of it (according to 3.111). Because we’re dealing with a free massless boson theory, the results from the subsection are applicable. It can be seen from 3.71 that the operator σz = σ_z1σ_z2 is primary and the conformal weights

are (h, ¯h) = (1/8, 1/8). This will lead to the answer hσz1(x)σz1(0)i

2 ∼ 1/(x)1/2 which is in agreement with the known (zero-temperature, equal time) result [60].

In a similar manner, the bosonic representation for σx can be

estab-lished. Recalling the Jordan-Wigner transformation for σx 3.79 and the

definitions 3.88,3.91: σx =σx1σx2 = −4a2ψ21ψ11ψ22ψ12= 3.92 =4a2ψ1+ψ2+ψ−1ψ2− (3.122)

Because ψ1+ψ2+ = −iR+R and ψ1−ψ2− = −iL+L (eq. 3.110 ), σxthe

bosoniza-tion rules 3.99 bring us to a simple result :

σx = −4a 2 π ∂zφ∂¯z ¯ φ. (3.123) 36

Thermalization/Relaxation in integrable and free field theories

Thermalization/Relaxation in

integrable and free field theories

Thermalization/Relaxation in

integrable and free field theories

Tereza Vakhtel

Abstract

Contents

Chapter

1

Introduction

Chapter

2

The question and the main result of

the thesis

2.1

Definition of thermalization in this thesis

∑

2.2

The no-go condition and its proof

∑

∑

∑

∑

∑

Chapter

3

Examples

3.1

2D Conformal field theories

3.1.1

Basic background

3.1.2

General results for 2D CFTs

3.1.3

Free massless boson theory in 2D

∑

∑

∑

∑

∑

∑

∑

∑

∏

∑

∑

∏

∏

∏

∑

3.2

Transverse field Ising model

∑

∏

∑

∑

3.2.1

Bosonization: basic rules and the intuition behind

them

3.2.2

Bosonization technique applied to the transverse field

Ising model

∏

∏

∏

∑

∑

∑

∑