Entanglement and Conformal Field Theory: Distribution in symmetry sectors

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Universit`a degli Studi di Pisa Corso di Laurea Magistrale in Fisica

Entanglement and Conformal Field Theory

Distribution in symmetry sectors

Relatori:

Prof. Pasquale Calabrese Prof. Mihail Mintchev

Candidato: Luca Capizzi

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Introduction

As recognized by Schr¨odinger [1], the characteristic trait of quantum mechanics is found in the entanglement.

Two parts, say A and B, of the same quantum system, may be correlated in a way that does not have a classic analogue: the entanglement encodes the possibility to affect the measurements of local observables in A by local operations in B. This ‘spooky’ non-locality gave rise to severe skepticism since the early days of quantum mechanics and it has no counterpart in the context of classical information.

Numerous criteria have been proposed to quantify the amount of entanglement in generic systems, as for example the Schmidt rank and the entanglement entropy (also known as von Neumann entropy)[4].

In the last few years, these concepts appeared in different, apparently unrelated, fields of physics. In condensed matter, there are complex ground states in various many-body systems (superconductors, ferromagnets, quantum Hall systems ,...) that show peculiar correlation properties.

The entanglement shared among qubits is also supposed to be a building block for the quantum computer and its supremacy respect to classical computations. Other applica-tions in quantum information include Shor’s algorithm, quantum teleportation and quan-tum cryptography ([5, 6, 7, 8]).

In the context of black hole, the Bekenstein-Hawking entropy is recognized as a measure of quantum correlation among the interior and exterior part of black hole (inside and outside the event horizon) ([14, 15, 16, 17]).

Furthermore, the efficiency of a DMRG (density matrix renormalization group) numerical simulation of a low energy system relies on a priori estimation of the effective degrees of freedom needed to describe the main properties of the many-body quantum state taken in exam: the entanglement entropy of the state is strictly related to the number of such degrees of freedom (see [9, 10, 11, 12]).

From a theoretical point of view, many questions have been rise on the scaling regime of the entanglement entropy. It is important to understand the dependence on the micro-scopical details and the properties of A and B.

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For extended systems with finite correlation length, the leading order of the entanglement entropy follows the area law : S is proportional to the volume of the interface between the two subsystems ([24], [25]). A naive explanation for this behavior is the observation that the bulk properties don’t contribute to, S whereas the correlation effects are localized near the interface, due to the finite correlation length.

Figure 1.1: The system is partitioned in regions A and B. For finite length correlation, the correlation effects are localized near the boundary (yellow region). [From [4]].

Now, we are going to explore what happens when the correlation length is infinite and the area law is typically violated. There is no a universal scaling law for S in extended systems of different dimensionality ([18, 19, 20]).

However, for a certain class of quantum 1D systems, conformal invariant systems, the entanglement entropy shows a remarkable universal behavior. If an infinite system is bi-partite in a finite segment of length ` and in the rest, it holds

S = c

3log ` + . . . , (1.1)

where c is the central charge of the underlying conformal field theory (CFT).

It is important to stress that c is the only parameter which appears in this formula and there is no dependence on microscopic details. Indeed, the measure of S is the most precise way to get the central charge c, since it does not require any other knowledge. The formula (1.1) was firstly derived by Holzhey et al. and then by Calabrese and Cardy [45] using CFT. Not only the CFT approach is useful to get the scaling behavior of the entanglement entropy of an infinite extended ground-state. It has also been developed a general formal-ism and many other quantities have been calculated: R´enyi entropy, relative entropy [71], entanglement negativity [87, 88, 89], and so on. Moreover, the CFT described the low-energy excitations around the ground state and a set of excited states can be investigated through the same formalism [58].

Various non-equilibrium situations have been taken in account (e.g. the quantum quenches [26]). The dynamics of conformal systems (and, in general, of integrable systems) is pe-culiar and the initial state converges very slowly to an equilibrium state which is not

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thermal. The connection between the thermodynamics and a unitary dynamics is sub-tle and probably not fully understood, but the interest in this topic grew up in the last decades with the realization of cold atoms experiments. It is believed that understanding the entanglement evolution unveils how thermodynamics emerges in isolated systems (see [26, 27, 28, 30, 31, 32]).

A complete list of applications of CFT in low-energy physics and the role of the entangle-ment is beyond the scope of this thesis.

Our main purpose is to understand what happens to the entanglement when the sys-tem has additional symmetries. Quantum chains, made by fermions or spins, often show an internal U (1)-symmetry (say the conservation of the number of particles or the total magnetization along a chosen axis) which allows to separate the set of the states in differ-ent symmetry sectors (represdiffer-entations of the underlying symmetry) [40].

If one has to deal with a quantum state of a closed system with a fixed number of particles, it is no longer true that the number of particles of a subsystem A has a fixed value: the conservation law refers only to the entire system. For each U (1)-symmetry sector of A there is a finite contribution to the total entanglement shared by A and the rest of the system, that corresponds to the different situations where A can be found, conditioned to have a certain number of particles.

The model we will consider to make CFT predictions is the massless free boson in 1D, also known as Luttinger liquid. This is motivated not only by theoretical issues (the full comprehension of the underlying CFT we have, the simple structure of the correlation functions...), but also by the fact that this model describes effectively many 1D systems, both free and interacting ones: the famous technique which allows this kind of correspon-dences is called bosonization [48].

There is a lot of theoretical results in the literature concerning the entanglement for the free boson: we want to mention only the distribution in symmetry sectors of the ground state [40, 63, 66] and the entanglement of low-energy excited states [58, 59].

Our goal is to discover the behavior of the distribution in symmetry sectors of the excited states, for which no CFT results have been known until now for the free boson.

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Chapter 2

Conformal Field Theory

For a special class of critical phenomena, expecially for second-order phase transitions, the critical temperature vanishes or is small compared to other relevant energy scales. The statistical fluctuations give rise to correlation which are not thermal, but mainly quantum-mechanical. A quantum description of the system is thus indispensable. One of the main feature of these system is the presence of gapless excitations: they are signals of a scale invariance of the underlying field theory. In many cases, the scale-invariance can be enhanced to a bigger symmetry, namely the conformal invariance. There are some quantities (e.g. critical exponents) which are universal near these critical points, and they can be computed exactly through analytical methods coming from the CFT [42].

The conformal field theory is thus the natural language to describe such criticalities which arise in statistical physics. CFT plays moreover a key role in the formulation of string theory and in QFT (a vanishing beta-function β(g) = 0 means that the theory is scale-invariant).

Most of the next sections are devoted to the peculiarities and the specific formalism of two-dimensional CFT’s.

2.1 Conformal transformations

We say that a certain transformation is conformal if it preserves the angles. This is equiv-alent to say that under the diffeomorphism x → x0 the metric changes by

g_µν0 (x0) = e2λ(x0)gµν(x(x0)), (2.1)

where e2λ(x0) is a local scale factor.

The set of conformal transformation in D dimension is a group and it obviously con-tains the group of rotation SO(D) (or the Poincar´e algebra, depending on the signature of gµν), which are obtained imposing a more strict condition λ(x) = 0.

In D = 1 the condition (2.1) is trivial: any diffeomorphism is conformal. In fact, the vector space associated with the possible metrics applied to a certain point x is one-dimensional and, therefore, any two metrics fields are proportional to each other.

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For D ≥ 3 the group is isomorphic to the finite-dimension Lie group SO(D + 1, 1). A part from SO(D) and the global dilation (λ(x) does not explicitly depend on x) another set of transformation in needed to generate the whole conformal group. It contains the special transformation that are parametrized by a vector bµ in the following way

xµ→ x

µ_{− b}µ_x2

1 − 2(b · x) + b2_x2. (2.2)

The case of interest D = 2 is nontrivial. Let us use complex coordinates    z ≡ x + iy, ¯ z ≡ x − iy. (2.3) It is easy to prove that the local condition (2.1) is satisfied by the holomorphic/antiholomorphic mappings, provided they are invertible at the point x.

It is important to say that not every local conformal transformation can be extended to a global transformation (invertible everywhere). The M¨obius transformations, generated by

z → az + b

cz + d (2.4)

are the only global conformal maps of the Riemann sphere (one needs the condtion ad − bc 6= 0 for the invertibility). This is a finite dimensional Lie group of (complex) dimension 3. However, we will not restrict on globally defined transformation, dealing only with local conditions through the Lie algebra associated to the group.

The Lie algebra associated to the holomorphic infinitesimal transformations (in 2 di-mensions) is called Witt algebra. It is generated by the elements {Ln}n which satisfy the

following commutation relations

[Lm, Ln] = (m − n)Lm+n. (2.5)

A simple Lie representation is given by the action of the Witt algebra on the holomorphic functions    z0≡ z + (z) = z +P nnzn+1, φ(z) → φ(z(z0)) ' φ(z) − (z)∂φ. (2.6) The set of generators {Ln}

Ln≡ −zn+1∂z (2.7)

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2.2 Quantum conformal symmetry

In quantum mechanics one is typically interested in central extensions of classical symme-try, due to the fact that the state vectors of the Hilbert space are identified up to a phase. The unique (non-trivial) central extension of Witt algebra is called Virasoro algebra; one can construct it by adding to the Witt algebra a central element c (it commutes with any other element), called central charge, and imposing the following commutation relation

[Lm, Ln] = (m − n)Lm+n+

c 12(m

3_{− m)δ}

m,−n. (2.8)

Every representation ρ of the Virasoro algebra such that ρ(c) = 0 can be seen as Witt representation: the relations (2.8) reduce to (2.5). However, in realistic physical models the conformal symmetry is typically implemented by a Virasoro representation in which ρ(c) 6= 0.

Given an irreducible representation ρ, one can prove by Schur lemma that c is represented by an operator proportional to the identity

ρ(c) = ˆc · id. (2.9)

For the sake of simplicity, we won’t make distinction between the central charge (c) and its value (ˆc).

From this point onward, we will always concentrate on 2-dimensional quantum field theory. We say that a local operator O(z, ¯z) is primary with conformal weights (h, ¯h) if its corresponding transformation law under a local conformal transformation is

O(z, ¯z) → O0(z0, ¯z0) ≡ dz dz0 h d¯z d¯z0 ¯h O(z(z0), ¯z(¯z0)). (2.10) If the previous equality holds only for global transformation, O(z, ¯z) is said quasi-primary.

Under a global scale transformation    z → az, ¯ z → a¯z (2.11)

of a scale factor a ∈ R, one has O0(z, z) = a−(h+¯h)(a−1z, a−1z).¯ Similarly, under a global rotation

   z → eiθz, ¯ z → e−iθz¯ (2.12)

of angle θ ∈ R, one obtains O0(z, z) = e−iθ(h−¯h)(e−iθz, eiθz).¯

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The richness of conformal invariance imposes a lot of nontrivial constraints on the theory, especially on the correlation functions. Let us define the n-points function of some local observable {Oj(z, ¯z)}

hO1(z1, ¯z1)...On(zn, ¯zn)i ≡

Z

Dφ e−S[φ]O1(z1, ¯z1)...On(zn, ¯zn), (2.13)

where φ denotes the set of all functionally independents fields in the theory and S[φ] is the Euclidean action.

If one assume conformal invariance on both S and Dφ, it is possible to prove that for primary operators Oj of weights (hj, ¯hj) it holds

hO1(z1, ¯z1)O2(z2, ¯z2)i ∝

δh1,h2δ¯h2,¯h2

(z1− z2)2h1(¯z1− ¯z2)2¯h1

. (2.14)

Moreover, a part from the case (hj, ¯hj) = 0 that requires Oj to be proportional to the

identity, the 1-point function vanishes

hOj(zj, ¯zj)i = 0. (2.15)

The formulas that are written above hold if the space-time is the plane. In different geometry, the short-distance behavior is the same but the large-distance functional form can be different. In the next chapters a cylindrical geometry will be taken in account; the equivalence of the complex plane (Riemann sphere) and the cylinder, up to conformal transformation, is sufficient to get the correlation functions of operators put on the cylin-der.

We mention that the functional form of 3-points functions can also be fixed by the conformal invariance (see [42]). However n ≥ 4 the constraint becomes weaker, due to the fact that the cross-ratio of four points is invariant, and the functional form cannot be determined a priori.

2.3 Tracelessness of energy-momentum

The trace of the energy-momentum Tµµis a scalar function and its value fixes some energy

scale. One would expect that conformal invariance is incompatible with a typical energy scale, so the physical-motivated guess would be

T_µµ= 0. (2.16)

This is true for a classical system (as we will prove later on). Although, we mention for completeness that the background-curvature is typically associated with a quantum

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anomaly (Weyl anomaly) that breaks the relation (2.16) at the quantum level.

Suppose we have a classical action S[φ, gµν] diffeomorphism-invariant (transforming

both φ and gµν); one could explicitly introduce the background metric dependence gµν

through minimal-coupling in order to study the case above. The energy-momentum Tµν is defined by Tµν ≡ 2π 2 p|g| δS δgµν , (2.17)

where the factor 2π is introduced for practical convenience.

The covariant conservation ∇µTµν = 0 follows from diffeomorphism-invariance; given the

flat background gµν = ηµν, it reduces to a canonical conservation law ∂µTµν = 0.

Now, let us impose that the local scale-transformation

gµν → e2λ(x)gµν(x), (2.18)

is a symmetry of our action. The associated infinitesimal off-shell change of S (recall that gµν is not a dynamical field!) is

0 = δλS = Z dDx δS δgµν δλgµν = Z dDxp|g|T_µµλ(x). (2.19) Because of the arbitrariness of λ we can conclude that

T_µµ= 0. (2.20)

Using complex coordinates in D = 2 dimensions, the energy momentum conservation is equivalent to    ∂zTz ¯¯z= 0, ∂¯zTzz = 0, (2.21) while the tracelessness can be expressed by Tz ¯z = 0.

The information contained in the energy momentum tensor can be encoded in two holomorphic/antiholomorphic functions respectively

   T (z) ≡ Tzz(z), ¯ T (¯z) ≡ ¯Tz ¯¯z(¯z). (2.22)

2.4 Ward Identities

The quantum counterpart of the Noether theorem is given by a set of equations known as Ward identities.

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Let us take a QFT described by an Euclidean action S[φ]. A symmetry of the quantum theory is a transformation

φ → φ + δφ, (2.23)

that leaves both the action S[φ] and the measure Dφ invariant.

Let Jµ(x) be the local generator of the symmetry. For any given set {O(xj)}j=1,...,n of

local operators the Ward identity takes this form − 1

2πh∂µJ

µ_(x)O

1(x1) . . . On(xn)i

=δ(x − x1)hδO1(x1) . . . On(xn)i + · · · + δ(x − xn)hO1(x1) . . . δOn(xn)i,

(2.24)

where the expected value h·i is weighted by e−S[φ]and δO is the infinitesimal transforma-tion of the corresponding operator ( the normalizatransforma-tion _2π1 is a matter of convention). The only difference with the Noether equation (∂µJµ= 0 ) is given by contact terms.

In two dimensions we can use complex coordinates and integrate in x the Ward equa-tion in a small region |x − x1| 1. Making use of Stokes theorem we obtain

i 2π I dzhJz(z, ¯z)O1(x1) . . . i − i 2π I d¯zhJz¯(z, ¯z)O1(x1) . . . i = hδO1(x1) . . . i. (2.25)

2.5 OPE

Let us take a local operator such that the infinitesimal transformation associated to δz = z, δ ¯z = ¯¯z (a scaling) is

δO = −(hO + z∂O) − ¯(¯hO + ¯z ¯∂O). (2.26) The Ward identity applied to the generator of conformal transformation T (z), tells us (after a straightforward algebra) that the OPE (operators product expansion) between T and O is    T (z)O(w, ¯w) = · · · +hO(w, ¯_(z−w)w)2 + ∂O(w, ¯w) z−w + . . . , ¯ T (¯z)O(w, ¯w) = · · · +hO(w, ¯¯_(¯_{z− ¯}_w)w)2 + ¯ ∂O(w, ¯w) ¯ z− ¯w + . . . (2.27) If there are no more singular terms, O is called primary (the definition is compatible with the previous one (2.14)).

In general the energy-momentum tensor is NOT primary. In fact, in a Virasoro repre-sentation with central charge c the OPE between T and itself is given by

T (z)T (w) = c 2

1

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and the weights of T (z) are (h, ¯h) = (2, 0).

One can prove that the transformation law associated to T is T0(z0) = (dz dz0) 2_{(T (z(z}0 )) − c 12S(z 0 , z)), (2.29) with S(z0, z) = (d_dz3z30)(dz 0 dz) −1₋ 3 2( d2_z0 dz2)2(dz 0 dz)

−2 _{called the schwartzian derivative. This}

in-duction is far from obvious, but it can be proved starting from the OPE between T and itself. The presence of the anomalous term proportional to cS(z0, z) lies in the quantum nature of symmetry: in [38] it is shown that the in the free boson theory the previous term comes precisely from the normal-ordered definition of T .

Probably the transformation law (2.29) is the most important formula of the 1+1 dimen-sional CFT in terms of its implications.

It is easy to prove that for global conformal transformation S(z0, z) vanishes, so one can deduce that T (z) is quasi-primary.

2.6 Operator formalism

The operator formalism, which distinguishes a time’s direction from a space’s one, appears somewhat arbitrary in the context of Euclidean QFT, since the path-integral approach does not make this distinction. There are (at least) two typical choices in the context of a flat space-time background: the canonical quantization and the radial quantization.

In the first case one parametrize a space-time points with a coordinate z = x + it; the second case, obtained by a conformal map form the previous one, uses the parametrization z = e2πL(t+ix) and identifies the points (x = 0, t) and (x = L, t). The radial quantization is

a traditional choice in the context of string theory.

Figure 2.1: Mapping from the cylinder to the complex plane.[From [42]].

Dealing with a local operator φ(z, ¯z) and a vacuum state |0i, we can associate the asymptotic ‘in’ state

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|φi ∼ lim

z,¯z→0φ(z, ¯z) |0i . (2.30)

The above map is called state-operator map.

The definition of the asymptotic ‘out’ states in radial quantization requires a careful anal-ysis: the infinite past point is z = 0 while the infinite future point is z = ∞. Dealing with a quasi-primary operator of weights (h, ¯h) we define

hφ| ∼ lim

z,¯z→0z¯

−2h_z−2¯h_{h0| φ(¯}_z−1_{, z}−1_), _(2.31)

in order to be consistent with the canonical quantization.

In the radial quantization context the time ordering of correlation functions becomes a radial ordering defined by

Rφ1(z)φ2(w) ≡    φ1(z)φ2(w) if |z| > |w|, φ2(w)φ1(z) if |z| < |w|. (2.32) If the fields are fermions a minus sign is added in front of the second expression.

In a CFT where the Virasoro algebra is represented by the operators {Ln}, one can

see that the energy momentum T (z) is built up from {Ln} as underlying modes

T (z) =X

n∈Z

Ln

z2+n. (2.33)

The requirement of the representation to be unitary, which is the common paradigm when the theory is described by a hermitian Hamiltonian, becomes L−n= L†n.

There are a couple of simple but important constraints that arise from unitarity [41]: both h, ¯h (conformal weights of any quasi-primary operator) and the central charge c have to be real and ≥ 0.

2.7 Free Boson

The free massless scalar field theory is a typical example of a CFT. Its associated Eu-clidean action is

S[φ] ≡ 1 8πK

Z

dxdτ (∂τϕ)2+ (∂xϕ)2. (2.34)

By minimizing the action, one can find the classical equation of motion

∂µ∂µϕ = 0, (2.35)

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Using the identity

∂µ∂µlog |x|2= 4πδ(x), (2.36)

it easy easy to calculate the propagator in complex coordinates

hϕ(z, ¯z)ϕ(w, ¯w)i = −K log |z − w|2. (2.37)

The field can be split in two sectors, the holomorphic and the antiholomorphic (equiva-lently, the right- and left-moving)

ϕ(z, ¯z) = φ(z) + ¯φ(¯z), (2.38)

in such a way that the correlation function may be factorized in two parts          hφ(z)φ(0)i = −K log z, h ¯φ(¯z) ¯φ(0)i = −K log ¯z, hφ(z) ¯φ(0)i = 0. (2.39)

Despite the fact that φ has classical dimension (h, ¯h) = (0, 0), it is not a primary operator (due to the logarithmic UV-divergence).

The tensor energy-momentum is found to be T = − 1

2K : ∂φ∂φ : . (2.40)

The OPE T (z)T (0) can be computed and one obtains the value of the central charge c = 1. Using φ one can define the vertex operator

Vα, ¯α(z, ¯z) ≡ : eiαφ(z)+i ¯α ¯φ(¯z) :, (2.41)

which is primary with conformal weight (h, ¯h) = (α2₂K,α¯2₂K). Most of the time we will deal only with the holomorphic sector, so it is natural to consider Vα ≡ Vα,0, in order to

simplify the notations.

The normal ordering avoids the emerging of infinite constant shift. For example lim

w→zhφ(z)φ(w)i = ∞, instead

: φ(z)φ(z) : ≡ lim

w→zφ(z)φ(w) − hφ(z)φ(w)i, (2.42)

has a well defined limit. If it is not specified otherwise, the formal powers of the fields are always to be considered normal ordered.

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Let us prove that Vα is primary by exploiting the OPE with T (this is a simple

appli-cation of Wick theorem)

T (z) : eiαφ(w) : = − 1 2K : ∂φ(z)∂φ(z) : ∞ X n=0 (iα)n n! : φ(w) n_: = α 2 2Kh∂φ(z)φ(w)i 2_V α(w) − iα Kh∂φ(z)φ(w)i∂zφ(z)Vα(w) = 1 (z − w)2 Kα2 2 Vα(w) + 1 z − w∂wVα(w) + ... (2.43)

The general correlation function of n vertex operators {Vαj}j vanishes if

P jαj 6= 0; otherwhise hVα1(z1)...Vαn(zn)i = Y i<j (zi− zj)Kαiαj. (2.44)

Another useful primary operator is the derivative operator ∂φ with weight (1, 0). Let us compute its OPE with T

T (z)∂φ(w) = − 1 2K : ∂φ(z)∂φ(z) : ∂φ(w) = − 1 Kh∂φ(z)∂φ(w)i∂φ(z) = 1 (z − w)2∂zφ(z) = 1 (z − w)2∂wφ(w) + 1 z − w∂ 2 wφ(w) + ... (2.45)

2.8 Free Fermion

The massless Dirac field is composed by two Weyl’s fields, which may be also Majorana’s in two dimensions Ψ(z, ¯z) = ψ(z) ¯ ψ(¯z) ! . (2.46)

For the sake of simplicity, we will focus only on the holomorphic field ψ. The chiral action for ψ is

S ≡ 1 2π

Z

d2x ψ ¯∂ψ, (2.47)

and the associated equations of motion are ¯

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A careful analysis shows that the UV behavior of the fermion propagator is hψ(z)ψ(0)i ∼ 1

z. (2.49)

This means that the conformal weight of ψ is (1₂, 0). The energy momentum is proportional to

T ∝: ψ∂ψ : . (2.50)

Taking care of the proportionality constant, one finds that c = 1₂.

However, the whole Dirac theory takes contributions from each chiral sector; the asso-ciated central charge is indeed c = 1 = 1₂ +1₂.

The fact that c = 1 for both Dirac theory and the free boson is not a coincidence! As we will show later on, through bosonization one can match the two theories in each other.

For the sake of completeness, we will remind some basic facts about the Dirac complex theory (without Majorana fermions) in two dimensions.

Let us define

Ψ ≡ ΨL ΨR

!

. (2.51)

We choose the following (chiral) representation of the Dirac Euclidean matrices

γ0 = 0 1 1 0 ! , γ1 = 0 −i i 0 ! . (2.52)

The massless Dirac Euclidean Lagrangian is usually (out of CFT context) defined as L ≡ Ψ†γ0γµ∂µΨ

= Ψ†_R(∂τ − i∂x)ΨR+ Ψ†_L(∂τ+ i∂x)ΨL.

(2.53)

A straightforward calculation leads to the Feynman propagators

hΨ_R(τ, x)Ψ†_R(0, 0)i = Z dωdk (2π)2 e−iωτ +ikx −iω + k = i 2π 1 x + iτ, hΨL(τ, x)Ψ†_L(0, 0)i = Z _dωdk (2π)2 e−iωτ +ikx −iω − k = i 2π 1 −x + iτ. (2.54)

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Chapter 3

Entropy

3.1 Pure and mixed states

The standard treatment of quantum mechanics requires the notion of pure states, defined as the rays (unit vectors identified up to a phase) of a Hilbert space H . The above paradigm is sufficient to describe the dynamics and the measurement statistics of a closed system.

Unfortunately, in every realistic physical realization of a system there are interactions with an external environment we cannot directly access, so one has to deal with open systems whenever there is a lack of information about the environment. However, most of the formalism built up with Hilbert spaces and operators as observables can be applied again if one extends the set of physical states.

We say that the C-linear operator ρ ∈ End(H ) (ρ : H → H ) is a state (equivalently a density operator ) if

• ρ is Hermitian;

• ρ ≥ 0, which means that its eigenvalues are ≥ 0; • trρ = 1.

The expectation value of any observable O respect to ρ is computed in the following way

hOi = tr(ρO). (3.1)

There are, of course, states which are not projectors (pure states) and they are called mixed states. A construction of a mixed state is made up by an ensemble of pure states {|Ψ_ii}_i with probabilities {pi}

ρ ≡X

i

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We mention that the previous axioms are sufficient to describe a state ρ in terms of an ensemble, but there are different ensembles which produce the same state ρ.

3.2 Shannon and Von Neumann Entropy

In the context of classical information, from a probability distribution {pi}i∈I of a certain

random variable X, which takes values in I, one can associate the Shannon entropy S(X) ≡ −X

i∈I

pilog pi. (3.3)

One defines von Neumann entropy of a mixed state ρ as the Shannon entropy associated to the spectrum of ρ; in other words

S(ρ) ≡ −tr[ρ log ρ]. (3.4)

There are many common properties shared by the Shannon entropy (classical) and the von Neumann entropy (quantum); however not every quantum information quantity has a classical counterpart: the problem is found in the non-commutativity of the quantum observables.

An important measure of entanglement between two systems A and B, related to the von Neumann entropy, is known as entanglement entropy. Taking a pure state of the composite Hilbert space |Ψi ∈HA⊗HB, one can associate the reduced density matrices

   ρA= trB|Ψi hΨ| ∈ End(HA), ρB= trA|Ψi hΨ| ∈ End(HB), (3.5)

where trA/B are the partial traces respect to the subsystem A/B.

Despite the fact that ρAand ρB are very different objects, acting on different Hilbert

spaces, one can show (by Schmidt decomposition) that the nonzero part of the spectrum of the two states is the same!

This observation leads to a definition of entanglement entropy

S ≡ S(ρA) = S(ρB), (3.6)

which depends only on the correlations between the two subsystems A and B.

3.3 Conditional entropy

Suppose X and Y are two (classical) random variables. The information content of X related to the information content of Y is measured by the conditional entropy.

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Let (X, Y ) be the joint random variable. The entropy of X conditional on knowing Y is defined by

S(X|Y ) ≡ S(X, Y ) − S(Y ). (3.7)

We stress that there is no notion of joint probability in quantum mechanics. Never-theless, whenever two quantum observables commute, there is a classical interpretation of the probability distributions; under these assumptions, we can use these concepts also in the quantum paradigm.

The definition given above is motivated by the following property

S(X|Y ) = hS(X|Y = y)iy. (3.8)

Proof:

Let p(x, y) the joint probability distribution of (X, Y ). It has p(x) and p(y) as marginal distributions    p(x) =P yp(x, y), p(y) =P xp(x, y). (3.9)

We recall that the conditional distribution is given by p(x|y) = p(x,y)_p(y) . By a straightforward calculation one obtains

hS(X|Y = y)i_y = −X

y

p(y)X

x

p(x|y) log p(x|y) =

−X

x,y

p(x, y) log p(x, y) +X

y

p(y) log p(y) = S(X, Y ) − S(Y ).

(3.10)

3.4 R´

enyi entropy

The n-R´enyi entropy, with n ≥ 0 and n 6= 1, is a generalization of the Shannon entropy. Let p(x) the probability distribution of random variable X; we define

S(n)(X) ≡ 1 1 − nlog

X

x

pn(x). (3.11)

The Shannon entropy is obtained by taking the limit n → 1.

The R´enyi entropy can be also considered a measure of ignorance. Indeed, S(n)(X) = 0 if there exists a single outcome x0 such that

   p(x0) = 1, p(x) = 0 if x 6= x0. (3.12)

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Moreover it reaches the maximum value when p(x) = p(x0) for all possible outcomes x, x0 of X.

In the context of quantum information, we can proceed as in the case of von Neumann entropy. One defines

S_A(n)= 1

1 − nlog tr(ρ

n

A), (3.13)

which is equivalent to compute in a classical way the R´enyi entropy of the spectrum of ρA.

We see again that there is a dependence only on the nonzero part of the spectrum of ρA.

3.5 Linear algebra

3.5.1 The polar and the singular value decomposition

Theorem 1 (Polar decomposition) Let A a linear operator on a finite dimensionial Hilbert spaceH . Then there exists unitary U and positive operators J and K such that

S = U J = KU, (3.14)

where J and K are defined by J = √

A†_{A and K =}√_AA†_{. Moreover, if A is invertible}

the U is unique.

Theorem 2 (Singular value decomposition) Let A be a square matrix. Then there ex-ist unitary matrices U and V , and a diagonal matrix D with non-negative entries such that

A = U DV, (3.15)

The diagonal elements of D are called the singular values of A.

3.5.2 The Schmidt decomposition

Theorem 3 Suppose |ψi is a pure state of a composite system AB (|ψi ∈ HA⊗HB).

Then there exist orthonormal states |iAi for system A and |iBi for system B such that

|ψi =X

i

µi|iAi |iBi , (3.16)

where µiare non-negative real numbers satisfyingPiµ2i = 1 known as Schmidt coefficients.

This result is very useful in quantum information theory. One can deduce that ρA/B := trB/A(|ψi hψ|) =

X

i

µ2_i i_A/B i_A/B

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Chapter 4

Entanglement Entropy and CFT

The entanglement entropy, like any other quantity related to the density matrix ρ, can be reconstructed from the direct knowledge of the spectrum of ρ. This task is extremely difficult for generic systems, due to the fact that the dimension of the Hilbert space grows exponentially with the size of the system. There are some specific techniques, as the en-tanglement Hamiltonian approach, which are able to simplify the problem in the case of gaussian states (e.g. a ground state of quadratic Hamiltonian).

The aim of this section is to exploit and prove the relation S(n)= c 6 1 +1 n log ` + ..., (4.1)

valid for the ground state of a 1+1 conformal system.

If one is able to calculate the quantity tr[ρn] for integer n and to make the analytic continuation for n > 1, it would be possible to find the entanglement entropy by making the limit n → 1+.

4.1 Path-integral approach

Let us suppose we want to describe a 1+1 quantum field theory which has a complete set of local commuting observables {φ(x)}x (the field).

The thermal density matrix is formally equivalent to a unitary evolution in imaginary time and its matrix elements can be seen as transmission amplitudes

hφ1|

1 Ze

−βH_|φ

0i . (4.2)

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hφ₁| 1 Ze −βH_|φ 0i = 1 Z Z φ(τ =β)=φ1 φ(τ =0)=φ0 Dφ(x, τ )e−SE[φ(x,τ )]_, _(4.3)

having defined the normalization factor Z =

Z

φ(τ =β)=φ(τ =0)

Dφ(x, τ )e−SE[φ(x,τ )]_, _(4.4)

which ensures that tr

e−βH

Z

= 1.

One can see pictorially this path-integral as a partition function over an infinite cylin-der with a circumference β having boundary conditions given by φ0 (at τ = 0) and by φ1

(at τ = β).

The reduced density matrix ρAover a subsystem A can be obtained by choosing

peri-odic boundary condition out of A; the complementary system of A will be denoted as B. In other words, the amplitude between two configurations of φA₀ and φA₁

φA 1 tr_B( 1 Ze −βH )φA₀ , (4.5)

is a path integral over the field configurations φ = (φA, φB) such that          φA(τ = 0) = φA₀, φA(τ = β) = φA₁, φB_{(τ = 0) = φ}B_{(τ = β).} (4.6)

It is then possible to calculate the matrix elements of ρn_A(for n ∈ N) by taking n copies of the cylinder and sewing together along the cut in A.

So we can interpret tr(ρn_A) as partition function on a certain surface. If A is made by m segments, we will denote the surface as Rn,m.

The approach given above can be also generalized in any dimension. For the sake of simplicity, we will focus only on 1 + 1 systems taking A as a segment (m = 1) of length ` and we will denote Rn,1as Rn. We mention that even for m = 2, the discussion becomes

extremely technical and only some specific models have been analyzed in the literature (e.g. the free boson [51]).

4.2 Twist fields

A peculiarity of 1+1 systems is that the boundary of a subsystem is given by a discrete set of points (2 points for a single interval).

It reasonable to expect, as we will show later, that certain operations at the boundary can be described by local operators. We will denote as twist fields such operators that are able to sew together the copies of the cylinder along the cut on A and we will describe the

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partition function of Rn as a correlation function of twist fields in R1.

Let us take n copies of the field φ defined on the cylinder and denote them as {φ_j}_j=1,...,n.

An equivalent way to describe the partition function of Rnis to put the following boundary

conditions on the field φj

   φj(x, τ = 0+) = φj+1(x, τ = 0−) x ∈ A, φj(x, τ = 0+) = φj(x, τ = 0−) x ∈ B. (4.7)

We define the twist field Tn(x, τ ) such that

Tn(x, τ ) · φi(x0, τ ) =    φi+1(x0, τ ) x0> x, φi(x0, τ ) otherwise. (4.8)

If A = [u1, v1] ∪ ... ∪ [um, vm], the partition function on Rn,m is proportional to

Z(Rn,m) ∝ hTn(u1)Tn−1(v1)...Tn(um)Tn−1(vm)i. (4.9)

Figure 4.1: A pictorial representation of the Riemann surface R3,1. Different sheets are

connected through the cut in [u, v]. [From [44]].

The technical issues for m ≥ 2 (see [52]) are related to the computation of a 2m-points correlation function, which is not a trivial task; however, for m = 1 the behavior is im-posed by conformal invariance.

Now, we have to prove that the twist field Tn is primary and to find its conformal

weights.

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and rotational invariance on the complex plane C one has

hT (z)i_C= 0. (4.10)

It is possible to map the Riemann surface Rn, with A = [u, v], to the complex plane C

through the following conformal transformation

ξ : Rn→ C, ξ : z 7→ ξ(z) = z − u z − v 1 n . (4.11)

Using the relation (2.29), a straightforward computation leads to

hT (z)iRn = c 24 1 − 1 n (u − v)2 (z − u)2_{(z − v)}2 (4.12)

The previous expectation value can be interpreted in the theory with n-copies of the field as a correlation a function between the stress-energy tensor and the twist fields in the complex plane [44] hT (z)iRn = 1 n hT_n(u)Tn(v)T (z)i hTn(u)Tn(v)i . (4.13)

We put the factor _n1 in order to avoid overcounting: in the n-copies theory, the total stress energy-momentum is T =P

jTj where Tj is the contribution related to φj.

Comparing the correlation function and the Ward identity of a primary operator, we conclude that both Tn and Tn−1 have conformal weights

hn= ¯hn= c 12 n − 1 n . (4.14)

The conformal invariance fixes the functional dependence (up to multiplicative constants) of the two-point function to be

trρn_A∼ hTn(u)Tn(v)i ∼ 1 v − u c 6(n− 1 n) . (4.15)

Plugging the previous result in the definition of R´enyi entropy, one obtains S_A(n)= 1 1 − nlog trρ n A= c 6 1 + 1 n log ` a+ cn. (4.16)

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depends on microscopical details of the model taken in account.

We notice that there is an explicit violation of the area-law, because of the long-range correlations, and that the leading term is universal (it depends only on the representation of the Virasoro algebra, through the value of c).

The equation (4.1) is a remarkable result which admit a straightforward generalization for finite systems, put on a ring on length L. By Wick rotation, the same trick can be used for the entanglement of infinite systems put in a bath at finite temperature β−1. The derivation follows from the existence of conformal mapping (the exponential map) between the infinite cylinder and the complex plane. One obtains

   S_A(n)(L, β = ∞) = c₆(1 + 1_n) log L_πsin π`_L + . . . , S_A(n)(L = ∞, β) = c₆(1 + 1_n) log β πsinh π` β + . . . (4.17)

As one should expect, in the ground state (β = ∞) there is an explicit symmetry ` → L−`. Moreover, for β ` one recover the extensiveness of the (thermal) entropy. The gener-alization at L and β both finite is not easy and not universal: it is impossible to map a cylinder in a torus, due to topological obstructions; the calculation has been performed only for specific models (see [14] for the massless Dirac fermion).

u v

` L

A

B

Figure 4.2: The system is put on a ring of length L and A is the segment [u, v] of length `

4.3 Entanglement spectrum

The knowledge of tr[ρn] is also useful for the reconstruction of the whole spectrum of ρ. If one is able to compute the trace of the resolvent operator

g(z) = tr 1 z − ρ , (4.18)

it is easy to get the distribution of the eigenvalues {λj}j of ρ by taking the imaginary part

of g(z)

Im g(z + i0+) = −πX

j

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On the other hand, g(z) can be formally expanded in a Laurent series g(z) = 1 z ∞ X n=0 tr[ρn]z−n. (4.20)

In many cases we have analyzed in the previous sections, we have seen a functional dependence of the form (ignoring the contribution of the non-universal constants cn)

tr[ρn] = e−b(n−n1), (4.21)

where b is a parameter which depends only on L, β, ` but not on n. Following the reference give in [53] we are able to calculate g(z)

g(z) = 1 z ∞ X k=0 bk k!Lik(e −b z−1), (4.22)

in terms of series involving the polylogarithm function Lik(z)

The sum can be explicitly made and it leads to P (λ) = δ(λ − λM) − log λMθ(λM − λ) λ q − log λ_MlogλM λ I1(2p− log λMlog(λM/λ)), (4.23)

with λM = e−b the maximum eigenvalue and In(z) the modified Bessel function of the

first kind .

The comprehension of of P (λ) is sufficient to calculate any quantity which depends on the (non-zero) spectrum of ρ

tr f (ρ) = Z

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Figure 4.3: s(M ) is the sum of the first M eigenvalues of ρA in the XX-model. The plot

refers to an infinite chain with different values of the length of the subsystem A (called `). The red line corresponds to the CFT predictions obtained from (4.23). [From [53]].

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Chapter 5

Boson-fermion correspondence

The equivalence of fermions and bosons is a peculiarity of 1+1 systems. We exploit this mapping to interpret some CFT-results related to the massless boson in terms of a lattice fermion chain; this is motivated by the possibility to check numerically all the CFT-predictions we will make.

5.1 Entanglement Hamiltonian

Let us consider a lattice free-model described by a hamiltonian H acting on the Hilbert spaceH . We will focus only on free fermion chains which conserve the number of parti-cle (this hypothesis is not fundamental). In this case, the Hamiltonian has the general form

H =X

i,j

Jijc†icj, (5.1)

where {cj, c†j} are fermion creation/annihilation operators and Jij is the hopping matrix.

Let |Ψi be an eigenstate, and let denote as Cij the correlation matrix defined by

Cij ≡ hΨ| c†_icj|Ψi . (5.2)

If i and j belong to the sublattice A, the expectation value can be also computed with the reduced density matrix ρA

Cij = tr ρAc † icj . (5.3)

The Wick property hc†_ic†_jckcli = hc † iclihc † jcki − hc † ickihc †

jcli is valid for the state |Ψi;

obviously this is true also for ρA (if i, j, k, l belong to A). By Wick’s theorem [54], ρA

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ρ = e−HA_,

HA=

X

i,j∈A

tijc†icj. (5.4)

HA is called entanglement Hamiltonian.

By matching the two descriptions in terms of |Ψi and ρA, one obtains a relation

be-tween the matrices tij and Cij

e−t = C

1 − C. (5.5)

The relation (5.5) shows that ρA can be reconstructed once one knows the correlation

matrix Cij. It could not be trivial to diagonalize analytically Cij on the subsystem A;

however, it is a simple numerical task because the dimension of C grows linearly with the dimension of A.

From the set {Ck}k of the eigenvalues of Cij, the R´enyi entropy of A is easily found

   S_A(n)= _1−n1 P klog(Ckn+ (1 − Ck)n), SA= −P_k(Cklog Ck+ (1 − Ck) log(1 − Ck)). (5.6)

5.2 Bosonization

The method of bosonization is a useful technique to map quantum operators of a fermion theory in terms of a bosonic one [48]. There are many interacting fermion models which are equivalent to a bosonic free theory; the details of the interaction are hidden in two parameters, namely the Luttinger parameter, denoted as K, and a velocity scale vF.

One can choose to measure space and time with the same units putting vF = 1, while the

adimensional K parameter is a fundamental quantity which dictates the functional form of the correlation functions.

The key idea is to find a field in the bosonic theory with the same anticommuting algebra of the fundamental fermion field. Namely, the vertex operator plays that role. From this point onward we will focus on free massless theories. The right choice of Lut-tinger parameter is K = 1 while the equivalence takes the following form

   ψ(z) ∼ e−iφ(z), ¯ ψ(¯z) ∼ ei ¯φ(¯z). (5.7)

It is important to stress that, a part from conventional proportionality constant, the only important things in this mapping are the conformal weights: (1₂, 0) for the right sector

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and (0,1₂) for the left one.

Dealing with complex fermion field Ψ (not Majorana), Ψ and Ψ†are different fields. One has to introduce Grassman variables, known as Klein factors, in front of the vertex oper-ator in order to ensure the right commutation relations between Ψ and Ψ†.

The chiral density fields Ψ†_RΨR and Ψ †

LΨL can be easily found by a straightforward

calculation. We choose the proportionality constants such that

   Ψ†_RΨR= −_2π1 ∂xφ, Ψ†_LΨL= −_2π1 ∂xφ.¯ (5.8)

In order to prove the previous relation, we have to regularize the product removing the (infinite) expectation value. Let us explicit the calculation for the right sector

Ψ†_R(x)ΨR(x) ∼ lim →0Ψ † R(x)ΨR(x + ) − hΨ†R(x)ΨR(x + )i = lim →0 1 2π h

: eiφ(x): : e−iφ(x−): −h: eiφ(x): : e−iφ(x−):i i = lim →0 1 2π : eiφ(x)−iφ(x−): 1 i − 1 i = − 1 2π∂xφ. (5.9)

We make use of the identity : eA: : eB:=: ehA+Bi : ehABi, which applies if A, B are boson fields.

One could ask what is the correspondent symmetry U (1)R × U (1)L in the bosonic

theory. By Noether theorem, there is conservation law to the translational symmetry

ϕ → ϕ + or, equivalently    φ → φ + , ¯ φ → ¯φ + . (5.10) The associated current jµ is conserved only on-shell

jµ∝ ∂µϕ = (∂tϕ, −∂xϕ), (5.11)

for instance ∂µjµ= ∂µ∂µϕ = 0.

The translational symmetry is related to the axial symmetry of the fermionic theory    ΨR→ e−iΨR, ΨL→ eiΨL. (5.12) This can be proved by exploiting the bosonization map. It is also physically reasonable, by notice that both symmetries depend mainly on a vanishing mass term m = 0.

On the other hand, the topological current

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is trivially conserved also off-shell. It corresponds to the U (1)R+L vector symmetry of the fermionic theory    ΨR→ e−iΨR, ΨL→ e−iΨL. (5.14) Both the topological current and the vector symmetry are preserved when a mass term is present.

5.3 XX Model

We consider a 1D spin chain described by the following Hamiltonian (see [60]) H = −X j 1 2[σ x jσj+1x + σ y jσ y j+1] + X j hσ_jz. (5.15)

σa_j (a = x, y, z) are the Pauli matrices put on the site j of the lattice and they obey the following commutation relations

[σa_i, σ_jb] = 2iabcσcδij. (5.16)

It is possible to map the previous model in a free fermion chain (see [21, 22]) and the following Jordan-Wigner transformation is needed

         σ_j+≡ σ x j+iσ y j 2 = f † je iπP l<jf † lfl_, σ_j−≡ σ x j−iσ y j 2 = fje −iπP l<jf † lfl_, σ_jz= 2f_j†fj− 1, (5.17)

where {fj, fj†} obey the standard (anti)commutation relation of fermion operators

         {f_i, f_j†} = δ_ij, {f_i†, f_j†} = 0, {fi, fj} = 0. (5.18)

Plugging (5.17) in (5.15) one obtains HXX = − X j [f_j†fj+1+ c.c] + 2h X j f_j†fj+ const.. (5.19)

The translational invariance allows to decouple the theory in momentum space. Let us introduce the Fourier transform of the fields

fk= 1 √ L X j fje−ikj k ∈ 2π LZ and k ∈ [−π, π]. (5.20)

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A simple calculation leads to

H = 2X

k

(h − cos k)f_k†fk. (5.21)

We recognise the one-particle spectrum

(k) = 2(h − cos k). (5.22)

The ground state of such Hamiltonian is described by a Fermi sea of fermions occupying all momentum states k whith (k) < 0 ( the positive energy states remain empty).

The low energy excitations near the Fermi momentum kF (defined by (kF) = 0) shows

a linear spectrum    kF = cos−1h, (±(kF + p)) = (kF) + vF · p + O(p2), (5.23)

where vF ≡ _dkd|k=kF is the group velocity at the Fermi momentum.

By changing h, which can be seen as a chemical potential, it is possible to modify the number of occupied (one particle-)states and the Fermi velocity vF. A part from that, the

physics is almost the same and we will focus mostly on h = 0 and kF = π₂ (half-filled chain).

In the continuum limit, the lattice operator fermion operator f (x) can be decomposed in two chiral Weyl fields which annihilates the groundstate of HXX

f (x) ∼ eikFx_Ψ

R(x) + e−ikFxΨL(x). (5.24)

Taking only the linear part of the spectrum near k = ±kF and focusing only on the

low energy excitations above the ground state, the same theory can be described in terms of

Ψ = ΨR ΨL

!

, (5.25)

subjected to the Dirac massless lagrangian.

The XX model is a lattice counterpart of the free boson, useful for the numerical checks. Indeed, we are able to compute the two-point function Cij = hfi†fji of the

ground-state (and some excited ground-states); through numerical diagonalization of Cij, restricted to a

sublattice A, it is possible to get the R´enyi entropies. In the continuum limit L → ∞, the results can be compared with those of CFT at c = 1.

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Fermi sea (k) k kF −k_F ΨR ΨL −k_F Fermi sea kF k (k)

Figure 5.1: The filling of the Fermi sea in the XX model (left) and in its linearized version (right). The equivalence holds for the low-energy excitations of the system and the separation between the left and right sector is valid in the continuum limit.

5.3.1 Ground state L = ∞

The ground state of the infinite chain (L = ∞) has been studied in great detail (see [60],[61]). We are going to analize the leading part of the entanglement spectrum through the entanglement Hamiltonian approach.

In momentum space the two-points function between lattice fermion operators is de-scribed by a filled Fermi-sea

hf_k†fk0i = θ(k_F − |k|)δ_kk0. (5.26)

Going to the real space, one obtains

Cjj0 ≡ hf† jfj0i = 1 L X k θ(kF − |k|)e−ik(j−j 0₎ → L→∞ Z kF −kF dk 2πe −ik(j−j0₎ = sin kF(j − j 0₎ π(j − j0₎ . (5.27) Let us restrict the matrix Cjj0 on the sublattice A, taking care of the following indeces

j, j0 = 0, ..., ` − 1.

The leading part of the spectrum {tk}kof tij (5.5) has been calculated analytically [62]

tk ' ±

π2k

log ` k = 1, 2, ... (5.28)

The correction to the previous formula is irrilevant for the scaling limit of the R´enyi en-tropy. We stress that in general k is not the momentum, but a label for the eigenstates (if A is the whole system we can choose k as the momentum).

Let nk be the mean number of fermions in the eigeinstate k, say

nk=

1

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One finds that the n-th R´enyi is S(n)(A) ≡ 1 1 − nlog tr[ρ n_(A)] ' 1 1 − n Z ∞ −∞ dk log[nn_k+ (1 − nk)n] = 1 1 − n Z ∞ −∞ dk log[1 + entk_{] − n log[1 + e}tk_] = 2log ` π2 1 n+ 1 [ Z ∞ 0 dk log1 + e−k] = 1 6 1 n+ 1 log `, (5.30)

as one should expect from CFT arguments.

The O(L0) corrections to previous expression have been explicitly calculated by Jin and Korepin [61]

  

S(n)= 1₆(1 +_n1) log(2`| sin kF|) + cn+ o(1),

cn= (1 +_n1) R∞ 0 dt t[ 1

1−n−2(_{n sinh t/n}1 − _{sinh t}1 )_{sinh t}1 −e −2t

6 ].

(5.31)

Figure 5.2: R´enyi entropies (n = 1, 2, 3) of the half-filled infinite XX chain. The continuous lines correspond to the predictions of Jin and Korepin [61]. When ` → ∞, the oscillating corrections to the previous predicition become suppressed.

We mention that going beyond CFT, there is a more accurate analysis made by Cal-abrese and Essler in [60]. They found subleading corrections in O(`−n2) which oscillate as

∼ cos(2k_F`).

5.3.2 Ground state, finite L

Putting the system in a ring of finite length L, the set of allowed momentum becomes a lattice. The correlation function Cjj0 depends both on the parity of L and the boundary

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conditions (periodic/antiperiodic). One expects to recover the infinite-size result in the limit when the distances are small compared to L. Although in this case one must be careful about how the Fermi momentum is defined, the leading part of the spectrum re-mained untouched by such technicalities.

We provide an explicit expression for Cjj0

Cjj0 = 1

L

sin kF(j − j0)

sinπ(j−j_L 0) , (5.32) that in practical applications will be used mostly with kF = π₂ (half-filling).

It is possible to match the CFT results (4.17) with the lattice formulation and one gets S(n)= 1 6 1 + 1 n log L π sin π` L + cn+ o(1). (5.33)

Figure 5.3: R´enyi entropies (n = 1, 2, 3) of the half-filled XX chain at finite L (L = 100). The continuous lines correspond to the prediction of Jin and Korepin [61]. The corrections are suppressed in the limit where x is kept fixed and L → ∞.

We anticipate that one has to analyze the correlation function Cjj0 of the low-energy

excited states (hole excitation, particle-hole excitation...), only a few terms must be added to the formula (5.32) (which holds for the ground-states): they are oscillating terms com-ing from the proximity of the Fermi momentum.

We are going to calculate explicitly Cjj0 when L is even; two different cases (when L

2 is

even or odd) will be take in consideration. The choice of the boundary conditions is such that the chain is periodic in the original formulation in terms of spin operators: going to the fermionic formulation, the non-local phase of the Jordan-Wigner transformation

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modifies the boundary conditions.

Let q = e−2πijL and m − 1 the integer which correspond to the largest occupied

mo-mentum.

If L₂ is even we have to fix antiperiodic boundary condition. The allowed momenta are k ∈ ±2π L 1 2, 3 2, ..., L − 1 2 . (5.34)

The calculation of Cj0 requires the knowledge of the finite geometric series

Cj0= 1 L m−1 X n=0 q12qn+ c.c. = 1 Lq 1 21 − q m 1 − q + c.c. = = 1 L qm2 − q− m 2 q12 − q− 1 2 (qm2 + q− m 2) = 1 L qm− q−m q12 − q− 1 2 = 1 L sin 2π_Lmj sin _Lπj . (5.35)

The largest occupied momentum is k = 2π_L(m − 1₂), while kF = k + 1₂2π_L.

In contrast, if L₂ is odd we have to fix periodic boundary condition and the allowed momenta are k ∈ 2π L 0, ±1, ±2, ...,L − 1 2 . (5.36)

The correlation function is thus

Cj0 = 1 L m−1 X n=−(m−1) qn= 1 L m−1 X n=0 qn+ c.c. − 1 ! = = 1 L qm−12 − q−m+ 1 2 q12 − q− 1 2 = 1 L sin 2π_L(m −1₂)j sin π_Lj . (5.37)

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Chapter 6

Excited states

In this chapter the previous results will be generalized to those excited states that corre-spond to primary fields in CFT.

We will focus on finite-size system of length L, defining x ≡ `

L, (6.1)

the ratio between the length of the segment A (`) and the length of the whole system (L). It was proved (see [58] [59]) that the excess of the R´enyi entropy of such excited states (with respect to the ground-state) shows a universal behavior, which depends only on the ratio x and the conformal property of the associated primary fields.

6.1 CFT approach

6.1.1 Path-integral

Let Υ be a primary field and ρΥ the associated density matrix

ρΥ≡ |Υi hΥ| . (6.2)

The elements of the density matrix ρΥ are given in the path-integral formalism by

inser-tion of Υ at the infinite past (τ = −∞) and ¯Υ at the infinite future (τ = +∞).

The same reasoning applies for ρΥ(A) ≡ trBρΥ, taking care of putting boundary

con-ditions on B. Similarly, the elements of ρn_Υ(A) can be obtained by putting Υ/ ¯Υ for each infinite past/future point z_k∓ which belongs to the k-th sheet of Rn (k = 0, ..., n − 1).

Taking properly into account the normalization constant, one finally obtains the fol-lowing ratio, expressed in terms of correlators in the Riemann surface

F_Υ(n)≡ tr ρ n Υ tr ρn GS = h Q kΥ(z − k) ¯Υ(z + k)iRn hΥ(z₀−) ¯Υ(z+₀)in R1 , (6.3)

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2π

w+₀ w−₀ w₁+ w₁− w+₂ w₂−

Υ Υ Υ

Υ† Υ† Υ†

Figure 6.1: The cylinder that corresponds to R3 with the insertion of the operators Υ,Υ†

in the points {w±_k}

and the non-universal contributions disappear. We explicitly notice that the state-operator correspondence maps the identity operator 1 to the ground-state |0i.

From F_Υ(n) it is easy to get the R´enyi entropy excess S_Υ(n)− S₁(n)= 1

1 − nlog F

(n)

Υ . (6.4)

We anticipate that F(n), being a dimensionless correlation function, depends on L and ` only through the ratio x.

6.1.2 Mapping to cylinder

We want to map Rn, having cut in [u, v], into the cylinder [0, 2π] × iR. Firstly, we define

a transformation from Rn to the complex plane C

ζ(z) ≡ −sin π(z−u) L sinπ(z−v)_L !1_n . (6.5)

Subsequently, we can compose with a transformation from C to the cylinder [0, 2π]×iR.

w(ζ) = −i log ζ. (6.6)

Composing the two transformation w(z) ≡ w(ζ(z)) one finally obtains

F_Υ(n)= n−2n(h+¯h)h Q

kΥk(wk−) ¯Υk(wk+)icyl

hΥ(w₀−) ¯Υ(w₀+)in_cyl , (6.7) where (h, ¯h) are the conformal weights of Υ and ¯Υ.

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   {w_k−= π(1+x)+2πk_n }_{k=0,...,n−1}, {w_k+= π(1−x)+2πk_n }k=0,...,n−1. (6.8) 6.1.3 Boson-fermion correspondence

Let us now consider some excited states of the XX model and let us give them an inter-pretation in the free boson theory [58].

We will focus on a certain class states which are both gaussian both translational invari-ant: they can be obtained by adding or removing particles in the momentum space.

There is a kind of excitations for which there are no holes in momentum space, as for the ground-state, and consecutive momenta differing by 2π_L are occupied. Such excitations are called compact and it turns out that in the thermodynamic limit the entropy is the same of the ground state entropy. The lowest energy sector is obtained by adding/removing a single fermion near the Fermi sea, and the correspondent operator in the Dirac theory are ΨR/L, Ψ

†

R/L. We have shown that such operators correspond to the vertex operators

V_{(α, ¯}_α) for some wieghts (α, ¯α): as we will see, the lack of energy excess can be shown by CFT arguments.

The other kind of excitations is obviously called noncompact. A first example, with the lowest possible energy, is provided by the particle-hole excitation and it corresponds to the destruction of the fermion below the Fermi point and the creation of a fermion immediately above it. There are two Fermi points, which belongs to right/left sectors, and therefore such excitations can be obtained by Ψ†_R/LΨR/L. The associated boson operators are i∂φ

and i ¯∂ ¯φ, and there is a full comprehension of them in a CFT context.

Unfortunately it is not present in literature a full description of the states of the fermion chain in terms of the one of the boson theory. On the other hand, the higher energy states are expected to be described by descendants (non primary) operators. In that case, the formalism we have developed so far must be adapted to treat this kind of operators (see [90] for the details).

6.1.4 Vertex operator

The hermitian conjugate of Vβ is

¯

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1 1

ΨR e−iφ

Ψ†_R eiφ Ψ†_RΨR i∂φ

Figure 6.2: Bosonization dictionary for the low-energy excitations Evaluating the expression (6.7) for Υ = Vβ (with h = Kβ

2

2 and ¯h = 0) one has [58]

F_V(n) β (x) =   n −n sin(πx)n sin(πx/n)n n−1 Y k=1 sin πk_n2(n−k) sinπ(x+k)_n n−ksinπ(−x+k)_n n−k    Kβ2 . (6.10)

Comparing the zeroes in x of the numerator and in the denominator, one finds that they are equal and have the same order. As analytic functions, this implies that F is a con-stant. If we consider the limit x → 1, it is possible to find the constant which turn out to be

F_V(n)

β (x) = 1. (6.11)

The same result applies for the antiholomorphic sector. In conclusion we can say that for any vertex operator

Υ = eiβφ+i ¯β ¯φ, (6.12)

the entropy of the excited state is the same of the one of the ground state, as we expect for compact excitations.

6.1.5 Derivative operator

Let us focus in the primary field i∂φ with wieghts (h, ¯h) = (1, 0). The two-point function in the plane is equal to

h(i∂φ)(z1)(i∂φ)(z2)i =

K (z1− z2)2

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From Wick’s theorem h n Y j=1 (i∂φ)(zj)i = Kndet 1 zi− zj . (6.14)

Going to the cylinder through the substitution zi− zj 7→ 1₂sinzi −zj

2 , we have

F_i∂φ(n)(x) = n−2n(sin πx)2n4ndet " 1 2 sinzi−zj 2 # , (6.15) where zi ∈ {w_k+, w−_k}k.

The determinant has been calculated for any n. It can be rewritten in a more compact form (see [56] for the details)

F_i∂φ(n)(x) = " 2 sin(πx) n n_Γ(1+n+n csc(πx) 2 ) Γ(1−n+n csc(πx)₂ ) #2 . (6.16)

Figure 6.3: R´enyi entropy ratio of the particle-hole excitation in the XX/free fermion model (of finite length N ), n = 2, 3 vs. the CFT prediction equations (continuous lines). nF is the number of fermions. For n = 2, a system with filling fraction n_NF = 1₆ is also

shown. [From [58]].

The limit n → 1 gives the entanglement entropy excess SΥ− SGS = lim n→1 1 1 − nlog F (n)_(x) = −2 log |2 sin(πx)| − 2ψ 1 2 sin(πx) − 2 sin(πx). (6.17)

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6.2 OPE

In the limit x 1 the terms Υ ¯Υ appearing in (6.7) can be approximated by the following OPE (see [59] and [58])

Υ ¯Υ = 1 + Ψ + ..., (6.18)

where Ψ is the operator with the smallest scaling dimension ∆Ψ= hΨ+ ¯hΨ.

One finds that

F_Υ(n)(x) = 1 + hΥ+ ¯hΥ 3 1 n − n (πx)2+ O(x2∆Ψ_). _(6.19)

If ∆Ψ6= 1, the limit x → 1 gives information about ∆Ψ.

The equation (6.19) can be used to find the R´enyi entropy excess

∆S = 2π

2

3 (hΥ+ ¯hΥ)(πx)

2_{+ ...} _(6.20)

As an example, we shall consider the CFT of the free boson with c = 1. For the vertex operator

Vα(z + )V−α(z) = eiα(φ(z)+∂φ(z)+...)e−iαφ(z)= 1 + iα∂φ(z) + ... (6.21)

Unfortunately Ψ ∼ (i∂φ), so ∆Ψ= 1 and the equation (6.19) is not very useful: the term

of order O(x2∆Ψ_{) cancels the other one and one finally obtains F}(n)_{= 1.}

Instead, considering the derivative excitation with weights (h, ¯h) = (1, 0), from (6.19) one obtains the x 1 limit of (6.16)

F_i∂φ(n)= 1 + 1 3(πx) 2 1 n− n + ... (6.22)

6.3 Relative entropy and trace distance

In the context of quantum information, different quantities have been introduced to quan-tify how much two states are different.

An interesting quantity is the so-called relative entropy that for two given density matrices ρ0, ρ1 is defined as [5, 71]

S(ρ1kρ0) ≡ tr(ρ1log ρ1) − tr(ρ1log ρ0). (6.23)

It has different properties that make it similar to a distance, except for the fact that it is asymmetrical.

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entropy in terms of path-integral S(ρ1kρ0) = lim n→1 1 1 − nlog tr(ρ1ρn−10 ) tr(ρn₁) . (6.24)

The same formalism we have developed in this chapter can be used to analyze states induced by primary operators of a CFT (of dimension 1 + 1). If ρ0, ρ1 are induced by

Υ0, Υ1 respectively, it holds tr(ρ1ρn−10 ) tr(ρn₁) = hΥ1(z−₀) ¯Υ1(z₀+)Q_k6=0Υ0(z_k−) ¯Υ0(z_k+)iRnhΥ1(z − 0 ) ¯Υ1(z + 0)i n−1 R1 hQ kΥ1(zk−) ¯Υ1(zk+)iRnhΥ0(z − 0 ) ¯Υ0(z0+)i n−1 R1 . (6.25)

Here are some results related to the bosonic theory [71]          S(ρVαkρVβ) = S(ρVβkρVα) = (α − β) 2_{(1 − πx cot(πx)),}

S(ρi∂φkρGS) = 2(log(2 sin(πx)) + 1 − πx cot(πx) + ψ0(csc(πx)₂ ) + sin(πx)),

S(ρi∂φkρVβ) = S(ρi∂φkρGS) + S(ρGSkρVβ).

(6.26)

Another concept, that is a distance in the mathematical sense of the term, is provided by the trace-distance

D1(ρ1, ρ0) =

1

2kρ1− ρ0k1, (6.27)

where kΛk1 is defined, whenever Λ is an Hermitian operator, as follows

kΛk1≡

X

λ∈SpecΛ

|λ|. (6.28)

It is however extremely difficult to evaluate analytically D1: even for Gaussian states there is no way to compute it from the two-point correlators (and for this reason there are many more results on relative entropies [70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86] ).

In Ref. [69] can be found a technical approach to the 2D CFT, which involves the OPE between twist fields Tn placed at a short distance from each other. The trace distance

between states relative to the free boson has been calculated in the x 1 regime    D1(ρVα, ρVβ) = x|α − β| + o(x), D1(ρi∂φ, ρVα) = |α|x + o(x). (6.29)

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Chapter 7

Symmetry resolution

Let us consider a quantum system which possesses an internal symmetry U (1). We want to split the total entanglement, and in general the spectrum of the reduced density matrix, into the contributions coming from disjoint symmetry sectors.

Let ρ ≡ |0i h0| be a pure state of a composite system A+B. If we want that state lies in a representation of the symmetry (as for example if the total system has a fixed number of particles) it necessary to impose the condition

[ρ, Q] = 0, (7.1)

where Q is the operator which generates the U (1)-symmetry.

Another crucial assumption, that is important when one has to deal with the bipartion A+B, is that the symmetry comes from local and separate degrees of freedom associated to A and B. More techically, we will suppose that the generator is Q = QA+ QB and

[QA, QB] = 0.

Tracing out the relation (7.1) over the subsystem B, we obtain

[ρA, QA] = 0. (7.2)

This means that ρA is block-diagonal with different blocks corresponding to different

eigenspaces of QA. The situation depicted above allows use to interpret the lack of

knowl-edge we have about Q, if we only are able to probe the subsystem A, in a classical way: QA can be seen as a random variable and ρA encodes its probability distribution.

So, let Πq denote the projector to the eigenspace of QAassociated to the eigenvalue q.

The probability of getting QA= q is

p(q) = trA[ρAΠq]. (7.3)

Entanglement and Conformal Field Theory: Distribution in symmetry sectors