The scaling relation of the quantum group entanglement entropy

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Irrationality and non-compactness

The systematic characterisation of many CFTs is often made possible thanks to the ratio-nality of the theories [60, Chapter 3]. A 2D conformal field theory is rational if it possesses a finite number of primary fields of some extended algebra thus simplifying drastically the analysis. The most famous examples of rational conformal field theories are the minimal models [1]. Conversely, irrational conformal field theories have an infinite number of primary fields and, because of their complexity, many aspects are not perfectly understood. On the lattice, many geometrical models are discrete regularisations of non-rational CFTs and are used to study them. For instance, most well-known logarithmic conformal field theories are non-rational (see [61] for an exception). Some irrational CFTs are called quasi-rational [62] if they are described by an infinite number of fusion rules but any fusion of two representa-tions decomposes on a finite sum of representations. Particular extreme cases of irrational models are the non-compact CFTs (see below) for which the set of primary fields is not even countable.
Many interesting systems have an infinite number of degrees of freedom and are called non-compact. A very simple example is the Brownian motion where each edge can be visited an arbitrary number of times. It is also the case for the supersymmetric formulation of the Chalker-Coddington model describing the plateau transition in the IQHE. In the continuum, a theory is said to be non-compact if described by a field living in a non-compact space. As an example, the Brownian motion is described [63], by a bosonic field with free Euclidean action Z S = d2x (r )2 : (1.21)
The field lives on the real axis, which is non-compact. In practice, these theories have a continuum of critical exponents, whereas in usual CFT, the set of conformal dimensions is discrete. The consequence on the lattice are very important. Given a lattice observable O, it can usually be written as a sum over primary operators and their descendants. In the case of a continuum of fields this sum becomes naturally an integral. As a consequence, a two-point function has the form Z 1 hO(0)O(r)ilattice = dx (x)r 2 0+x2 (1.22) x=0 where plays the role of a non-universal density and 0 is the smallest conformal dimension appearing in the decomposition of O on conformal fields. This is very different to what is observed for compact theories where the largest contribution dominates all the subleading terms. Here, for large distances r, the correlation functions have logarithmic corrections hO(0)O(r)ilattice r 2 0 (log r) (1.23).
where depends on the precise behaviour of in the vicinity of x = 0. Let us emphasize that the logarithmic part in the correlation function has a quite different origin than the one encountered in logarithmic conformal field theory. It comes from the lattice discretisa-tion whereas, for LCFTs, the logarithm correlation functions are intrinsic properties of the continuum.

Entanglement in non-unitary critical systems

Ideas coming from both quantum information theory and field theory have profoundly af-fected our understanding of quantum systems at criticality. Let us consider a partition of a quantum system into two parts A and B. The quantity called entanglement entropy SA (or equivalently SB) is a measurement of the entanglement between A and B. It has many physical implications and, in particular, important consequences in numerical simulations. Methods such as DMRG or more generally tensor networks [80,81] were developed thanks to the improving understanding of entanglement. They are now applied to strongly correlated systems with great success. For non-critical (gapped) systems, the entanglement entropy satisfies the so called area law [82–85] SA k Area(@A) (2.1).
where k is non-universal. In other words, the entanglement entropy grows as the size of the boundary between A and B. Indeed, in a gapped system, correlation functions decay exponentially. As a consequence, on a lattice, a site in A and a site in B are entangled only if they are close to each other. Globally, the entanglement between A and B is located near their boundary and thus SA grows with the size of @A. Of course, in the case of critical (gapless) systems, this simple argument breaks down. Critical systems have long-range correlation functions that decay only algebraically. Therefore, the area law is not satisfied anymore. It was found [86, 87], in the case of 1 + 1D critical systems where A in an interval of length L in a infinite system, that SA c L ; L a (2.2) log 3 a.
where a is a lattice cutoff and c the central charge of the associated CFT. This scaling relation involves a universal quantity and opens many possible connections between quantum information and conformal field theory [88, 89]. It is a very natural question to ask whether this result holds for non-unitary systems. Different approaches [90] have been investigated in order to derive (2.2). In several cases [91, 92], it is expected for non-unitary systems to have a modified scaling relation of the form SA ceff log L (2.3). where ceff is the effective central charge.

Conformal field theory interpretation

Let us first give a brief presentation of some connections between the entanglement entropy and conformal field theory, following the ideas of Calabrese and Cardy [94]. In order to provide a connection between the entanglement entropy and conformal field theory, let us consider a 1 + 1D quantum system at finite inverse temperature = 1=T described by a Hamiltonian H. Its density matrix reads = 1 e H (2.10).
with Z = Tr e H the partition function, appearing here to ensure the right normalisation Tr = 1. Given two states j i, j 0i, the matrix element ( ; 0) = h j j 0i is the overlap between and 0 after a propagation, in imaginary time evolution, at a time = . It can be represented by the picture 2.1a.

The non-unitary case: first observations

The naive extension to non-unitary case is now discussed. Before considering a specific model, we discuss the apparent issues with the scaling (2.22) and review some results found in the literature. First, as it was hinted earlier, the definition of the density matrix must distinguish between right and left eigenvectors. Indeed, the field theory interpretation holds only if it is possible to write the density matrix as the zero temperature limit of the evolution operator in imaginary time. This definition may seem curious from the point of view of pure quantum information. Indeed the von Neumann entropy measures the entanglement within a given quantum state and it is, a priori, acceptable to study a naive entropy where = j0Rih0Rj.
A second apparent difference comes from the prefactor of the scaling law (2.22). In a unitary CFT, a non trivial theory has a strictly positive central charge hence equation (2.22) is in a perfect agreement with the fact that the von Neumann entropy is a positive quantity. However in a non-unitary system, the central charge can be zero or negative. The simplest cases of such systems are the non-unitary minimal models. A famous member of this class of model is the Yang-Lee model [96, 97] with c = 22=5. The minimal models M(p; p0) are a series of conformal field theories with a finite number of primary fields with integer p and p0 coprime such that 2 p < p0. The central charge is given by c = 1 6(p p0)2 (2.23) pp0.

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Quantum group entanglement entropy

This first section presents our approach to the entanglement entropy in the XXZ model. A new quantity, called quantum group entanglement entropy is introduced. This choice is first motivated by a simple case on two sites for the vertex model. The same calculations are performed in the loop model. In particular, it is shown that the entanglement entropy has a straightforward interpretation with loop connectivities. General definitions are then given and motivated by the correspondance between the two representations. Then a more complex example on four sites is detailed. In the end of the section, a few properties of this modified entanglement entropy are given. First we show that the definition respects the Uqsl(2) symmetry of the model and discuss the several required properties of an entropy.

The scaling relation of the quantum group entanglement en-tropy

This section presents the derivation of the scaling relation for the quantum group entangle-ment entropy. We start with a brief reminder on Coulomb gas and the computation of the scaling law of the entanglement is performed in this formalism. The quantum group entan-glement entropy is shown to behave as expected in unitary conformal field theory with the true central charge. Numerical analysis using DMRG is given at the end of this section.

Table of contents :

1 Introduction to non-unitary critical phenomena 
1.1 Universality and CFT
1.2 The quantum Hall effect
1.3 Geometric systems and polymers
1.4 Non-unitary features
1.4.1 General considerations
1.4.2 Non-unitary representations of the Viraso algebra and negative conformal dimensions
1.4.3 Indecomposability and logarithmic correlators
1.4.4 Irrationality and non-compactness
1.4.5 PT symmetry and RG-flow
1.5 The plan of this manuscript
2 Entanglement in non-unitary critical systems 
2.1 Entanglement entropy
2.1.1 Definitions
2.1.2 Conformal field theory interpretation
2.1.3 The non-unitary case: first observations
2.2 The XXZ spin chain
2.2.1 Potts model
2.2.2 Loop model formulation
2.2.3 The six-vertex model and the XXZ Hamiltonian.
2.2.4 Quantum group
2.3 Quantum group entanglement entropy
2.3.1 Pedagogical example on 2 sites
2.3.2 Entanglement in the loop model and Markov Trace
2.3.3 Definition of the quantum group entanglement entropy and motivations
2.3.4 A more complex example: 2M = 4 sites
2.3.5 Properties of the entropy
2.4 The scaling relation of the quantum group entanglement entropy
2.4.1 A brief introduction to Coulomb Gas
2.4.2 The replica trick and the modified scaling relation
2.4.3 Numerical analysis
2.5 Extensions
2.5.1 Restricted Solid-on-Solid models
2.5.2 A supersymmetric example
2.5.3 Entanglement entropy in the non-compact case
2.6 Comparisons and conclusion
2.6.1 Entanglement in non-unitary minimal models
2.6.2 The null-vector conditions in the cyclic orbifold
3 Truncations of non-compact loop models 56
3.1 The Chalker-Coddington model
3.1.1 Definition as a one-particle model
3.1.2 Supersymmetric formulation
3.1.3 The supersymmetric gl(2j2) spin chain
3.1.4 Exact results and critical exponents
3.2 The first truncation as a loop model
3.2.1 Truncations as a loop model: the case M = 1
3.2.2 An integrable deformation
3.2.3 Symmetries
3.2.4 Comparison
3.2.5 A word on the dense phase
3.2.6 Lattice observables in the network model
3.3 Higher truncations
3.3.1 The second truncation
3.3.2 Generalisation
3.3.3 Preliminary numerical results
3.4 Truncations of the Brownian motion
3.4.1 Brownian motion as a supersymmetric spin chain
3.4.2 Equivalence between oriented/unoriented lattice
3.4.3 The first truncation: self-avoiding walks
3.4.4 Hamiltonian limit
3.4.5 Symmetries in the continuum limit
3.4.6 Higher truncation of the Brownian motion
3.4.7 The multicritical point of the second truncation
4 A flow between class A and class C 
4.1 Lattice model interpolating between class A and class C
4.1.1 The Spin Quantum Hall Effect as a network model
4.1.2 Second quantisation and the Hamiltonian limit
4.1.3 Choosing an interpolation
4.1.4 Loop formulation of the model
4.1.5 Percolation as a two-colours loop model
4.2 The untruncated model
4.2.1 Symmetries
4.2.2 Lyapunov exponents
4.3 Truncations
4.3.1 The phase diagram
4.3.2 Symmetries
4.3.3 The dense phase
4.3.4 Critical exponents of the critical dilute phase
5 Operators in the Potts model 
5.1 Observables in the Q-state Potts model
5.1.1 Potts model and Fortuin-Kasteleyn clusters
5.1.2 Definitions and representation theory of SQ
5.1.3 Observables of one spin
5.1.4 Observables of two spins
5.1.5 Procedure for general representations
5.1.6 Internal structure and LCFT
5.2 Correlation functions
5.2.1 Symmetric observables of two spins
5.2.2 Anti-symmetric observables of two spins
5.2.3 Observables with mixed symmetry: [Q 􀀀 3; 2; 1]
5.2.4 Generic case
5.3 Physical interpretation
5.3.1 Primal and secondary operators
5.3.2 Critical exponents on a cylinder
5.3.3 Numerics
5.3.4 Spin
5.4 Logarithmic correlations in 3D percolation
Conclusion 

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