Non-unitary representations of the Viraso algebra and negative conformal dimensions

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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 pp0.

The XXZ spin chain

The Potts model is introduced in this section. Its mappings to a loop model and a vertex model are discussed. We show explicitly the connection between both and derive the quantum Hamiltonian associated. The XXZ Hamiltonian is recovered with Hermiticity breaking boundary terms. It is symmetric under the action of the quantum group Uqsl(2), a deformation of sl(2).

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.

Pedagogical example on 2 sites

Let us start the discussion with the simple example of 2M = 2 spin, for pedagogical purposes. The XXZ Hamiltonian is H = 􀀀e1, with e1 the unique TL generator given equation (2.41). The parameter q = ei is chosen such that 2 [0; =2[, leading to a positive loop fugacity n = 2 cos . In the sector of zero magnetisation, there are 2 distinct eigenenergies E0 = 􀀀(q + q􀀀1) = 􀀀n and E1 = 0. The right ground state, defined as Hj0Ri = E0j0Ri reads j0Ri = 1 p   (q􀀀1=2j « #i 􀀀 q1=2j # »i).

Definition of the quantum group entanglement entropy and motivations

The general definition for the modified entanglement entropy in the XXZ spin chain is now discussed. For a chain, with open boundary conditions and L sites, let us consider a subsystem A made of M neighbour spins. Its complement B is made of two parts, BL on the left and BR on theright, so that B = BL [BR and H = HBL HA HBR. The right and left ground states are distinguished so the density matrix reads ~ = j0Rih0Lj; (2.71).
with j0Ri and h0Lj the right and left ground states. The modified reduced density ~A matrix is defined as ~A = TrB q2SzB L 􀀀2SzB R ~ (2.72) where SzB  = 1 2 P i2BL zi and SzB R = 1 2 P i2BR z i are the magnetisations in the respective subsystems. The Quantum Group Entanglement Entropy, or QG EE, is defined by ~ SA = 􀀀Tr q2SzA ~A log ~A: (2.73).
The phases ensure that all loops, in the mapping to the geometrical representation, have the right fugacity. On a N-sheeted Riemann surface, a loop can propagate on the successive replicas and close on itself without being contractible. This is also the case on a cylinder. The figure 2.5a shows an example of configuration.

The scaling relation of the quantum group entanglement entropy

This section presents the derivation of the scaling relation for the quantum group entanglement 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 entanglement 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.

A brief introduction to Coulomb Gas

The Coulomb gas approach is useful to describe the continuum limit of two-dimensional loop models [28, 31, 105]. In particular, it is particularly powerful to describe systems such as the Potts model (like in this chapter) or the O(n) model. Suppose the system is on a cylinder, the Coulomb gas method describes a soup of oriented loops, that are level lines of a compactified height field. In the continuum, it is described by a field (x) with the action S = g 4 Z d2x (r)2 : (2.98) which is the Euclidean action of a free compactified boson (x)+2 = (x). The parameter g controls the rigidity of the height surface and is related to the loop fugacity by n = 􀀀2 cos g.

The replica trick and the modified scaling relation

We now claim that for the critical quantum group invariant XXZ chain with Hamiltonian H = 􀀀 P ei, the Rényi and Von-Neumann entropies scale as expected in a conformal field theory, with the true central charge. The simplest argument for this relies on a field theoretic analysis. We follow the Cardy and Calabrese [94] replica calculation extended to the non unitary case, the density operator is ~ = j0Rih0Lj.
With N replicas in the Coulomb Gas formalism, there are N bosonic fields 1; : : : N. An essential complication arises because of the cut: the loops winding N times around one of the extremities should still have weight n, while, due to the collection of phases gathered in the winding, the complex Boltzmann weights conspire to give them the weight ~n = 2 cosN .
This issue was discussed earlier and a picture was given figure 2.5b. The problem of the noncontractible loops fugacity can be repaired by the introduction of electric charges at the two extremities of the cut. In practice, in the field theory, vertex operators exp[iel;r(1+: : :+N)] are inserted at the left and right extremities of the cut. The charges el;r are respectively the charges of the vertex operators inserted at the left and right of the cuts and are tuned onveniently to compensate the extra phases. An oriented loop surrounding both extremities first gathers a weight eie0 due to the complex turning weights with a sign depending on the orientation. The two vertex operators provide the additional weight ei(el+er). The total complex weight must satisfy ei(e0+el+er) = eie0 or ei(e0+el+er) = e􀀀ie0.

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 
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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