Search

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $${\mathfrak{gl}(1|1)}$$ spin-chain and its continuum limit-the $${c=-2}$$ symplectic fermions theory-and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, ) and Gainutdinov et al. (Nucl Phys B 871:289-329, ). Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL, goes over in the continuum limit to a bigger algebra than $${\boldsymbol{\mathcal{V}}}$$ , the product of the left and right Virasoro algebras. This algebra, $${\mathcal{S}}$$ -which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field $${S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}$$ , with a symmetric form $${S_{\alpha\beta}}$$ and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL in the $${\mathfrak{gl}(1|1)}$$ spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of $${\mathfrak{sp}_{N-2}}$$ . The semi-simple part of JTL is represented by $${U \mathfrak{sp}_{N-2}}$$ , providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL image represented in the spin-chain. On the continuum side, simple modules over $${\mathcal{S}}$$ are identified with 'fundamental' representations of $${\mathfrak{sp}_\infty}$$ . [ABSTRACT FROM AUTHOR]

Entanglement entropy (EE) contains signatures of many universal properties of conformal field theories (CFTs), especially in the presence of boundaries or defects. In particular, topological defects are interesting since they reflect internal symmetries of the CFT and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, no lattice computation of EE has been available. Here, we present an ab initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE depends, as expected, on the geometric arrangement of the subsystem with respect to the defect, we find that zero-energy modes give rise to crucial finite-size corrections. Importantly, contrary to the field-theory predictions, the universal subleading term in the EE when the defect lies at the edge of the subsystem arises entirely due to these zero-energy modes and is not directly related to the modular S matrix of the Ising CFT.

Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to "nonunitary quantum mechanics," which has seen growing interest from areas as diverse as open quantum systems, noninteracting electronic disordered systems, or nonunitary conformal field theory (CFT). We propose and investigate such an extension here, by focusing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of Rényi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well-understood cases. The example of the sl(2|1) alternating spin chain for percolation is discussed in detail.

The possibility of extending the Liouville conformal field theory from values of the central charge c≥25 to c≤1 has been debated for many years in condensed matter physics as well as in string theory. It was only recently proven that such an extension-involving a real spectrum of critical exponents as well as an analytic continuation of the Dorn-Otto-Zamolodchikov-Zamolodchikov formula for three-point couplings-does give rise to a consistent theory. We show in this Letter that this theory can be interpreted in terms of microscopic loop models. We introduce in particular a family of geometrical operators, and, using an efficient algorithm to compute three-point functions from the lattice, we show that their operator algebra corresponds exactly to that of vertex operators V_{α[over ^]} in c≤1 Liouville theory. We interpret geometrically the limit α[over ^]→0 of V_{α[over ^]} and explain why it is not the identity operator (despite having conformal weight Δ=0).

It is well known that the ground states of a Fermi liquid with and without a single Kondo impurity have an overlap that decays as a power law of the system size, expressing the Anderson orthogonality catastrophe. Ground states with two different values of the Kondo couplings have, however, a finite overlap in the thermodynamic limit. This overlap, which plays an important role in quantum quenches for impurity systems, is a universal function of the ratio of the corresponding Kondo temperatures, which is not accessible using perturbation theory or the Bethe ansatz. Using a strategy based on the integrable structure of the corresponding quantum field theory, we propose an exact formula for this overlap, which we check against extensive density matrix renormalization group calculations.