Jérôme Dubail

Conformal field theory at central charge c=0: A measure of the indecomposability (b) parameters

Abstract A good understanding of conformal field theory (CFT) at c = 0 is vital to the physics of disordered systems, as well as geometrical problems such as polymers and percolation. Steady progress has shown that these CFTs should be logarithmic, with indecomposable operator product expansions, and indecomposable representations of the Virasoro algebra. In one of the earliest papers on the subject, V. Gurarie introduced a single parameter b to quantify this indecomposability in terms of the logarithmic partner t of the stress–energy tensor T . He and A. Ludwig conjectured further that b = − 5 8 for polymers and b = 5 6 for percolation. While a lot of physics may be hidden behind this parameter — which has also given rise to a lot of discussions — it had remained very elusive up to now, due to the lack of available methods to measure it experimentally or numerically, in contrast say with the central charge. We show in this paper how to overcome the many difficulties in trying to measure b . This requires control of a lattice scalar product, lattice Jordan cells, together with a precise construction of the state L − 2 | 0 〉 . The final result is that b = 5 6 for polymers. For percolation, we find that b = − 5 8 within an XXZ or supersymmetric representation. In the geometrical representation, we do not find a Jordan cell for L 0 at level two (finite-size Hamiltonian and transfer matrices are fully diagonalizable), so there is no b in this case.

Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases

Conformal field theory (CFT) has been extremely successful in describing large-scale universal effects in one-dimensional (1D) systems at quantum critical points. Unfortunately, its applicability in condensed matter physics has been limited to situations in which the bulk is uniform because CFT describes low-energy excitations around some energy scale, taken to be constant throughout the system. However, in many experimental contexts, such as quantum gases in trapping potentials and in several out-of-equilibrium situations, systems are strongly inhomogeneous. We show here that the powerful CFT methods can be extended to deal with such 1D situations, providing a few concrete examples for non-interacting Fermi gases. The systems inhomogeneity enters the field theory action through parameters that vary with position; in particular, the metric itself varies, resulting in a CFT in curved space. This approach allows us to derive exact formulas for entanglement entropies which were not known by other means.

Inhomogeneous quenches in a free fermionic chain: Exact results

We consider the non-equilibrium physics induced by joining together two tight-binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be different. We present a number of exact results, focusing on two-point correlators and the Loschmidt echo (return probability). For the non-interacting case, we identify through an exact derivation the regime in which a semiclassical ansatz is valid. We present a number of analytical results beyond semiclassics, such as the approach to the non-equilibrium steady state and the appearance of Tracy-Widom distributions at the front of the light cone. The light cone behavior is quantified through a series expansion in time, and this description is shown to be valid for interacting systems as well. Results on the Loschmidt echo, presented for finite and zero interactions, illustrate that the physics is different from both local and global quenches.

Conformal two-boundary loop model on the annulus

Abstract We study the two-boundary extension of a loop model—corresponding to the dense phase of the O ( n ) model, or to the Q = n 2 state Potts model—in the critical regime − 2 n ⩽ 2 . This model is defined on an annulus of aspect ratio τ . Loops touching the left, right, or both rims of the annulus are distinguished by arbitrary (real) weights which moreover depend on whether they wrap the periodic direction. Any value of these weights corresponds to a conformally invariant boundary condition. We obtain the exact seven-parameter partition function in the continuum limit, as a function of τ , by a combination of algebraic and field theoretical arguments. As a specific application we derive some new crossing formulae for percolation clusters.

Inhomogeneous field theory inside the arctic circle

Motivated by quantum quenches in spin chains, a one-dimensional toy-model of fermionic particles evolving in imaginary-time from a domain-wall initial state is solved. The main interest of this toy-model is that it exhibits the arctic circle phenomenon, namely a spatial phase separation between a critically fluctuating region and a frozen region. Large-scale correlations inside the critical region are expressed in terms of correlators in a (euclidean) two-dimensional massless Dirac field theory. It is observed that this theory is inhomogenous: the metric is position-dependent, so it is in fact a Dirac field theory in curved space. The technique used to solve the toy-model is then extended to deal with the transfer matrices of other models: dimers on the honeycomb and square lattice, and the six-vertex model at the free fermion point (

Edge state inner products and real-space entanglement spectrum of trial quantum Hall states

\Delta=0

Real-space entanglement spectrum of quantum Hall systems

). In all cases, explicit expressions are given for the long-range correlations in the critical region, as well as for the underlying Dirac action. Although the setup developed here is heavily based on fermionic observables, the results can be translated into the language of height configurations and of the gaussian free field, via bosonization. Correlations close to the phase boundary and the generic appearance of Airy processes in all these models are also briefly revisited in the appendix.

Conformal boundary conditions in the critical O(n) model and dilute loop models

We consider the trial wavefunctions for the Fractional Quantum Hall Effect (FQHE) that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e. short-range correlations only) inside the quantum Hall droplet, and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk 2d chiral CFT which underlies the trial wavefunction. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation. This approach is then used to analyze the entanglement spectrum (ES) of the ground state obtained with a bipartition A\cupB in real-space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization group arguments. We prove that generalized screening implies that the entanglement Hamiltonian H_E = - log {\rho}_A is isospectral to an operator that is local along the cut between A and B. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wavefunctions given by conformal blocks and Tensor Product States, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of px + ipy paired superfluids are treated in a similar fashion in the appendix.

Universal behavior of a bipartite fidelity at quantum criticality

We study the real-space entanglement spectrum for fractional quantum Hall systems, which maintains locality along the spatial cut, and provide evidence that it possesses a scaling property. We also consider the closely-related particle entanglement spectrum, and carry out the Schmidt decomposition of the Laughlin state analytically at large size.

Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids

Abstract We study the conformal boundary conditions of the dilute O ( n ) model in two dimensions. A pair of mutually dual solutions to the boundary Yang–Baxter equations are found. They describe anisotropic special transitions, and can be interpreted in terms of symmetry breaking interactions in the O ( n ) model. We identify the corresponding boundary condition changing operators, Virasoro characters, and conformally invariant partition functions. We compute the entropies of the conformal boundary states, and organize the flows between the various boundary critical points in a consistent phase diagram. The operators responsible for the various flows are identified. Finally, we discuss the relation to open boundary conditions in the O ( n ) model, and present new crossing probabilities for Ising domain walls.