arXiv subjects

Romain Couvreur

Publications and source records attributed to Romain Couvreur.

6 recordsLinked to original sources

arXiv2022

What is the group of a quantum group? From $SL_q(2)$ to MPO representations of $SL(2)$

Generalized symmetries realized by matrix product operators (MPOs) have so far been tied to discrete data leaving continuous symmetries outside the framework. Quantum groups fill this gap: starting from the RTT relations of $SL_q(2)$ at an odd root of unity $q^d=1$, we construct a three-parameter family of periodic-boundary MPOs of bond dimension $d^2$ that multiply according to the group law of $SL(2)$. This gives Lusztig's quantum Frobenius map a concrete operator form, and endows the XXZ chain on the ring at $\Delta=(q+q^{-1})/2$ with an intrinsically non-local $SU(2)$ symmetry. Fusion of the local tensors is generically semisimple. On special loci it is non-semisimple, and yields reducible but indecomposable tensors with Jordan blocks. Differentiating $\mathcal{D}(g)$ at the identity produces the $\mathfrak{sl}(2)$ generators as $d$-body operators: the MPO algebra supplies the exponential map for quantum group algebras.

cond-mat.stat-mech↗

arXiv2018

On truncations of the Chalker-Coddington model

The supersymmetric reformulation of physical observables in the Chalker-Coddington model (CC) for the plateau transition in the integer quantum Hall effect leads to a reformulation of its critical properties in terms of a 2D non-compact loop model or a 1D non-compact $gl(2|2)$ spin chain. Following a proposal by Ikhlef, Fendley and Cardy, we define and study a series of truncations of these loop models and spin chains, involving a finite and growing number of degrees of freedom per site. The case of the first truncation is solved analytically using the Bethe-ansatz. It is shown to exhibit many of the qualitative features expected for the untruncated theory, including a quadratic spectrum of exponents with a continuous component, and a normalizable ground state below that continuum. Quantitative properties are however at odds with the results of simulations on the CC model. Higher truncations are studied only numerically. While their properties are found to get closer to those of the CC model, it is not clear whether this is a genuine effect, or the result of strong finite-size corrections.

cond-mat.stat-mech↗

arXiv2018

Observation of non-scalar and logarithmic correlations in 2D and 3D percolation

Percolation, a paradigmatic geometric system in various branches of physical sciences, is known to possess logarithmic factors in its correlators. Starting from its definition, as the $Q\rightarrow1$ limit of the $Q$-state Potts model with $S_Q$ symmetry, in terms of geometrical clusters, its operator content as $N$-cluster observables has been classified. We extensively simulate critical bond percolation in two and three dimensions and determine with high precision the $N$-cluster exponents and non-scalar features up to $N \! =\! 4$ (2D) and $N \! =\! 3$ (3D). The results are in excellent agreement with the predicted exact values in 2D, while such families of critical exponents have not been reported in 3D, to our knowledge. Finally, we demonstrate the validity of predictions about the logarithmic structure between the energy and two-cluster operators in 3D.

cond-mat.stat-mech↗

arXiv2017

Non-scalar operators for the Potts model in arbitrary dimension

We investigate the operator content of the Q-state Potts model in arbitrary dimension, using the representation theory of the symmetric group. In particular we construct all possible tensors acting on N spins, corresponding to given symmetries under $S_Q$ and $S_N$, in terms of representations involving any Young diagram. These operators transform non-trivially under the group of spatial rotations, with a definite conformal spin. The two-point correlation functions are then computed, and their physical interpretation is given in terms of Fortuin-Kasteleyn clusters propagating between two neighbourhoods of each N spins. In two dimensions, we obtain analytically the critical exponent corresponding to each operator. In the simplest and physically most relevant cases, we confirm the values of the critical exponent and the conformal spin by numerical measurements, using both Monte Carlo simulations and transfer matrix diagonalisations. Our classification partially provides the structure of Jordan cells of the dilatation operator in arbitrary dimensions, which in turn gives rise to logarithmic correlation functions.

cond-mat.stat-mech↗

arXiv2016

Entanglement in non-unitary quantum critical spin chains

Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to non-unitary quantum mechanics, which has seen growing interest from areas as diverse as open quantum systems, non-interacting electronic disordered systems, or non-unitary conformal field theory (CFT). We propose and investigate such an extension here, by focussing 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.

cond-mat.stat-mech↗

arXiv2014

Kac boundary conditions of the logarithmic minimal models

We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models ${\cal LM}(p,p')$ with $1\le p<p'$ and $p,p'$ coprime. Working in a strip geometry, we consider the $(r,s)$ boundary conditions, which are organized into infinitely extended Kac tables labeled by $r,s=1,2,3,...$. They are conjugate to Virasoro Kac representations with conformal dimensions $Δ_{r,s}$ given by the usual Kac formula. On a finite strip of width $N$, built from a square lattice, the associated integrable boundary conditions are constructed by acting on the vacuum $(1,1)$ boundary with an $s$-type seam of width $s-1$ columns and an $r$-type seam of width $ρ-1$ columns. The $r$-type seam contains an arbitrary boundary field $ξ$. The usual fusion construction of the $r$-type seam relies on the existence of Wenzl-Jones projectors restricting its application to $r\leρ<p'$. This limitation was recently removed by Pearce, Rasmussen and Villani who further conjectured that the conformal boundary conditions labeled by $r$ are realized, in particular, for $ρ=ρ(r)=\lfloor \frac{rp'}{p}\rfloor$. In this paper, we confirm this conjecture by performing extensive numerics on the commuting double row transfer matrices and their associated quantum Hamiltonian chains. Letting $[x]$ denote the fractional part, we fix the boundary field to the specialized values $ξ=\fracπ{2}$ if $[\fracρ{p'}]=0$ and $ξ=[\frac{ρp}{p'}]\fracπ{2}$ otherwise. For these boundary conditions, we obtain the Kac conformal weights $Δ_{r,s}$ by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes $N\le 32-ρ-s$. Additionally, by solving local inversion relations, we obtain general analytic expressions for the boundary free energies allowing for more accurate estimates of the conformal data.

hep-th↗