Continuum limit and symmetries of the periodic spin chain

Abstract

This paper is the first in a series devoted to the study of logarithmic conformal field theories (LCFT) in the bulk. Building on earlier work in the boundary case, our general strategy consists in analyzing the algebraic properties of lattice regularizations (quantum spin chains) of these theories. In the boundary case, a crucial step was the identification of the space of states as a bimodule over the Temperley-Lieb (TL) algebra and the quantum group U q sℓ(2). The extension of this analysis in the bulk case involves considerable difficulties, since the U q sℓ(2) symmetry is partly lost, while the TL algebra is replaced by a much richer version (the Jones-Temperley-Lieb -JTL -algebra). Even the simplest case of the gℓ(1|1) spin chain -corresponding to the c = −2 symplectic fermions theory in the continuum limit -presents very rich aspects, which we will discuss in several papers.In this first work, we focus on the symmetries of the spin chain, that is, the centralizer of the JTL algebra in the alternating tensor product of the gℓ(1|1) fundamental representation and its dual. We prove that this centralizer is only a subalgebra of U q sℓ(2) at q = i that we dub U odd q sℓ(2). We then begin the analysis of the continuum limit of the JTL algebra: using general arguments about the regularization of the stress energy-tensor, we identify families of JTL elements going over to the Virasoro generators L n ,L n in the continuum limit. We then discuss the sℓ(2) symmetry of the (continuum limit) symplectic fermions theory from the lattice and JTL point of view.The analysis of the spin chain as a bimodule over U odd q sℓ(2) and JT L N is discussed in the second paper of this series.

Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra

“…For the reader's convenience we summarise here some of the notations used in this paper. They are, as far as possible, the same as those used in [27,28,[37][38][39][40][41][42].…”

“…where the conformal amplitude has been defined in (37). More generally, we can obtain the corresponding results for subdominant contributions from the conformal blocks, corresponding to the amplitude multiplying a term of the type ξ NξN in the square bracket of (35).…”

A fusion for the periodic Temperley-Lieb algebra and its continuum limit

“…We use then the fact that restriction of the quotient (to a subalgebra) is the quotient of restrictions. 7 We then need just to combine properly the terms on the right-hand side using (4.24) and the rules in (5.2) in order to kill the corresponding terms and thus get the quotient in the end. In the following we suppose N even, whence j is integer.…”

Dissipation-driven integrable fermionic systems: from graded Yangians to exact nonequilibrium steady states

“…Physical significance of non-unitary representations in integrable theories is on the other hand far less understood and has not been much explored in the literature, although a few prominent examples are worth mentioning. Most notably, the logarithmic CFTs are based on (non-unitary) reducible indecomposable representations of Virasoro algebra [133,134] and are known to capture various phenomena in statistical physics ranging from critical dense polymers [135], symplectic fermions [136,137], critical percolation [138][139][140] to Gaussian disordered systems [141,142]. It is perhaps instructive to add that non-compact spin chains are also found in the hadron scatting in QCD, which is in the Regge regime governed by the s = 0 non-compact isotropic Heisenberg magnet [143,144].…”