54 research outputs found
Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
For generic values of q, all the eigenvectors of the transfer matrix of the
U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be
constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin.
However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe
equations acquire continuous solutions, and the transfer matrix develops Jordan
cells. Hence, there appear eigenvectors of two new types: eigenvectors
corresponding to continuous solutions (exact complete p-strings), and
generalized eigenvectors. We propose general ABA constructions for these two
new types of eigenvectors. We present many explicit examples, and we construct
complete sets of (generalized) eigenvectors for various values of p and N.Comment: 50pp, 2 figures, v2: few typos are fixed, Nucl. Phys. B (2016
The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre
We introduce a family of factorisable ribbon quasi-Hopf algebras for
a positive integer: as an algebra, is the semidirect product of
with the direct sum of a Grassmann and a Clifford
algebra in generators. We show that is ribbon equivalent to the
symplectic fermion category that was computed by the third author from
conformal blocks of the corresponding logarithmic conformal field theory. The
latter category in turn is conjecturally ribbon equivalent to representations
of , the even part of the symplectic fermion vertex operator super
algebra.
Using the formalism developed in our previous paper we compute the projective
-action on the centre of as obtained from
Lyubashenko's general theory of mapping class group actions for factorisable
finite ribbon categories. This allows us to test a conjectural non-semisimple
version of the modular Verlinde formula: we verify that the
-action computed from agrees projectively with that on
pseudo trace functions of .Comment: 75pp; typos fixed, references update
Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of
finite length N. For the case that q is a root of unity, we propose a formula
for the number of admissible solutions of the Bethe ansatz equations in terms
of dimensions of irreducible representations of the Temperley-Lieb algebra; and
a formula for the degeneracies of the transfer matrix eigenvalues in terms of
dimensions of tilting sl(2)_q-modules. These formulas include corrections that
appear if two or more tilting modules are spectrum-degenerate. For the XX case
(q=exp(i pi/2)), we give explicit formulas for the number of admissible
solutions and degeneracies. We also consider the cases of generic q and the
isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8
are presented. Our results are consistent with the Bethe ansatz solution being
complete.Comment: 34 pages; v2: reference added; v3: two more references added and
minor correction
Integrability of rank-two web models
We continue our work on lattice models of webs, which generalise the
well-known loop models to allow for various kinds of bifurcations
[arXiv:2101.00282, arXiv:2107.10106]. Here we define new web models
corresponding to each of the rank-two spiders considered by Kuperberg
[arXiv:q-alg/9712003]. These models are based on the , and Lie
algebras, and their local vertex configurations are intertwiners of the
corresponding -deformed quantum algebras. In all three cases we define a
corresponding model on the hexagonal lattice, and in the case of also on
the square lattice. For specific root-of-unity choices of , we show the
equivalence to a number of three- and four-state spin models on the dual
lattice.
The main result of this paper is to exhibit integrable manifolds in the
parameter spaces of each web model. For on the unit circle, these models
are critical and we characterise the corresponding conformal field theories via
numerical diagonalisation of the transfer matrix.
In the case we find two integrable regimes. The first one contains a
dense and a dilute phase, for which we have analytic control via a Coulomb gas
construction, while the second one is more elusive and likely conceals
non-compact physics. Three particular points correspond to a three-state spin
model with plaquette interactions, of which the one in the second regime
appears to present a new universality class. In the case we identify four
regimes numerically. The case is too unwieldy to be studied numerically
in the general case, but it found analytically to contain a simpler sub-model
based on generators of the dilute Birman-Murakami-Wenzl algebra.Comment: 69 page
Dynamics of the Hypoxia—Induced Tissue Edema in the Rat Barrel Cortex in vitro
Cerebral edema is a major, life threatening complication of ischemic brain damage. Previous studies using brain slices have revealed that cellular swelling and a concomitant increase in tissue transparency starts within minutes of the onset of metabolic insult in association with collective anoxic spreading depolarization (aSD). However, the dynamics of tissue swelling in brain slices under ischemia-like conditions remain elusive. Here, we explored the dynamics of brain tissue swelling induced by oxygen-glucose deprivation (OGD) in submerged rat barrel cortex slices. Video monitoring of the vertical and horizontal position of fluorescent dye-filled neurons and contrast slice surface imaging revealed elevation of the slice surface and a horizontal displacement of the cortical tissue during OGD. The OGD-induced tissue movement was also associated with an expansion of the slice borders. Tissue swelling started several minutes after aSD and continued during reperfusion with normal solution. Thirty minutes after aSD, slice borders had expanded by ~130 μm and the slice surface had moved up to attain a height of ~70 μm above control levels, which corresponded to a volume increase of ~30%. Hyperosmotic sucrose solution partially reduced the OGD-induced slice swelling. Thus, OGD-induced cortical slice tissue swelling in brain slices in vitro recapitulates many features of ischemic cerebral edema in vivo, its onset is tightly linked to aSD and it develops at a relatively slow pace after aSD. We propose that this model of cerebral edema in vitro could be useful for the exploration of the pathophysiological mechanisms underlying ischemic cerebral edema and in the search for an efficient treatment to this devastating condition
The puzzle of bulk conformal field theories at central charge c=0
Non-trivial critical models in 2D with central charge c=0 are described by
Logarithmic Conformal Field Theories (LCFTs), and exhibit in particular mixing
of the stress-energy tensor with a "logarithmic" partner under a conformal
transformation. This mixing is quantified by a parameter (usually denoted b),
introduced in [V. Gurarie, Nucl. Phys. B 546, 765 (1999)], and which was first
thought to play the role of an "effective" central charge. The value of b has
been determined over the last few years for the boundary versions of these
models: for percolation and for
dilute polymers. Meanwhile, the existence and value of for the bulk theory
has remained an open problem. Using lattice regularization techniques we
provide here an "experimental study" of this question. We show that, while the
chiral stress tensor has indeed a single logarithmic partner in the chiral
sector of the theory, the value of b is not the expected one: instead, b=-5 for
both theories. We suggest a theoretical explanation of this result using
operator product expansions and Coulomb gas arguments, and discuss the physical
consequences on correlation functions. Our results imply that the relation
between bulk LCFTs of physical interest and their boundary counterparts is
considerably more involved than in the non-logarithmic case.Comment: 5 pages, published versio
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