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261
result(s) for
"Jacobsen, Jesper"
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Torus partition function of the six-vertex model from algebraic geometry
A
bstract
We develop an efficient method to compute the torus partition function of the six-vertex model exactly for finite lattice size. The method is based on the algebro-geometric approach to the resolution of Bethe ansatz equations initiated in a previous work, and on further ingredients introduced in the present paper. The latter include rational
Q
-system, primary decomposition, algebraic extension and Galois theory. Using this approach, we probe new structures in the solution space of the Bethe ansatz equations which enable us to boost the efficiency of the computation. As an application, we study the zeros of the partition function in a partial thermodynamic limit of
M
×
N
tori with
N
≫
M
. We observe that for
N
→ ∞ the zeros accumulate on some curves and give a numerical method to generate the curves of accumulation points.
Journal Article
Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra
A
bstract
We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional
Q
-state Potts model conformal field theory. In a recent work [
1
], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on infinite cylinders. We find that the spectra in the scaling limit are much richer than those proposed in [
1
]: they involve in particular fields with conformal weight
h
r,s
where
r
is dense on the real axis.
Journal Article
Non-invertible symmetries and RG flows in the two-dimensional O(n) loop model
A
bstract
In a recent paper, Gorbenko and Zan [
1
] observed that
O
(
n
) symmetry alone does not protect the well-known renormalization group flow from the dilute to the dense phase of the two-dimensional
O
(
n
) model under thermal perturbations. We show in this paper that the required “extra protection” is topological in nature, and is related to the existence of certain non-invertible topological defect lines. We define these defect lines and discuss the ensuing topological protection, both in the context of the
O
(
n
) lattice model and in its recently understood continuum limit, which takes the form of a conformal field theory governed by an interchiral algebra.
Journal Article
Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap
A
bstract
Based on the spectrum identified in our earlier work [
1
], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the
Q
-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight
h
r,
1
, with
r
∈ ℕ
*
, and are related to the underlying presence of the “interchiral algebra” introduced in [
2
]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field
Φ
12
D
in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.
Journal Article
The two upper critical dimensions of the Ising and Potts models
A
bstract
We derive the exact actions of the
Q
-state Potts model valid on any graph, first for the spin degrees of freedom, and second for the Fortuin-Kasteleyn clusters. In both cases the field is a traceless
Q
-component scalar field Φ
α
. For the Ising model (
Q
= 2), the field theory for the spins has upper critical dimension
= 4, whereas for the clusters it has
= 6. As a consequence, the probability for three points to be in the same cluster is not given by mean-field theory for
d
within 4
< d <
6. We estimate the associated universal structure constant as
. This shows that some observables in the Ising model have an upper critical dimension of 4, while others have an upper critical dimension of 6. Combining perturbative results from the
ϵ
= 6 –
d
expansion with a non-perturbative treatment close to dimension
d
= 4 allows us to locate the shape of the critical domain of the Potts model in the whole (
Q
,
d
) plane.
Journal Article
Emerging Jordan blocks in the two-dimensional Potts and loop models at generic Q
A
bstract
It was recently suggested — based on general self-consistency arguments as well as results from the bootstrap [
1
,
2
–
3
] — that the CFT describing the
Q
-state Potts model is logarithmic for generic values of
Q
, with rank-two Jordan blocks for
L
0
and
L
¯
0
in many sectors of the theory. This is despite the well-known fact that the lattice transfer matrix (or Hamiltonian) is diagonalizable in (arbitrary) finite size. While the emergence of Jordan blocks only in the limit
L
→ ∞ is perfectly possible conceptually, diagonalizability in finite size makes the measurement of logarithmic couplings (whose values are analytically predicted in [
2
,
3
]) very challenging. This problem is solved in the present paper (which can be considered a companion to [
2
]), and the conjectured logarithmic structure of the CFT confirmed in detail by the study of the lattice model and associated “emerging Jordan blocks.”
Journal Article
On zero-remainder conditions in the Bethe ansatz
A
bstract
We prove that physical solutions to the Heisenberg spin chain Bethe ansatz equations are exactly obtained by imposing two zero-remainder conditions. This bridges the gap between different criteria, yielding an alternative proof of a recently devised algorithm based on
QQ
relations, and solving its minimality issue.
Journal Article
Geometric algebra and algebraic geometry of loop and Potts models
A
bstract
We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.
Journal Article
Lattice regularisation of a non-compact boundary conformal field theory
A
bstract
Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions.
The present paper follows a series of previous works on a ℤ
2
-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable
D
2
2
vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.
Journal Article