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Register a new userConfinement in the q-state Potts model: an RG-TCSA study
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In the ferromagnetic phase of the q-state Potts model, switching on an external magnetic field induces confinement of the domain wall excitations. For the Ising model (q = 2) the spectrum consists of kink-antikink states which are the analogues of mesonic states in QCD, while for q = 3, depending on the sign of the field, the spectrum may also contain three-kink bound states which are the analogues of the baryons. In recent years the resulting hadron spectrum was described using several different approaches, such as quantum mechanics in the confining linear potential, WKB methods and also the Bethe-Salpeter equation. Here we compare the available predictions to numerical results from renormalization group improved truncated conformal space approach (RG-TCSA). While mesonic states in the Ising model have already been considered in a different truncated Hamiltonian approach, this is the first time that a precision numerical study is performed for the 3-state Potts model. We find that the semiclassical approach provides a very accurate description for the mesonic spectrum in all the parameter regime for weak magnetic field, while the low-energy expansion from the Bethe-Salpeter equation is only valid for very weak fields where it gives a slight improvement over the semiclassical results. In addition, we confirm the validity of the recent predictions for the baryon spectrum obtained from solving the quantum mechanical three-body problem.
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The bootstrap determination of the geometrical correlation functions in the two-dimensional Potts model proposed in a paper [arXiv:1607.07224] was later shown in [arXiv:1809.02191] to be incorrect, the actual spectrum of the model being considerably more complex than initially conjectured. We provide in this paper a geometrical interpretation of the four-point functions built in [arXiv:1607.07224], and explain why the results obtained by these authors, albeit incorrect, appeared so close to those of their numerical simulations of the Potts model. Our strategy is based on a cluster expansion of correlation functions in RSOS minimal models, and a subsequent numerical and algebraic analysis of the corresponding $s$-channel spectrum, in full analogy with our early work on the Potts model [arXiv:1809.02191]. Remarkable properties of the lattice amplitudes are uncovered, which explain in particular the truncation of the spectrum of [arXiv:1809.02191] to the much simpler one of the RSOS models, and which will be used in a forthcoming paper to finally determine the geometric four-point functions of the Potts model itself.
We extend a recent analysis of the $q$-states Potts model on an ensemble of random planar graphs with $pleqslant q$ allowed, equally weighted, spins on a connected boundary. In this paper we explore the $(q<4,pleqslant q)$ parameter space of finite-sheeted resolvents and derive the associated critical exponents. By definition a value of $q$ is allowed if there is a $p=1$ solution, and we reproduce the long-known result that $q= 2(1+cos{frac{m}{n} pi})$ with $m,n$ coprime. In addition we find that there are two distinct sequences of solutions, one of which contains $p=2$ and $p=q/2$ while the other does not. The boundary condition $p=3$ appears only for $q=3$ which also has a $p=3/2$ boundary condition; we conjecture that this new solution corresponds in the scaling limit to the New boundary condition, discovered on the flat lattice by Affleck et al. We also explore Kramers-Wannier duality for $q=3$ in this context and explicitly construct the known boundary conditions; we show that the mixed boundary condition is dual to a boundary condition on dual graphs that corresponds to Affleck et als identification of the New boundary condition on fixed lattices. On the other hand we find that the mixed boundary condition of the dual, and the corresponding New boundary condition of the original theory are not described by conventional resolvents.
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 [M. Picco, S. Ribault and R. Santachiara, SciPost Phys. 1, 009 (2016); arXiv:1607.07224], 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 [arXiv:1607.07224]: they involve in particular fields with conformal weight $h_{r,s}$ where $r$ is dense on the real axis.
In this paper, we apply the form factor bootstrap approach to branch point twist fields in the $q$-state Potts model for $qleq 3$. For $q=3$ this is an integrable interacting quantum field theory with an internal discrete $mathbb{Z}_3$ symmetry and therefore provides an ideal starting point for the investigation of the symmetry resolved entanglement entropies. However, more generally, for $qleq 3$ the standard Renyi and entanglement entropies are also accessible through the bootstrap programme. In our work we present form factor solutions both for the standard branch point twist field with $qleq 3$ and for the composite (or symmetry resolved) branch point twist field with $q=3$. In both cases, the form factor equations are solved for two particles and the solutions are carefully checked via the $Delta$-sum rule. Using our analytic predictions, we compute the leading finite-size corrections to the entanglement entropy and entanglement equipartition for a single interval in the ground state.
We study the spectrum of the scaling Lee-Yang model on a finite interval from two points of view: via a generalisation of the truncated conformal space approach to systems with boundaries, and via the boundary thermodynamic Bethe ansatz. This allows reflection factors to be matched with specific boundary conditions, and leads us to propose a new (and non-minimal) family of reflection factors to describe the one relevant boundary perturbation in the model. The equations proposed previously for the ground state on an interval must be revised in certain regimes, and we find the necessary modifications by analytic continuation. We also propose new equations to describe excited states, and check all equations against boundary truncated conformal space data. Access to the finite-size spectrum enables us to observe boundary flows when the bulk remains massless, and the formation of boundary bound states when the bulk is massive.
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