Inhomogeneous Gaussian free field inside the interacting arctic curve

Abstract

The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e., models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e., not free, not determinantal) six-vertex model, and provide evidence for the following two claims:(i) the critical fluctuations are given by a Gaussian Free Field (GFF), as in the free case, but (ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant K becomes position-dependent, K → K(x).The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of K and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green's function of the Laplacian ∆ = ∇ · ∇ inside the critical domain, but instead, of the Green's function of a generalized Laplacian ∆ = ∇ · 1 K ∇ parametrized by the function K. Surprisingly, we also find that there is a change of regime when ∆ ≤ −1/2, with K becoming singular at one point.

Entanglement entropies of inhomogeneous Luttinger liquids

“…Notice however that, as discussed in detail in Refs. [22,23,52], promoting the model to inhomogeneous v(x) and K(x) leads to possible ambiguities. Different choices are in principle allowed: for example, substituting 1/K(∂ xφ ) 2 in the homogeneous Hamiltonian (5) by [∂ x ( 1/K(x)φ)] 2 instead of 1/K(x)(∂ xφ ) 2 would lead to another inhomogeneous Hamiltonian which would look equally valid.…”

Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

“…In the thermodynamic limit the numerical data are expected to converge to the CFT predictions for G c n,A (ρ 1 ||ρ 0 ) in Eq. (48). The numerical results are shown in Figure 7 for different number of particles N and for all possible pairs of states.…”

“…This curved CFT approach has already been employed for many applications: the entanglement entropies [40], the entanglement hamiltonian [41], and some correlation functions [40,42] have been calculated for many different situations in inhomogeneous freefermion models; the field theory description of the rainbow model was also unveiled [43], spin chains with gradients were studied [44], and the presence of curved lightcones has been investigated [45]. All these applications refer to free models, but some results for interacting systems are also available [46][47][48][49]. The goal of this work is to understand whether and in which form the universality features of the entanglement in low-lying excited states persist in inhomogeneous settings, as a Figure 1.…”

“…Different colours correspond to different N . The continuous curves are the CFT predictions in Eq (48)…”

Arctic Curve of the Free-Fermion Six-Vertex Model in an L-Shaped Domain

“…The thermodynamics of the six-vertex model with fixed boundary conditions has attracted much attention in recent years, in particular, as an example of a system exhibiting (in an appropriate scaling limit) spatial phase separation phenomena [1][2][3][4][5][6][7][8][9][10][11][12]. The model can be regarded as a nontrivial generalization of dimer models, and in particular, under specific geometry and boundary conditions, of the famous problem of domino tilings of the Aztec diamond, where the celebrated Arctic Circle phenomenon was discovered [13].…”