Nina Javerzat, Sebastian Grijalva, Alberto Rosso, Raoul Santachiara
SciPost Phys. 9, 050 (2020) ·
published 12 October 2020
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We consider discrete random fractal surfaces with negative Hurst exponent
$H<0$. A random colouring of the lattice is provided by activating the sites at
which the surface height is greater than a given level $h$. The set of
activated sites is usually denoted as the excursion set. The connected
components of this set, the level clusters, define a one-parameter ($H$) family
of percolation models with long-range correlation in the site occupation. The
level clusters percolate at a finite value $h=h_c$ and for $H\leq-\frac{3}{4}$
the phase transition is expected to remain in the same universality class of
the pure (i.e. uncorrelated) percolation. For $-\frac{3}{4}<H< 0$ instead,
there is a line of critical points with continously varying exponents. The
universality class of these points, in particular concerning the conformal
invariance of the level clusters, is poorly understood. By combining the
Conformal Field Theory and the numerical approach, we provide new insights on
these phases. We focus on the connectivity function, defined as the probability
that two sites belong to the same level cluster. In our simulations, the
surfaces are defined on a lattice torus of size $M\times N$. We show that the
topological effects on the connectivity function make manifest the conformal
invariance for all the critical line $H<0$. In particular, exploiting the
anisotropy of the rectangular torus ($M\neq N$), we directly test the presence
of the two components of the traceless stress-energy tensor. Moreover, we probe
the spectrum and the structure constants of the underlying Conformal Field
Theory. Finally, we observed that the corrections to the scaling clearly point
out a breaking of integrability moving from the pure percolation point to the
long-range correlated one.
Sebastian Grijalva, Jacopo De Nardis, Veronique Terras
SciPost Phys. 7, 023 (2019) ·
published 20 August 2019
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We study the open XXZ spin chain in the anti-ferromagnetic regime and for
generic longitudinal magnetic fields at the two boundaries. We discuss the
ground state via the Bethe ansatz and we show that, for a chain of even length
L and in a regime where both boundary magnetic fields are equal and bounded by
a critical field, the spectrum is gapped and the ground state is doubly
degenerate up to exponentially small corrections in L. We connect this
degeneracy to the presence of a boundary root, namely an excitation localized
at one of the two boundaries. We compute the local magnetization at the left
edge of the chain and we show that, due to the existence of a boundary root,
this depends also on the value of the field at the opposite edge, even in the
half-infinite chain limit. Moreover we give an exact expression for the large
time limit of the spin autocorrelation at the boundary, which we explicitly
compute in terms of the form factor between the two quasi-degenerate ground
states. This, as we show, turns out to be equal to the contribution of the
boundary root to the local magnetization. We finally discuss the case of chains
of odd length.
