Jean Michel Maillet, Giuliano Niccoli, Louis Vignoli
SciPost Phys. 9, 060 (2020) ·
published 27 October 2020

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We construct quantum Separation of Variables (SoV) bases for both the
fundamental inhomogeneous $gl_{\mathcal{M}\mathcal{N}}$ supersymmetric
integrable models and for the inhomogeneous Hubbard model both defined with
quasiperiodic twisted boundary conditions given by twist matrices having
simple spectrum. The SoV bases are obtained by using the integrable structure
of these quantum models,i.e. the associated commuting transfer matrices,
following the general scheme introduced in [1]; namely, they are given by set
of states generated by the multiple action of the transfer matrices on a
generic covector. The existence of such SoV bases implies that the
corresponding transfer matrices have non degenerate spectrum and that they are
diagonalizable with simple spectrum if the twist matrices defining the
quasiperiodic boundary conditions have that property. Moreover, in these SoV
bases the resolution of the transfer matrix eigenvalue problem leads to the
resolution of the full spectral problem, i.e. both eigenvalues and
eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a
trivial overall normalization) eigenvector whose wavefunction in the SoV bases
is factorized into products of the corresponding transfer matrix eigenvalue
computed on the spectrum of the separate variables. As an application, we
characterize completely the transfer matrix spectrum in our SoV framework for
the fundamental $gl_{12}$ supersymmetric integrable model associated to a
special class of twist matrices. From these results we also prove the
completeness of the Bethe Ansatz for that case. The complete solution of the
spectral problem for fundamental inhomogeneous $gl_{\mathcal{M}\mathcal{N}}$
supersymmetric integrable models and for the inhomogeneous Hubbard model under
the general twisted boundary conditions will be addressed in a future
publication.
SciPost Phys. 9, 054 (2020) ·
published 20 October 2020

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A large family of diffusive models of transport that has been considered in
the past years admits a transformation into the same model in contact with an
equilibrium bath. This mapping holds at the full dynamical level, and is
independent of dimension or topology. It provides a good opportunity to discuss
questions of time reversal in out of equilibrium contexts. In particular,
thanks to the mapping one may define the freeenergy in the nonequilibrium
states very naturally as the (usual) free energy of the mapped system.
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 oneparameter ($H$) family
of percolation models with longrange 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 stressenergy 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
longrange correlated one.