Jean Michel Maillet, Giuliano Niccoli, Louis Vignoli
SciPost Phys. 9, 086 (2020) ·
published 11 December 2020

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Using the framework of the quantum separation of variables (SoV) for higher
rank quantum integrable lattice models [1], we introduce some foundations to go
beyond the obtained complete transfer matrix spectrum description, and open the
way to the computation of matrix elements of local operators. This first
amounts to obtain simple expressions for scalar products of the socalled
separate states (transfer matrix eigenstates or some simple generalization of
them). In the higher rank case, left and right SoV bases are expected to be
pseudoorthogonal, that is for a given SoV covector, there could be more than
one nonvanishing overlap with the vectors of the chosen right SoV basis. For
simplicity, we describe our method to get these pseudoorthogonality overlaps
in the fundamental representations of the $\mathcal{Y}(gl_3)$ lattice model
with $N$ sites, a case of rank 2. The nonzero couplings between the covector
and vector SoV bases are exactly characterized. While the corresponding
SoVmeasure stays reasonably simple and of possible practical use, we address
the problem of constructing left and right SoV bases which do satisfy standard
orthogonality. In our approach, the SoV bases are constructed by using families
of conserved charges. This gives us a large freedom in the SoV bases
construction, and allows us to look for the choice of a family of conserved
charges which leads to orthogonal covector/vector SoV bases. We first define
such a choice in the case of twist matrices having simple spectrum and zero
determinant. Then, we generalize the associated family of conserved charges and
orthogonal SoV bases to generic simple spectrum and invertible twist matrices.
Under this choice of conserved charges, and of the associated orthogonal SoV
bases, the scalar products of separate states simplify considerably and take a
form similar to the $\mathcal{Y}(gl_2)$ rank one case.
SciPost Phys. 9, 082 (2020) ·
published 7 December 2020

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We introduce a framework for calculating dynamical correlations in the LiebLiniger model in arbitrary energy eigenstates and for all space and time, that combines a Lehmann representation with a $1/c$ expansion. The $n^{\rm th}$ term of the expansion is of order $1/c^n$ and takes into account all $\lfloor \tfrac{n}{2}\rfloor+1$ particlehole excitations over the averaging eigenstate. Importantly, in contrast to a ``bare" $1/c$ expansion it is uniform in space and time. The framework is based on a method for taking the thermodynamic limit of sums of form factors that exhibit non integrable singularities. We expect our framework to be applicable to any local operator.\\
We determine the first three terms of this expansion and obtain an explicit expression for the densitydensity dynamical correlations and the dynamical structure factor at order $1/c^2$. We apply these to finitetemperature equilibrium states and nonequilibrium steady states after quantum quenches. We recover predictions of (nonlinear) Luttinger liquid theory and generalized hydrodynamics in the appropriate limits, and are able to compute subleading corrections to these.