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We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group $U_q(mathcal L(mathfrak{sl}_2))$ related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.
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In this work we study an elliptic solid-on-solid model with domain-wall boundaries having the elliptic quantum group $mathcal{E}_{p, gamma}[widehat{mathfrak{gl}_2}]$ as its underlying symmetry algebra. We elaborate on results previously presented by the author and extend our analysis to include continuous families of single determinantal representations for the models partition function. Interestingly, our families of representations are parameterized by two continuous complex variables which can be arbitrarily chosen without affecting the partition function.
We consider a multichannel wire with a disordered region of length $L$ and a reflecting boundary. The reflection of a wave of frequency $omega$ is described by the scattering matrix $mathcal{S}(omega)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $mathcal{Q}=-mathrm{i}, mathcal{S}^daggerpartial_omegamathcal{S}$ is another important matrix encoding temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, $mathcal{S}=mathrm{e}^{2mathrm{i}kL}mathcal{U}_Lmathcal{U}_R$ (with $mathcal{U}_L=mathcal{U}_R^mathrm{T}$ in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix: $widetilde{mathcal{Q}} =mathcal{U}_R,mathcal{Q},mathcal{U}_R^dagger = (2L/v),mathbf{1}_N -mathrm{i},mathcal{U}_L^daggerpartial_omegabig(mathcal{U}_Lmathcal{U}_Rbig),mathcal{U}_R^dagger$, where $k$ is the wave vector and $v$ the group velocity. We demonstrate that $widetilde{mathcal{Q}}$ can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, $Ltoinfty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for $mathcal{Q}$s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, the exponential functional representation is used to calculate the first moments $langlemathrm{tr}(mathcal{Q})rangle$, $langlemathrm{tr}(mathcal{Q}^2)rangle$ and $langlebig[mathrm{tr}(mathcal{Q})big]^2rangle$. Finally we derive a partial differential equation for the resolvent $g(z;L)=lim_{Ntoinfty}(1/N),mathrm{tr}big{big( z,mathbf{1}_N - N,mathcal{Q}big)^{-1}big}$ in the large $N$ limit.
We study the joint probability density of the eigenvalues of a product of rectangular real, complex or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously have been discussed for infinite matrix size. Moreover we derive the joint probability densities of the eigenvalues. To illustrate our results we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights we show that the product of complex random matrices yield a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.
We continue our exercises with the universal $R$-matrix based on the Khoroshkin and Tolstoy formula. Here we present our results for the case of the twisted affine Kac--Moody Lie algebra of type $A^{(2)}_2$. Our interest in this case is inspired by the fact that the Tzitzeica equation is associated with $A^{(2)}_2$ in a similar way as the sine-Gordon equation is related to $A^{(1)}_1$. The fundamental spin-chain Hamiltonian is constructed systematically as the logarithmic derivative of the transfer matrix. $L$-operators of two types are obtained by using q-deformed oscillators.
Standard derivations of the functional integral in non-equilibrium quantum field theory are based on the discrete time representation. In this work we derive the non-equilibrium functional integral for non-interacting bosons and fermions using a continuum time approach by accounting for the statistical distribution through the boundary conditions and using them to evaluate the Greens function.
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