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  • 1.
    Atai, Farrokh
    et al.
    KTH, School of Engineering Sciences (SCI), Physics, Mathematical Physics.
    Langmann, Edwin
    KTH, School of Engineering Sciences (SCI), Physics, Mathematical Physics.
    Series Solutions of the Non-Stationary Heun Equation2018In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 14, article id 011Article in journal (Refereed)
    Abstract [en]

    We consider the non-stationary Heun equation, also known as quantum Painleve VI, which has appeared in different works on quantum integrable models and conformal field theory. We use a generalized kernel function identity to transform the problem to solve this equation into a differential-difference equation which, as we show, can be solved by efficient recursive algorithms. We thus obtain series representations of solutions which provide elliptic generalizations of the Jacobi polynomials. These series reproduce, in a limiting case, a perturbative solution of the Heun equation due to Takemura, but our method is different in that we expand in non-conventional basis functions that allow us to obtain explicit formulas to all orders; in particular, for special parameter values, our series reduce to a single term.

  • 2.
    Langmann, Edwin
    et al.
    KTH, School of Engineering Sciences (SCI), Physics, Mathematical Physics.
    Lebowitz, Joel L.
    Mastropietro, Vieri
    Moosavi, Per
    KTH, School of Engineering Sciences (SCI), Physics, Mathematical Physics.
    Time evolution of the Luttinger model with nonuniform temperature profile2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 95, no 23, article id 235142Article in journal (Refereed)
    Abstract [en]

    We study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile T (x). As a specific example we consider the case when T (x) is equal to T-L (T-R) far to the left (right). Using a series expansion in epsilon = 2(T-R -T-L)/(T-L + T-R), we compute the energy density, the heat current density, and the fermion two-point correlation function for all times t >= 0. For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of T (x). For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in epsilon. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on T-L,T-R agrees with numerical results for the XXZ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.

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