148 research outputs found
Time evolution of the Luttinger model with nonuniform temperature profile
No full textWe 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
Steady States and Universal Conductance in a Quenched Luttinger Model
No full textWe obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian Hλ with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time t > 0 via a Hamiltonian Hλ′ which differs from Hλ by the strength of the interaction. Asymptotically in time, as t→ ∞, after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current I and has an effective chemical potential difference μ+- μ- between right- (+) and left- (−) moving fermions obtained from the two-point correlation function. Both I and μ+- μ- depend on λ and λ′. Only for the case λ= λ′= 0 does μ+- μ- equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, G= I/ (μ+- μ-) , has a universal value equal to the conductance quantum e2/ h for the spinless case
Algorithms to solve the (quantum) Sutherland model
No full textWe give a self-contained presentation and comparison of two different algorithms to explicitly solve quantum many body models of indistinguishable particles moving on a circle and interacting with two-body potentials of 1/sin(2)-type. The first algorithm is due to Sutherland and well-known; the second one is a limiting case of a novel algorithm to solve the elliptic generalization of the Sutherland model. These two algorithms are different in several details. We show that they are equivalent, i.e., they yield the same solution and are equally simple.</p
Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones
No full textThere exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, explicitly. Of course there is a catch to this result: if one insists on these eigenfunctions to be square integrable then the corresponding Hamiltonian is necessarily non-hermitean (and thus provides an example of an exactly solvable PT-symmetric quantum-many body system), and if one insists on the Hamiltonian to be hermitean then the eigenfunctions are singular and thus not acceptable as quantum mechanical eigenfunctions. The standard Calogero-Sutherland Hamiltonian is special due to the existence of an integral operator which allows to transform these singular eigenfunctions into regular ones
AN EXPLICIT SOLUTION OF THE (QUANTUM) ELLIPTIC CALOGERO-SUTHERLAND MODEL
No full textWe present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders in the nome of the elliptic functions, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explicit formulas for an elliptic deformation of the Jack polynomials.</p
Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model
No full textWe present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter lambda, we also obtain dual relations involving mixed correlation functions of anyons with two different statistics parameters lambda and 1/lambda. We also give alternative, elementary proofs of these identities by direct computations.</p
Generalized Yang–Mills actions from Dirac operator determinants
No full textWe consider the quantum effective action of Dirac fermions on four-dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a Dirac operator on flat R-4 twisted by generalized Yang-Mills fields. According to physics folklore, the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action. We present a simple explicit computation proving this fact and extending it to the chiral case. We use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo-differential operators and which yields an expansion of the logarithm of Dirac operators in local and quasi-gauge invariant polynomials of decreasing scaling dimension.</p
Source Identity and Kernel Functions for Elliptic Calogero–Sutherland Type Systems
No full textKernel functions related to quantum many-body systems of Calogero-Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh-Feigin-Veselov-Sergeev-type deformations of the elliptic Calogero-Sutherland model for special parameter values.</p
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