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arXiv:hep-th/0405264 · Journal of Statistical Mechanics Theory and Experiment

Finite temperature quantum field theory with impurities

Mihail Mintchev, P. Sorba

Published Jul 2, 2004• 15 citations
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Abstract

We apply the concept of reflection–transmission (RT) algebra, originally developed in the context of integrable systems in 1+1 space–time dimensions, to the study of finite temperature quantum field theory with impurities in higher dimensions. We consider a scalar field in ( s +1)+1 space–time dimensions, interacting with impurities localized on s -dimensional hyperplanes, but without self-interaction. We discuss first the case s = 0 and extend afterwards all results to s >0. Constructing the Gibbs state over an appropriate RT algebra, we derive the energy density at finite temperature and establish the correction to the Stefan–Boltzmann law generated by the impurity. The contribution of the impurity bound states is taken into account. The charge density profiles for various impurities are also investigated.

DOI
10.1088/1742-5468/2004/07/p07001

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We apply the concept of reflection–transmission (RT) algebra, originally developed in the context of integrable systems in 1+1 space–time dimensions, to the study of finite temperature quantum field theory with impurities in higher dimensions. We consider a scalar field in ( s +1)+1 space–time dimensions, interacting with impurities localized on s -dimensional hyperplanes, but without self-interaction. We discuss first the case s = 0 and extend afterwards all results to s >0. Constructing the Gibbs state over an appropriate RT algebra, we derive the energy density at finite temperature and establish the correction to the Stefan–Boltzmann law generated by the impurity. The contribution of the impurity bound states is taken into account. The charge density profiles for various impurities are also investigated.
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We establish the exact solution of the nonlinear Schrödinger equation with a delta-function impurity, representing a pointlike defect which reflects and transmits. We solve the problem both at the classical and the second quantized levels. In the quantum case the Zamolodchikov–Faddeev algebra, familiar from the case without impurities, is substituted by the recently discovered reflection-transmission (RT) algebra, which captures both particle–particle and particle–impurity interactions. The off-shell quantum solution is expressed in terms of the generators of the RT algebra and the exact scattering matrix of the theory is derived.
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The Kondo problem of two impurities in a 1D strongly correlated electron system within the framework of the open boundary Hubbard chain is solved and the impurities, coupled to the ends of the electron system, are introduced by their scattering matrices with electrons so that the boundary matrices satisfy the reflecting integrability condition. The finite-size correction of the ground-state energy is obtained due to the impurities. Exact expressions for the low-temperature specific heat contributed by the charge and spin parts of the magnetic impurities are derived. The Pauli susceptibility and the Kondo temperature are given explicitly. The Kondo temperature is inversely proportional to the density of electrons.
We explore the reflection–transmission quantum Yang–Baxter equations, arising in factorized scattering theory of integrable models with impurities. The physical origin of these equations is clarified and three general families of solutions are described in detail. Explicit representatives of each family are also displayed. These results allow us to establish, for the first time, a direct relationship with the different previous works on the subject and make evident the advantages of the reflection–transmission algebra as a universal approach to integrable systems with impurities.
Two-dimensional quantum gases provide a distinctive setting in which enhanced thermal fluctuations, finite-size effects, and two-body bound-state formation are intrinsically intertwined. In this work, we study a single attractive impurity immersed in a finite, weakly interacting two-dimensional Bose gas, where finite size stabilizes a nonzero condensate fraction by introducing an infrared momentum scale, thereby enabling a Bogoliubov description of the bath. Using a hybrid approach that combines finite-temperature many-body scattering theory with input from path-integral Monte Carlo, we analyze the impurity quasiparticle energy across the condensate and normal regimes. The infrared scale generates a phonon-activation temperature below which the impurity energy remains nearly temperature independent. Once the resolved phonon modes become thermally populated, their contribution competes with condensate depletion, producing a nonmonotonic temperature dependence of the polaron energy. These results suggest that attractive Bose polarons may serve as sensitive probes of finite-size thermal fluctuations, phonon dressing, and bound-state physics in low-dimensional Bose gases.