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Critical Theories of the Dissipative Hofstadter Model

Curtis G. Callan, Felce, A, Denise E. Freed

Published Jan 1, 1993• 18 citations
OpenAlex

Abstract

It has recently been shown that the dissipative Hofstadter model (dissipative quantum mechanics of an electron subject to uniform magnetic field and periodic potential in two dimensions) exhibits critical behavior on a network of lines in the dissipation/magnetic field plane. Apart from their obvious condensed matter interest, the corresponding critical theories represent non-trivial solutions of open string field theory, and a detailed account of their properties would be interesting from several points of view. A subject of particular interest is the dependence of physical quantities on the magnetic field since it, much like $\\theta_{\\rm QCD}$, serves only to give relative phases to different sectors of the partition sum. In this paper we report the results of an initial investigation of the free energy, $N$-point functions and boundary state of this type of critical theory. Although our primary goal is the study of the magnetic field dependence of these quantities, we will present some new results which bear on the zero magnetic field case as well.

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It has recently been shown that the dissipative Hofstadter model (dissipative quantum mechanics of an electron subject to uniform magnetic field and periodic potential in two dimensions) exhibits critical behavior on a network of lines in the dissipation/magnetic field plane. Apart from their obvious condensed matter interest, the corresponding critical theories represent non-trivial solutions of open string field theory, and a detailed account of their properties would be interesting from several points of view. A subject of particular interest is the dependence of physical quantities on the magnetic field since it, much like thetarmQCD\\theta_{\\rm QCD}thetarmQCD​, serves only to give relative phases to different sectors of the partition sum. In this paper we report the results of an initial investigation of the free energy, NNN-point functions and boundary state of this type of critical theory. Although our primary goal is the study of the magnetic field dependence of these quantities, we will present some new results which bear on the zero magnetic field case as well.
The dissipative Hofstadter model describes quantum particles moving in two dimensions subject to a uniform magnetic field, a periodic potential and a dissipative force. We discuss the dissipative Hofstadter model in the framework of the boundary state formulation in string theory and construct exact boundary states for the model at the magic points by using the fermion representation. The dissipative Hofstadter model at magic points is shown to be equivalent to the critical boundary sine-Gordon model.
We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable u=e−4Ku=e^{-4K}u=e−4K, where K=J/(kBT)K=J/(k_BT)K=J/(kB​T), for nonzero external magnetic field, HHH. Exact results are given for the phase diagram in the uuu plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real h=H/(kBT)h=H/(k_BT)h=H/(kB​T), these results provide new insights into features of our earlier study of this case. We also consider complex h=H/(kBT)h=H/(k_BT)h=H/(kB​T) and μ=e−2hμ=e^{-2h}μ=e−2h. Calculations of complex-uuu zeros of the partition function on sections of the square lattice are presented. For the case of imaginary hhh, i.e., μ=eiθμ=e^{iθ}μ=eiθ, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the uuu plane. We find that in this case, the phase boundary Bu{\cal B}_uBu​ contains a real line segment extending through part of the physical ferromagnetic interval 0≤u≤10 \le u \le 10≤u≤1, with a right-hand endpoint urheu_{rhe}urhe​ at the temperature for which the Yang-Lee edge singularity occurs at μ=e±iθμ=e^{\pm iθ}μ=e±iθ. Conformal field theory arguments are used to relate the singularities at urheu_{rhe}urhe​ and the Yang-Lee edge.
We study two aspects of the problem of a particle moving on a lattice while subject to dissipation, often called the "Schmid model." First, a correspondence between the Schmid model and boundary sine-Gordon field theory is explored, and a new method is applied to the calculation of the partition function for the theory. Second, a traditional condensed matter formulation of the problem in one spatial dimension is extended to the case of an arbitrary two-dimensional Bravais lattice. A well-known mathematical analogy between one-dimensional dissipative quantum mechanics and string theory provides an equivalence between the Schmid model at the critical point and boundary sine-Gordon theory, which describes a free bosonic field subject to periodic interaction on the boundaries. Using the tools of conformal field theory, the partition function is calculated as a function of the temperature and the renormalized coupling constants of the boundary interaction. The method pursues an established technique of introducing an auxiliary free boson, fermionizing the system, and constructing the boundary state in fermion variables. However, a different way of obtaining the fermionic boundary conditions from the bosonic theory leads to an alternative renormalization for the coupling constants that occurs at a more natural level than in the established approach. Recent renormalization group analyses of the extension of the Schmid model to a two-dimensional periodic potential have yielded interesting new structure in the phase diagram for the mobility. We extend a classic one-dimensional, finite temperature calculation to the case of an arbitrary two-dimensional Bravais lattice. The duality between weak-potential and tightbinding lattice limits is reproduced in the two-dimensional case, and a perturbation expansion in the potential strength used to verify the change in the critical dependence of the mobility on the strength of the dissipation. With a triangular lattice the possibility of third order contributions arises, and we obtain some preliminary expressions for their contributions to the mobility.