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arXiv:0705.3928 · Journal of the Korean Physical Society

Dissipative Hofstadter Model at the Magic Points and Critical Boundary Sine-Gordon Model

Seungmuk Ji, Ja-Yong Koo, Taejin Lee

Published May 31, 2007• 3 citations
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Abstract

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.

DOI
10.3938/jkps.50.54

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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.
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.
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.
We study a junction of three quantum wires enclosing a magnetic flux. This is the simplest problem of a quantum junction between Tomonaga-Luttinger liquids in which Fermi statistics enter in a nontrivial way. We present a direct connection between this problem and the dissipative Hofstadter problem, or quantum Brownian motion in two dimensions in a periodic potential and an external magnetic field, which in turn is connected to open string theory in a background electromagnetic field. We find nontrivial fixed points corresponding to a chiral conductance tensor leading to an asymmetric flow of the current.
We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.
We study the effects of dissipative boundaries in many-body systems at continuous quantum transitions, when the parameters of the Hamiltonian driving the unitary dynamics are close to their critical values. As paradigmatic models we consider fermionic wires subject to dissipative interactions at the boundaries, associated with pumping or loss of particles. They are induced by couplings with a Markovian baths, so that the evolution of the system density matrix can be described by a Lindblad master equation. We study the quantum evolution arising from variations of the Hamiltonian and dissipation parameters, starting at t=0t=0t=0 from the ground state of the Hamiltonian at, or close to, the critical point. Two different dynamic regimes emerge: (i) an early-time regime for times $t\ensuremath{\sim}L$, where the competition between coherent and incoherent drivings develops a dynamic finite-size scaling, obtained by extending the scaling framework describing the coherent critical dynamics of the closed system, to allow for the boundary dissipation; and (ii) a large-time regime for $t\ensuremath{\sim}{L}^{3}$ whose dynamic scaling describes the late quantum evolution leading to the $t\ensuremath{\rightarrow}\ensuremath{\infty}$ stationary states.