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Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

Ian Affleck, Masaki Oshikawa, Hubert Saleur

TL;DR

We present a boundary CFT solution for quantum Brownian motion on a triangular lattice, connecting it to a c=2 BCFT and a conformal embedding into Ising, Tricritical Ising and Potts sectors. Boundary states are constructed and classified via fusion, with calculable mobilities and ground-state degeneracies, and integrable flows between fixed points analyzed by the thermodynamic Bethe ansatz, yielding precise g-factor ratios. The work identifies four principal fixed points (Dirichlet, Neumann, Yi–Kane Y, and a new W state) with distinct transport properties, including Hall-like responses, and exhibits a non-bosonic boundary state within a free-boson framework. It further generalizes to higher dimensions and links QBM to boundary Potts and Kondo problems, illuminating deep connections between dissipative quantum dynamics and impurity physics.

Abstract

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2 dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to non-trivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.

Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

TL;DR

We present a boundary CFT solution for quantum Brownian motion on a triangular lattice, connecting it to a c=2 BCFT and a conformal embedding into Ising, Tricritical Ising and Potts sectors. Boundary states are constructed and classified via fusion, with calculable mobilities and ground-state degeneracies, and integrable flows between fixed points analyzed by the thermodynamic Bethe ansatz, yielding precise g-factor ratios. The work identifies four principal fixed points (Dirichlet, Neumann, Yi–Kane Y, and a new W state) with distinct transport properties, including Hall-like responses, and exhibits a non-bosonic boundary state within a free-boson framework. It further generalizes to higher dimensions and links QBM to boundary Potts and Kondo problems, illuminating deep connections between dissipative quantum dynamics and impurity physics.

Abstract

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2 dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to non-trivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.

Paper Structure

This paper contains 27 sections, 309 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Boundary Conformal Field Theory
  3. Fusion Approach to Boundary CFT
  4. Dirichlet Neumann, and Rotated Boundary Conditions For Free Bosons
  5. Dirichlet and Neumann Boundary Conditions From Fusion: c=1 Case
  6. Quantum Brownian motion and conformal field theory
  7. Strong Corrugation Limit
  8. Weak Corrugation Limit
  9. Groundstate Degeneracy and Boson Compactification
  10. quantum brownian motion on a triangular lattice
  11. Phase Diagram of the Boundary 3-State Potts Model
  12. Back to Quantum Brownian Motion
  13. Construction of boundary states via a conformal embedding
  14. Conformal embedding
  15. Fusion in the Potts sector
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Cylinder of length $l$, circumference $\beta$ with boundary conditions $A$ and $B$ at the two ends.
  • Figure 2: Triangular lattice with A,B and C sublattices marked. Note that the A points also form a triangular lattice with spacing $\sqrt{3}a$ and orientation rotated by $90^0$. The B and C points form a honeycomb lattice.
  • Figure 3: Phase diagram and RG flows of the triangular lattice QBM model with hopping strength $t$ and potential $v_A$. Neumann and Dirichlet b.c.'s are imposed on the dual fields, $\tilde{\vec{\phi}}$. As pointed out in Ref. ( YiKane) and discussed in Sec. VII, the "localized on B or C" phase corresponds to th weak coupling fixed point in the 3-channel Kondo problem.