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arXiv:hep-th/9507053

Massless Boundary Sine-Gordon at the Free Fermion Point: Correlation and Partition Functions with Applications to Quantum Wires

Robert Konik

Published Jul 9, 1995• 1 citation
arXivHTMLPDF

Abstract

In this report we compute the boundary states (including the boundary entropy) for the boundary sine-Gordon theory. From the boundary states, we derive both correlation and partition functions. Through the partition function, we show that boundary sine-Gordon maps onto a doubled boundary Ising model. With the current-current correlators, we calculate for finite system size the ac-conductance of tunneling quantum wires with dimensionless free conductance 1/2 (or, alternatively interacting quantum Hall edges at filling fraction 1/2). In the dc limit, the results of C. Kane and M. Fisher, Phys. Rev. B46 (1992) 15233, are reproduced.

DOI
10.48550/arxiv.hep-th/9507053

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In this report we compute the boundary states (including the boundary entropy) for the boundary sine-Gordon theory. From the boundary states, we derive both correlation and partition functions. Through the partition function, we show that boundary sine-Gordon maps onto a doubled boundary Ising model. With the current-current correlators, we calculate for finite system size the ac-conductance of tunneling quantum wires with dimensionless free conductance 1/2 (or, alternatively interacting quantum Hall edges at filling fraction 1/2). In the dc limit, the results of C. Kane and M. Fisher, Phys. Rev. B46 (1992) 15233, are reproduced.
Abstract We investigate the free-Fermion point of a boundary sine-Gordon model with nondiagonal boundary interactions for the ground state using auxiliary functions obtained from T − Q equations of a corresponding inhomogeneous open spin- 1 2 XXZ chain with nondiagonal boundary terms. In particular, we obtain the Casimir energy. Our result for the Casimir energy is shown to agree with the result from the TBA approach. The analytical result for the effective central charge in the ultraviolet (UV) limit is also verified from the plots of effective central charge for intermediate values of volume.
We study the properties of the double-frequency sine–Gordon model in the vicinity of the Ising quantum phase transition displayed by this model. Using a mapping onto a generalised lattice quantum Ashkin-Teller model, we obtain critical and nearly-off-critical correlation functions of various operators. We discuss applications of the double-sine-Gordon model to one-dimensional physical systems, like spin chains in a staggered external field and interacting electrons in a staggered potential. PACS: 71.10.Pm; 71.10.Fd; 75.10.Jm
Motivated by the realization of coupled Luttinger liquids in a split condensate for quantum sine-Gordon (SG) model [Haller et al., Nature 466, 597 (2010); Schweiglet et al., Nature 545, 323 (2017); Nat. Phys. 17, 559 (2021)], here we study the phase transition in coupled condensates using the bosonization method. We show that when these condensates have different velocities ui{u}_{i}ui​ and Luttinger parameters Ki{K}_{i}Ki​, the physics is described by a generalized SG model, which is also exactly solvable. We use various approaches, including the Wilson renormalization group, correlation function, variational wave function, and the equivalent Coulomb gas methods, demonstrating the phase transition at 2=1/(4K1)+1/(4K2)2=1/(4{K}_{1})+1/(4{K}_{2})2=1/(4K1​)+1/(4K2​). The physical meaning of this boundary is explained by the density-phase uncertainty principle, showing how the many-body interactions drive the coupled condensates from two gapless phases to one gapless phase. The relative phase between the condensates may fundamentally affect the band structure, yet it will not affect this boundary. Furthermore, we also discuss the relevance of our result in the two-dimensional models for the Berezinskii-Kosterlitz-Thouless transition, from which the physical meaning of this boundary is self-evident. Our results can be used to study the phase transitions in one-dimensional coupled models with Fermions and Bosons, and even the Kane-Fisher model with impurities, and study their phase transitions, non-Gaussian dynamics, and thermalization, which may be of great interest in quantum simulation.
We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation.
The massless boundary sine-Gordon (SG) model is the only interacting impurity model with a known exact solution out-of-equilibrium, yet existing so far only for integer values of the sine Gordon coupling λλλ [Phys. Rev. Lett. {\bf74}, 3005 (1995)]. We present here a full exact solution for arbitrary rational values of λλλ, at arbitrary voltage VVV and temperature TTT. We use the "string" solutions of the bulk SG model, here regarded as genuine quasiparticles avoiding charge diffusion in momentum space. We carefully present the finite voltage and temperature thermodynamics of this gas of interacting exotic quasiparticles, whose very nature depends on subtle arithmetic properties of the rational SG parameter λλλ, and explicitly check that the string representation is thermodynamically complete. By considering a Loschmidt echo, we derive the exact transmission probability of strings on the impurity. We obtain the exact universal scaling function for the electrical current I(V,T)I(V,T)I(V,T). Our results are in excellent agreement with recent experimental out-of-equilibrium data and question the reality of these exotic quasiparticles.