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arXiv:cond-mat/9512075 · Physical review. B, Condensed matter

Exact critical properties of the multicomponent interacting fermion model with boundaries

Satoshi Fujimoto, Norio Kawakami

Published Aug 15, 1996• 29 citations
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Abstract

Exact critical properties of the one-dimensional SU(N) interacting fermion model with open boundaries are studied by using the Bethe ansatz method. We derive the surface critical exponents of various correlation functions using boundary conformal field theory. They are classified into two types, i.e., the exponents for the chiral SU(N) Tomonaga-Luttinger liquid and those related to the orthogonality catastrophe. We discuss a possible application of the results to the photoemission (absorption) in the edge state of the fractional quantum Hall effect. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.5784

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Exact critical properties of the one-dimensional SU(N) interacting fermion model with open boundaries are studied by using the Bethe ansatz method. We derive the surface critical exponents of various correlation functions using boundary conformal field theory. They are classified into two types, i.e., the exponents for the chiral SU(N) Tomonaga-Luttinger liquid and those related to the orthogonality catastrophe. We discuss a possible application of the results to the photoemission (absorption) in the edge state of the fractional quantum Hall effect. \textcopyright{} 1996 The American Physical Society.
Using the principles of the conformal quantum-field theory and the finite size corrections of the energy of the ground and various excited states, we calculate the boundary critical exponents of single- and multicomponent Bethe-Ansatz soluble models. The boundary critical exponents are given in terms of the dressed-charge matrix which has the same form as that of systems with periodic boundary conditions and is uniquely determined by the Bethe-ansatz equations. A Luttinger liquid with open boundaries is the effective low-energy theory of these models. As applications of the theory, the Friedel oscillations due to the boundaries and the tunneling conductance through a barrier are also calculated. The tunneling conductance is determined by a nonuniversal boundary exponent which governs its power law dependence on temperature and frequency. \textcopyright{} 1996 The American Physical Society.
We show that one-dimensional quantum systems with gapless degrees of freedom and open boundary conditions form a universality class of quantum critical behavior, which we propose to call ``bounded Luttinger liquids.'' They share the following properties with ordinary (periodic) Luttinger liquids: the absence of fermionic quasiparticle excitations, charge-spin separation, and anomalous power-law correlations with exponents whose scaling relations are parametrized by a single coupling constant per degree of freedom, ${K}_{\ensuremath{\nu}}.$ The values of ${K}_{\ensuremath{\nu}}$ are independent of boundary conditions, but the representation of the critical exponents in terms of these ${K}_{\ensuremath{\nu}}'\mathrm{s}$ depends on boundary conditions. We illustrate these scaling relations by exploring general rules for boundary critical exponents derived earlier using the Bethe ansatz solution of the one-dimensional Hubbard model together with boundary conformal field theory, and the theory of Luttinger liquids in finite-size systems. We apply this theory to the photoemission properties of the organic conductors (TMTSF)2X,(\mathrm{TMTSF}{)}_{2}X,(TMTSF)2​X, where TMTSF is tetramethyltetraselenafulvalene, and X=ClO9,X={\mathrm{ClO}}_{9},X=ClO9​, PF6,{\mathrm{PF}}_{6},PF6​, ReO4,\mathrm{Re}{\mathrm{O}}_{4},ReO4​, and discuss to what extent the assumption of finite strands with open boundaries at the sample surface can reconcile the experimental results with independent information on the Luttinger-liquid state in these materials.
Exact critical exponents of the Green functions for pseudo-fermions and slave bosons in the SU(NNN) Anderson model with U→∞U\rightarrow\inftyU→∞ are obtained by using the Bethe ansatz solution and boundary conformal field theory. They are evaluated exactly for mixed valence systems and Kondo systems with crystalline fields. The results agree with the prediction of Menge and Müller-Hartmann, which coincide with those of the X-ray problem. Some implication of our results in one-dimensional chiral systems is also discussed.
Using the Bethe ansatz we study the spectrum of boundary bound states of the SU ( N ) generalization of the one-dimensional Hubbard model with open boundaries and applied boundary potentials. The number of electrons bound by the boundary field is found from the surface contribution to the energy. Finite size corrections to the low-lying energies in this system and use of the predictions of boundary conformal field theory allows us to study the exponents related to the orthogonality catastrophe and absorption edges.
Using conformal field theory the exact critical exponents are obtained for the SU(N) generalization of the U\ensuremath{\rightarrow}\ensuremath{\infty} Hubbard model and the supersymmetric t-J model in one dimension. For the Hubbard model the critical exponent of the momentum distribution function is given as \ensuremath{\theta}=(1-1/N)2{)}^{2})2/2 while for the supersymmetric t-J model \ensuremath{\theta} varies continuously in the range 0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}(1-1/N)2{)}^{2})2/2 depending on the electron concentration. The critical exponents of various correlation functions are shown to satisfy the universal scaling relations. The SU(N) t-J model in the range 0J/t\ensuremath{\le}2 belongs to the SU(N) Luttinger liquid for any finite N.