Exact boundary critical exponents and tunneling effects in integrable models for quantum wires

Abstract

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.

Luttinger liquids with boundaries: Power-laws and energy scales

“…Led by this observation and analogy to PBC several authors have generalized the above results to all models of LL's [11,12]. Using the numerically exact DMRG algorithm we investigated whether this is a legitimate generalization for two models with a short range interaction: The lattice model of spinless fermions with nearest neighbor interaction and the 1D Hubbard model [13,14]. In both cases we were able to explicitly verify that the final suppression of the spectral weight at small energies is consistent with the prediction of bosonization, although for the Hubbard model only at energies surprisingly close to the chemical potential, respectively for very long chains.…”

“…As for PBC the interacting model with OBC can be solved exactly by the Bethe ansatz, but similar to PBC not much about correlation functions can be learned directly from the solution. Information about boundary exponents can be obtained if conformal invariance is assumed [13,15].…”

“…As usual in scattering theory [34] we define the off-shell T matrix T (ω) which obeys the Lippmann-Schwinger equation 13) with the bare retarded Green's function…”

Surface energy of the one-dimensional supersymmetric t − J model with unparallel boundary fields

“…This fact enables us to calculate the surface energy (4.7) and (4.10), which is the same as that for the case of parallel boundary fields [23]. Moreover, it implies that the inhomogeneous term in (2.18) surely gives some contributions to the other physical qualities such as the boundary conformal charge, which are related to the coefficients in the expansion of energy E in terms of the powers of L −1 (namely, the coefficient of L −1 corresponds to the conformal charge [38]). The method used in this paper can be generalized to study the thermodynamic limit and surface energy of other models related to rational R-matrices, such as the spin-s XXX chain or the su(n) spin chain with unparallel boundary fields.…”

“…In the region of ξ < 0 and 1/2 < ξ ′ < 1, one of the Bethe roots at the ground state goes to ( 1 2 − ξ ′ )i when the system-size L tends to infinity [26,[37][38][39]. We note the value of this Bethe root is related with the boundary parameter ξ ′ .…”