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arXiv:0712.3439 · Journal of Statistical Mechanics Theory and Experiment

Ground state fidelity of Luttinger liquids: a wavefunctional approach

John Ove Fjærestad

Published Jul 10, 2008• 33 citations
arXivHTMLPDF

Abstract

We use a wavefunctional approach to calculate the fidelity of ground states in the Luttinger liquid universality class of one-dimensional gapless quantum many-body systems. The ground state wavefunctionals are discussed using both the Schrödinger (functional differential equation) formulation and a path integral formulation. The fidelity between Luttinger liquids with Luttinger parameters K and K ′ is found to decay exponentially with system size, and to obey the symmetry F ( K , K ′) = F (1/ K ,1/ K ′) as a consequence of a duality in the bosonization description of Luttinger liquids.

DOI
10.1088/1742-5468/2008/07/p07011

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We use a wavefunctional approach to calculate the fidelity of ground states in the Luttinger liquid universality class of one-dimensional gapless quantum many-body systems. The ground state wavefunctionals are discussed using both the Schrödinger (functional differential equation) formulation and a path integral formulation. The fidelity between Luttinger liquids with Luttinger parameters K and K ′ is found to decay exponentially with system size, and to obey the symmetry F ( K , K ′) = F (1/ K ,1/ K ′) as a consequence of a duality in the bosonization description of Luttinger liquids.
A general relation between quantum phase transitions and the second derivative of the fidelity (or the ``fidelity susceptibility'') is proposed. The validity and the limitation of the fidelity susceptibility in characterizing quantum phase transitions are thus established. Moreover, based on the bosonization method, general formulas of the fidelity and the fidelity susceptibility are obtained for a class of one-dimensional gapless systems known as the Tomonaga-Luttinger liquid. Applying these formulas to the one-dimensional spin-1∕21∕21∕2 XXZXXZXXZ model, we find that quantum phase transitions, even of the Beresinskii-Kosterlitz-Thouless type, can be signaled by the fidelity susceptibility.
Using eigen-functional bosonization method, we study quantum many-particle systems, and show that the quantum many-particle problems end in to solve the differential equation of the phase fields which represent the particle correlation strength. Thus, the physical properties of these systems are completely determined by the differential equation of the phase fields. We mainly focus on the study of D-dimensional electron gas with/without transverse gauge fields, two-dimensional electron gas under an external magnetic field, D-dimensional boson systems, a D-dimensional Heisenberg model and a one-band Hubbard model on a square lattice, and give their exact (accurate for Heisenberg model) functional expressions of the ground state energy and action, and the eigen-functional wave functions of the fermions/bosons. With them, we can calculate a variety of correlation functions of the systems, such as single particle Green's functions and their ground state wave functions. In present theoretical framework, we can unifiably represent the Landau Fermi liquid, non-Fermi liquid (D≥2D\geq 2D≥2) and Tomonaga-Luttinger liquid.
Gapless many-body quantum systems in one spatial dimension are universally described by the Luttinger liquid effective theory at low energies. Essentially, only two parameters enter the effective low-energy description, namely, the speed of sound and the Luttinger parameter. These are highly system dependent and their calculation requires accurate nonperturbative solutions of the many-body problem. Here we present a simple theoretical method that only uses collisional information to extract the low-energy properties of spinless one-dimensional systems. Our results are in remarkable agreement with available results for integrable models and from large-scale Monte Carlo simulations of one-dimensional helium and hydrogen isotopes. Moreover, we estimate theoretically the critical point for spinodal decomposition in one-dimensional 4He^{4}\mathrm{He}4He and show that the exponent governing the divergence of the Luttinger parameter near the critical point is exactly 1/2, in excellent agreement with Monte Carlo simulations.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.
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.
The Haldane hypothesis about the universality of the Luttinger liquid (LL) behavior in conducting one-dimensional fermion systems is checked numerically for the spinless fermion model with next-nearest-neighbor interactions. It is shown that for the large enough interactions the ground state can be gapless due to frustrations but at the same time might not belong to the same universality class as a simple LL. The exponents of the correlation functions for this unusual conducting state are found numerically by a finite-size method.