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Journal of Physics C Solid State Physics

One-dimensional Fermi system and plane xy model

Paul B. Wiegmann

Published Apr 28, 1978• 171 citations
Publisher

Abstract

The behaviour of correlation functions of the one-dimensional Fermi system at large distances has been investigated by reducing it to the Sine-Gordon model. The latter is dual to the low-temperature phase of a plane xy model. In this case the fermion operator is the product of the order and disorder parameters. The transitions to the coupling constants between different states of the Fermi system states of the Fermi system may be considered as phase transitions equivalent to xy models. The elasticity modulus and spin susceptibility serves as superfluid density. The question on the phase transition of the xy model is discussed.

DOI
10.1088/0022-3719/11/8/019

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The behaviour of correlation functions of the one-dimensional Fermi system at large distances has been investigated by reducing it to the Sine-Gordon model. The latter is dual to the low-temperature phase of a plane xy model. In this case the fermion operator is the product of the order and disorder parameters. The transitions to the coupling constants between different states of the Fermi system states of the Fermi system may be considered as phase transitions equivalent to xy models. The elasticity modulus and spin susceptibility serves as superfluid density. The question on the phase transition of the xy model is discussed.
We discuss the phase transitions of fermions in one dimension with a narrow Feshbach resonance described by the boson–fermion resonance model. By means of the bosonization technique, we derive a low-energy Hamiltonian of the system and show that a strongly correlated state exists, where the order parameters of the Bose condensation and superfluidity decay with the same critical exponent. We also show that density fluctuations near the Fermi wavevector are strongly suppressed as a consequence of a spin gap and a gap against the formation of phase slips. We find a Luther–Emery point where the phase slips and the spin excitations can be described in terms of pseudofermions, providing closed form expressions of the density–density correlations and the spectral functions. The relevance of our results for experiments with ultracold atomic gases subject to one-dimensional confinement is also discussed.
The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.
The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.
The authors study a one dimensional model system of fermions which proves to be exactly soluble in the thermodynamic limit. It is shown that the thermodynamic functions for the model system are the same as for a system of noninteracting fermions in a harmonic oscillator well. As a special case of the model system, with a particular choice for the coupling constant, it reduces to a particular two-species system of fermions interacting via pairwise harmonic forces. For this case of the two-species system the free energy per particle, the pressure the specific heat at constant volume, and the isothermal compressibility are calculated in the canonical ensemble in closed form. It is shown that this system does not undergo a phase transition.
The intent of my project is to determine if the proposal of sliding phases in XY layered systems has physical ground. It will be done by comparing numerical and analytical results for a layered XY models. Sliding phases were first proposed in the context of DNA complexes and then extended to XY models, 1D coupled wires and superfluid films. The existence of the sliding phase would mean that there is a phase transition from 3D to 2D behavior. Such systems have been studied both in the clean case and with disorder. The idea of the sliding phases is based on applying the criterion of relevancy of Josephson coupling between layers along the line of the Sine-Gordon model. Such a criterion is, strictly speaking, valid for a non-compact variable while the XY model is defined for compact one. Thus, it is not clear if this approximation has any validity. Our analysis of the numerical data obtained within Monte Carlo simulations of a model showed that the scaling behavior of the Josephson coupling disagrees qualitatively (and quantitatively) with the flow equation derived from the renormalization group (RG) prediction. Instead of the 3D to 2D phase transition predicted by the RG we see the behavior consistent with the system retaining its 3D character. Our approach is based on the language of duality. It allows reformulating the model in terms of the objects amenable to effective large scale simulations. Furthermore, using this language it becomes possible to formulate a general criterion for the sliding phases to exist – which conspicuously demonstrates that the RG approach is incorrect when applied to the situations of the dimensional reduction. The dual approach allows finding the asymptotically exact analytical solution of the model for clean and disordered cases. It is also suggested how to achieve sliding phases – by introducing effective gauge type interaction between Josephson currents mediated by some soft mode which does not affect currents along the planes.
The spin-correlation functions of the one-dimensional XYXYXY model are studied in the presence of a constant magnetic field. We find that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.