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arXiv:0811.2419 · Europhysics Letters (EPL)

Large-distance asymptotic behavior of the correlation functions of 1D impenetrable anyons at finite temperatures

O. I. Pâţu, V. E. Korepin, D. V. Averin

Published May 1, 2009• 21 citations
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Abstract

The large-distance asymptotic behavior of the field-field correlators has been computed for one-dimensional impenetrable anyons at finite temperatures. The asymptotic behavior agrees with the predictions of conformal field theory at low temperatures and reproduces the known results for impenetrable bosons and free fermions in appropriate limits. We have also obtained an integrable system of partial nonlinear differential equations which completely characterizes the 2-point correlation functions. The system is the same as for bosons but with different initial conditions.

DOI
10.1209/0295-5075/86/40001

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The large-distance asymptotic behavior of the field-field correlators has been computed for one-dimensional impenetrable anyons at finite temperatures. The asymptotic behavior agrees with the predictions of conformal field theory at low temperatures and reproduces the known results for impenetrable bosons and free fermions in appropriate limits. We have also obtained an integrable system of partial nonlinear differential equations which completely characterizes the 2-point correlation functions. The system is the same as for bosons but with different initial conditions.
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This work presents the derivation of the large time and distance asymptotic behavior of the field-field correlation functions of impenetrable one-dimensional anyons at finite temperature. In the appropriate limits of the statistics parameter, we recover the well-known results for impenetrable bosons and free fermions. In the low-temperature (usually expected to be the "conformal") limit, and for all values of the statistics parameter away from the bosonic point, the leading term in the correlator does not agree with the prediction of the conformal field theory, and is determined by the singularity of the density of the single-particle states at the bottom of the single-particle energy spectrum.
The exact large time and distance behavior of the field-field correlators has been computed for one-dimensional impenetrable anyons at finite temperatures. The result reproduces known asymptotics for impenetrable bosons and free fermions in the appropriate limits of the statistics parameter. The obtained asymptotic behavior of the correlators is dominated by the singularity in the spectral density of the quasiparticle states at the bottom of the band, and differs from the predictions of the conformal field theory. One can argue, however, that the anyonic response to the low-energy probes is still determined by the subleading conformal terms in the asymptotic expansion.
We have investigated the properties of a model of 1D anyons interacting through a δ-function repulsive potential. The structure of the quasi-periodic boundary conditions for the anyonic field operators and the many-anyon wavefunctions is clarified. The spectrum of the low-lying excitations including the particle–hole excitations is calculated for periodic and twisted boundary conditions. Using the ideas of the conformal field theory we obtain the large-distance asymptotics of the density and field correlation function at the critical temperature T = 0 and at small finite temperatures. Our expression for the field correlation function extends the results in the literature obtained for harmonic quantum anyonic fluids.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.