© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/0508650 · Physical Review B

Conformal properties of one-dimensional quantum systems with long-range interactions

Carlos Naón, Mariano J. Salvay, Marta L. Trobo

Published Dec 13, 2005• 4 citations

Abstract

We investigate conformal properties of a one-dimensional quantum system with a long-range, Coulomb-like potential of the form $1∕{\ensuremath{\mid}x\ensuremath{\mid}}^{\ensuremath{\sigma}}$, with $\ensuremath{\sigma}>0$. We compute the conformal anomaly $c$ as function of $\ensuremath{\sigma}$. We obtain $c=0$ for $\ensuremath{\sigma}<1$ (Wigner crystal) and $c=1$ for $\ensuremath{\sigma}>1$ (Luttinger liquid). By studying the scaling regime of the system when it is deformed by the inclusion of a term $t\ensuremath{\rho}(x)$ [with $\ensuremath{\rho}(x)$ a scaling operator], we show that, for $\ensuremath{\sigma}>1$, the correlation length $\ensuremath{\xi}$ gets an additional interaction dependent factor. This leads to a necessary redefinition of $\ensuremath{\xi}$ in order to avoid an unphysical dependence of the central charge on the coupling constant.

DOI: 10.1103/physrevb.72.245110

Related papers

Conformal properties of a periodically driven 1D quantum system

Anı́bal Iucci, Carlos Naón

In this paper, we investigate conformal properties of a one-dimensional quantum system in the presence of an oscillating, time-dependent barrier. We compute the conformal anomaly c. We consider two methods: the well-known low-temperature expansion of the free-energy, and the analysis of the scaling regime under a deformation by a scaling operator [Formula: see text], that explicitly breaks conformal symmetry. The first method allows us to show that, as a consequence of the oscillations, c becomes dependent on the corresponding amplitude. Then, by combining these two approaches, we show that the correlation length [Formula: see text] gets an additional interaction dependent factor.

Modern Physics Letters A•Apr 20, 2025•0 citations

Effects of long-range electronic interactions on a one-dimensional electron system

Sylvain Capponi, Didier Poilblanc, Thierry Giamarchi

The effects of a long-range electronic potential on a one-dimensional chain of spinless fermions are investigated by numerical techniques (exact diagonalization of rings with up to 30 sites complemented by finite size analysis) and analytic calculations. Due to a competition between the ${2k}_{F}$ oscillations in the density and the (very slow) log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) length scales. Despite some similarities, we found that this quasimetallic regime is not of the Luttinger liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth crossover to a strongly localized charge density wave.

arXiv:cond-mat/9909360•Physical review. B, Condensed matter•May 15, 2000•47 citations

Effects of long range electronic interactions on a one-dimensional electron system

Sylvain Capponi, Didier Poilblanc, Thierry Giamarchi

The effects of a long range electronic potential on a one dimensional chain of spinless fermions are investigated by numerical techniques and analytic calculations. Due to a competition between the 2kF oscillations in the density and the log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) lengthscales. We found that this quasi-metallic regime is not of the Luttinger Liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth cross-over to a strongly localized charge density wave.

Journal de Physique IV (Proceedings)•Mar 1, 2000•1 citation

Conformal Field Theory Approach to One-Dimensional Quantum Liquids

Norio Kawakami, Sung-Kil Yang

We review the critical behavior of 1D highly correlated electron systems and 1D quantum many-body systems with long-range interactions. These models exhibit the common universal long-distance properties which are characterized as Luttinger liquids. The microscopic foundation of Luttinger liquids is formulated on the basis of conformal field theory and Bethe-ansatz solutions.

Progress of Theoretical Physics Supplement•Jan 1, 1992•19 citations

Test of conformal invariance in a one-dimensional quantum liquid with long-range interactions

Rudolf A. Römer, Bill Sutherland

We study the momentum distribution of one-dimensional Bose and Fermi systems with long-range interaction g/${\mathit{r}}^{2}$ for the values g=-1/2, 0, 4, singled out by random-matrix theory. By making explicit use of the predictions of conformal invariance in periodic quantum chains of finite length, we can show that the critical exponents of the momentum distribution are independent of density and in excellent agreement with estimates obtained from c=1 conformal finite-size scaling analysis. Furthermore we show how the complete functional form of the one-particle reduced density matrix in a finite system considerably simplifies the numerical extrapolation of critical exponents for conformally invariant systems.

arXiv:cond-mat/9212021•Physical review. B, Condensed matter•Sep 1, 1993•12 citations

Long-range interactions in a one-dimensional electron system

Stefano Bellucci A

We study the unscreened Coulomb interaction in a one-dimensional electron system at low-energy. We use renormalization group methods and a GW approximation, in order to analyze the model. This yields both a strong wavefunction renormalization and a renormalization of the Fermi velocity. The significance of the effects depends on the filling level of the Fermi system. Despite the long-range character of the interaction, the system still falls into the Luttinger liquid universality class, since the effective couplings remain bounded at arbitrarily low energies.

arXiv:hep-th/9810181•Oct 22, 1998•0 citations

We investigate conformal properties of a one-dimensional quantum system with a long-range, Coulomb-like potential of the form

1∕{\ensuremath{\mid}x\ensuremath{\mid}}^{\ensuremath{\sigma}}

, with

\ensuremath{\sigma}>0

. We compute the conformal anomaly

c

as function of

\ensuremath{\sigma}

. We obtain

c=0

for

\ensuremath{\sigma}<1

(Wigner crystal) and

c=1

for

\ensuremath{\sigma}>1

(Luttinger liquid). By studying the scaling regime of the system when it is deformed by the inclusion of a term

t\ensuremath{\rho}(x)

[with

\ensuremath{\rho}(x)

a scaling operator], we show that, for

\ensuremath{\sigma}>1

, the correlation length

\ensuremath{\xi}

gets an additional interaction dependent factor. This leads to a necessary redefinition of

\ensuremath{\xi}

in order to avoid an unphysical dependence of the central charge on the coupling constant.

In this paper, we investigate conformal properties of a one-dimensional quantum system in the presence of an oscillating, time-dependent barrier. We compute the conformal anomaly c. We consider two methods: the well-known low-temperature expansion of the free-energy, and the analysis of the scaling regime under a deformation by a scaling operator [Formula: see text], that explicitly breaks conformal symmetry. The first method allows us to show that, as a consequence of the oscillations, c becomes dependent on the corresponding amplitude. Then, by combining these two approaches, we show that the correlation length [Formula: see text] gets an additional interaction dependent factor.

The effects of a long-range electronic potential on a one-dimensional chain of spinless fermions are investigated by numerical techniques (exact diagonalization of rings with up to 30 sites complemented by finite size analysis) and analytic calculations. Due to a competition between the

\ensuremath{\xi}

oscillations in the density and the (very slow) log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) length scales. Despite some similarities, we found that this quasimetallic regime is not of the Luttinger liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth crossover to a strongly localized charge density wave.

The effects of a long range electronic potential on a one dimensional chain of spinless fermions are investigated by numerical techniques and analytic calculations. Due to a competition between the 2kF oscillations in the density and the log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) lengthscales. We found that this quasi-metallic regime is not of the Luttinger Liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth cross-over to a strongly localized charge density wave.

We study the momentum distribution of one-dimensional Bose and Fermi systems with long-range interaction g/

{\mathit{r}}^{2}

for the values g=-1/2, 0, 4, singled out by random-matrix theory. By making explicit use of the predictions of conformal invariance in periodic quantum chains of finite length, we can show that the critical exponents of the momentum distribution are independent of density and in excellent agreement with estimates obtained from c=1 conformal finite-size scaling analysis. Furthermore we show how the complete functional form of the one-particle reduced density matrix in a finite system considerably simplifies the numerical extrapolation of critical exponents for conformally invariant systems.

We study the unscreened Coulomb interaction in a one-dimensional electron system at low-energy. We use renormalization group methods and a GW approximation, in order to analyze the model. This yields both a strong wavefunction renormalization and a renormalization of the Fermi velocity. The significance of the effects depends on the filling level of the Fermi system. Despite the long-range character of the interaction, the system still falls into the Luttinger liquid universality class, since the effective couplings remain bounded at arbitrarily low energies.