Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Fractional Charge and Correlation Exponents for Interacting Electrons in One Dimension

Eugene B. Kolomeisky, Joseph P. Straley

Published Jun 12, 1995• 5 citations
Publisher

Abstract

Close to the transition between incommensurate conductor and commensurate insulator, charge transport in a system of one-dimensional interacting spin-1/2 fermions moving in a periodic potential is done by solitons (i.e., the few extra particles or holes) having a charge $Q$ which is a fraction of the elementary charge $e$. We show that $Q$ is directly related to the limiting value at commensuration of the exponent ${g}_{c}^{*}$ of the charge-charge correlation function: ${g}_{c}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(Q/e{)}^{2}$. We give some simple rules for determining both charge and spin correlation exponents and point out their dependence on the parity of the order of the underlying umklapp scattering.

DOI
10.1103/physrevlett.74.4891

Related papers

Phase diagram and correlation exponents for interacting fermions in one dimension

Eugene B. Kolomeisky, Joseph P. Straley

A theory is given for the ground-state properties of short-ranged interacting spin-\textonehalf{} fermions moving in a periodic potential, based on a combination of renormalization-group ideas, the harmonic (Luttinger) liquid approach, and the symmetry properties of the Hubbard model, for a wide range of interactions and arbitrary fillings. For rational filling factors the outcome depends nontrivially on the parity of the order of the $2{k}_{F}$ umklapp scattering: even values fall into a universality class characterized by a separation of charge and spin degrees of freedom in the long-wavelength limit, while the action for the case of odd orders does not have this property, thus leading to distinct phase diagrams. It is argued that even when the long-wavelength action does exhibit a spin-charge separation it is not always an accurate approximation, particularly for the case of strong repulsion between electrons of the same spin. This leads to a classification of the possible ground states, associated correlation exponents, and phase transitions. The translational correlation exponents upon approaching insulator phases (existing at exact commensuration) by changing electron density can be related to the fractional charge of the solitons, using the Landauer rule.

Reviews of Modern Physics•Jan 1, 1996•48 citations

The metal-insulator transition in one dimension

Heinz J. Schulz

The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities $u_ρ$ and $u_σ$ of the charge and spin excitations, as well as by a parameter $K_ρ$ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.

arXiv:cond-mat/9412036•Dec 7, 1994•7 citations

Interaction, disorder, and commensurability in one and nearly one dimension

Eugene B. Kolomeisky

The interplay between Anderson localization and Mott localization is considered in the vicinity of one dimension for a collection of interacting spinless fermions or repulsive bosons. We demonstrate that in the absence of commensurability effects an Anderson localization transition does take place for d\ensuremath{\le}1. It is characterized by a critical dynamical exponent z having value z=2-d for d\ensuremath{\rightarrow}1-0, in disagreement with the previous claims that z always equals d. We find that there is only a localized phase for d>1 and expect that this similarity between free fermions and interacting quantum particles will be valid up to two dimensions. We also consider the competition between the effects of commensurability, disorder, and interaction in exactly one dimension. The outcome strongly depends on the order of umklapp scattering responsible for the formation of the Mott insulator in the pure case. If it is \ensuremath{\Delta}k=2${\mathit{k}}_{\mathit{F}}$ scattering and the disorder is sufficiently weak, the Mott phase survives and the Anderson phase intervenes almost everywhere between the Mott insulator and the conductor, except at one special symmetry-enhanced point corresponding to the fixed commensurate filling: here, as in the pure case, the Mott transition belongs to the Kosterlitz-Thouless universality class. For sufficiently strong disorder, or for any disorder and higher-order scattering, the path from the Mott insulator phase (if it survives) to the conductor always crosses the Anderson phase.

Physical review. B, Condensed matter•Aug 15, 1993•4 citations

Fractional Charge in Transport through a 1D Correlated Insulator of Finite Length

Vadim V. Ponomarenko, Naoto Nagaosa

Transport between two Fermi liquid reservoirs through a one channel wire of length $L$ is examined when the 1D electron system has an energy gap $2M\ensuremath{\gg}{v}_{c}/L$ ( ${v}_{c}$ is the charge velocity in the wire). In the spinless case transformation of reservoir electrons into solitons of fractional charge entails a crossover from a Fermi liquid regime of integer charge transfer below the crossover energy ${T}_{x}\ensuremath{\ll}{T}_{L}$ to an insulating regime of the fractional charge transfer in the current vs voltage, conductance vs temperature, and the shot noise. Similar behavior is predicted for a spin Mott insulator of filling factor $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n/({2m}^{\ensuremath{'}})$ ( ${n,m}^{\ensuremath{'}}$ are positive integers).

arXiv:cond-mat/9806136•Physical Review Letters•Aug 30, 1999•10 citations

Correlation functions of one-dimensional strongly interacting two-component gases

Ovidiu I. Pâţu

We address the problem of calculating the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions. The model considered in this paper describes particles with fractional statistics and in appropriate limits reduces to the Gaudin-Yang model or the spinor Bose gas. In the case of impenetrable particles we derive a Fredholm determinant representation for the temperature-, time-, and space-dependent correlation functions which is very easy to implement numerically and constitutes the starting point for the analytical investigation of the asymptotics. Making use of this determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter, while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry. Due to the exponential decay the momentum distribution $n(k)$ at zero temperature does not present algebraic singularities but the tails obey the universal decay ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{4}$ with the amplitude $C$ given by Tan's contact. As a function of the statistics parameter the contact is a monotonic function reaching its minimum for the fermionic system and the maximum for the bosonic system.

arXiv:1910.10042•Physical Review A•Dec 23, 2019•22 citations

Interaction-range effects for fermions in one dimension

Martin Hohenadler, Stefan Weßel, Maria Daghofer +1 more

Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter ${K}_{\ensuremath{\rho}}$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at ${K}_{\ensuremath{\rho}}=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant $2{k}_{\mathrm{F}}$ spin correlations to dominant $4{k}_{\mathrm{F}}$ charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.

arXiv:1201.3626•Physical Review B•May 9, 2012•30 citations
Close to the transition between incommensurate conductor and commensurate insulator, charge transport in a system of one-dimensional interacting spin-1/2 fermions moving in a periodic potential is done by solitons (i.e., the few extra particles or holes) having a charge QQQ which is a fraction of the elementary charge eee. We show that QQQ is directly related to the limiting value at commensuration of the exponent gc∗{g}_{c}^{*}gc∗​ of the charge-charge correlation function: gc∗=(Q/e)2{g}_{c}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(Q/e{)}^{2}gc∗​=(Q/e)2. We give some simple rules for determining both charge and spin correlation exponents and point out their dependence on the parity of the order of the underlying umklapp scattering.
A theory is given for the ground-state properties of short-ranged interacting spin-\textonehalf{} fermions moving in a periodic potential, based on a combination of renormalization-group ideas, the harmonic (Luttinger) liquid approach, and the symmetry properties of the Hubbard model, for a wide range of interactions and arbitrary fillings. For rational filling factors the outcome depends nontrivially on the parity of the order of the 2kF2{k}_{F}2kF​ umklapp scattering: even values fall into a universality class characterized by a separation of charge and spin degrees of freedom in the long-wavelength limit, while the action for the case of odd orders does not have this property, thus leading to distinct phase diagrams. It is argued that even when the long-wavelength action does exhibit a spin-charge separation it is not always an accurate approximation, particularly for the case of strong repulsion between electrons of the same spin. This leads to a classification of the possible ground states, associated correlation exponents, and phase transitions. The translational correlation exponents upon approaching insulator phases (existing at exact commensuration) by changing electron density can be related to the fractional charge of the solitons, using the Landauer rule.
The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities uρu_ρuρ​ and uσu_σuσ​ of the charge and spin excitations, as well as by a parameter KρK_ρKρ​ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.
The interplay between Anderson localization and Mott localization is considered in the vicinity of one dimension for a collection of interacting spinless fermions or repulsive bosons. We demonstrate that in the absence of commensurability effects an Anderson localization transition does take place for d\ensuremath{\le}1. It is characterized by a critical dynamical exponent z having value z=2-d for d\ensuremath{\rightarrow}1-0, in disagreement with the previous claims that z always equals d. We find that there is only a localized phase for d>1 and expect that this similarity between free fermions and interacting quantum particles will be valid up to two dimensions. We also consider the competition between the effects of commensurability, disorder, and interaction in exactly one dimension. The outcome strongly depends on the order of umklapp scattering responsible for the formation of the Mott insulator in the pure case. If it is \ensuremath{\Delta}k=2kF{\mathit{k}}_{\mathit{F}}kF​ scattering and the disorder is sufficiently weak, the Mott phase survives and the Anderson phase intervenes almost everywhere between the Mott insulator and the conductor, except at one special symmetry-enhanced point corresponding to the fixed commensurate filling: here, as in the pure case, the Mott transition belongs to the Kosterlitz-Thouless universality class. For sufficiently strong disorder, or for any disorder and higher-order scattering, the path from the Mott insulator phase (if it survives) to the conductor always crosses the Anderson phase.
Transport between two Fermi liquid reservoirs through a one channel wire of length LLL is examined when the 1D electron system has an energy gap $2M\ensuremath{\gg}{v}_{c}/L$ ( vc{v}_{c}vc​ is the charge velocity in the wire). In the spinless case transformation of reservoir electrons into solitons of fractional charge entails a crossover from a Fermi liquid regime of integer charge transfer below the crossover energy ${T}_{x}\ensuremath{\ll}{T}_{L}$ to an insulating regime of the fractional charge transfer in the current vs voltage, conductance vs temperature, and the shot noise. Similar behavior is predicted for a spin Mott insulator of filling factor $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n/({2m}^{\ensuremath{'}})$ ( ${n,m}^{\ensuremath{'}}$ are positive integers).
We address the problem of calculating the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions. The model considered in this paper describes particles with fractional statistics and in appropriate limits reduces to the Gaudin-Yang model or the spinor Bose gas. In the case of impenetrable particles we derive a Fredholm determinant representation for the temperature-, time-, and space-dependent correlation functions which is very easy to implement numerically and constitutes the starting point for the analytical investigation of the asymptotics. Making use of this determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter, while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry. Due to the exponential decay the momentum distribution n(k)n(k)n(k) at zero temperature does not present algebraic singularities but the tails obey the universal decay ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{4}$ with the amplitude CCC given by Tan's contact. As a function of the statistics parameter the contact is a monotonic function reaching its minimum for the fermionic system and the maximum for the bosonic system.
Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter ${K}_{\ensuremath{\rho}}$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at ${K}_{\ensuremath{\rho}}=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant 2kF2{k}_{\mathrm{F}}2kF​ spin correlations to dominant 4kF4{k}_{\mathrm{F}}4kF​ charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.