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

arXiv:1605.00275 · Physical review. B./Physical review. B

Comment on “Galilean invariance at quantum Hall edge”

Judith Hoeller, N. Read

Published May 23, 2016• 5 citations
arXivHTMLPDF

Abstract

In a recent paper by S. Moroz, C. Hoyos, and L. Radzihovsky [Phys. Rev. B 91, 195409 (2015)], it is claimed that the conductivity at low frequency $\ensuremath{\omega}$ and small wave vector $q$ along the edge of a quantum Hall system (that possesses Galilean invariance along the edge) contains a universal contribution of order ${q}^{2}$ that is determined by the orbital spin per particle in the bulk of the system, or alternatively by the shift of the ground state. (These quantities are known to be related to the Hall viscosity of the bulk.) In this Comment we calculate the real part of the conductivity, integrated over $\ensuremath{\omega}$, in this regime for the edge of a system of noninteracting electrons filling either the lowest, or the lowest $\ensuremath{\nu} (\ensuremath{\nu}=1,\phantom{\rule{0.16em}{0ex}}2,...)$, Landau level(s), and show that the ${q}^{2}$ term is nonuniversal and depends on details of the confining potential at the edge. In the special case of a linear potential, a form similar to the prediction is obtained; it is possible that this corrected form of the prediction may also hold for fractional quantum Hall states in systems with special forms of interactions between electrons.

DOI
10.1103/physrevb.93.197401

Related papers

Thermoelectric Hall conductivity of fractional quantum Hall systems on a disk

Zi-Yi Fang, Dan Ye, Yuyu Zhang +1 more

For fractional quantum Hall states on a finite disk, we study the thermoelectric transport properties under the influence of the presence of an edge and its reconstruction. In a recent study on a torus [Phys. Rev. B 101, 241101(R) (2020)], Sheng and Fu found a universal non-Fermi liquid power-law scaling of the thermoelectric conductivity ${\ensuremath{\alpha}}_{xy}\ensuremath{\propto}{T}^{\ensuremath{\eta}}$ for the gapless composite Fermi liquid state. The exponent $\ensuremath{\eta}\ensuremath{\sim}0.5$ appears an independence of the filling factors and the details of the interactions. In the presence of an edge, we find the properties of the edge spectrum dominants the low-temperature behaviors and a nonuniversal scaling behavior in low temperature is observed. To consider individually the effect of the edge states, the real space entanglement spectrum of the model wave function, which contains only the spectrum of edge excitation is employed. Its spectrum energies could be tuned by varying the radius of the subsystem. In the non-Abelian Moore-Read state, the Majorana neutral edge mode is found to have more significant effects than that of the charge mode in the low temperature.

arXiv:2102.00934•Physical review. B./Physical review. B•Jun 25, 2021•3 citations

Bulk-boundary correspondence in the quantum Hall effect

Andrea Cappelli A, Lorenzo Maffi

Abstract We present a detailed microscopic study of edge excitations for n filled Landau levels. We show that the higher-level wavefunctions possess a non-trivial radial dependence that should be integrated over for properly defining the edge conformal field theory. This analysis let’s us clarify the role of the electron orbital spin s in the edge theory and to discuss its universality, thus providing a further instance of the bulk-boundary correspondence. We find that the values s i for each level, parameterize a Casimir effect or chemical potential shift that could be experimentally observed. These results are generalized to fractional and hierarchical fillings by exploiting the W-infinity symmetry of incompressible Hall fluids.

arXiv:1801.03759•Journal of Physics A Mathematical and Theoretical•Jul 3, 2018•13 citations

Magnetization of quantum Hall systems

R. Haussmann

The magnetic properties of a two-dimensional interacting electron system in a strong magnetic field are considered in the quantum Hall regime, where the motion of the electrons is restricted to the lowest Landau level. We apply an approach based on the many-particle quantum-field theory which we developed recently for the spin-polarized system in the fractional quantum Hall regime, and which treats the problem of the degeneracy of a partially filled Landau level properly. Via a gauge transformation of the fermion Green's function, we achieve a dressing of the propagator with multiple collective excitations. Our theory allows a calculation of the magnetic polarization $M(T)$ for higher temperatures $T$, where $M(T)\ensuremath{\lesssim}0.4$ is not too large. For filling factor $\ensuremath{\nu}=1$ we obtain a magnetic polarization $M(T)$ which agrees quite well with the data of a NMR experiment by Barrett [Phys. Rev. Lett. 74, 5112 (1995)].

Physical review. B, Condensed matter•Oct 15, 1997•18 citations

Low-temperature conductivity in a narrow half-filled Hubbard chain

Jean-Pierre Gallinar

In this paper, the low-temperature (${k}_{B}T\ensuremath{\ll}U$) frequency-dependent $\ensuremath{\sigma}(\ensuremath{\omega})$ conductivity in a half-filled Hubbard chain is calculated to the lowest order in the parameter $\frac{t}{U}$ and in closed form to all orders in $\frac{t}{\ensuremath{\omega}}\ensuremath{-}U$. Here $t$ is the nearest-neighbor hopping integral and $U$ the intra-atomic Coulomb interaction. The problem is formulated in terms of the number of random walks of an electron and a hole which leave the spin configuration unchanged, at temperatures such that ${k}_{B}T\ensuremath{\ll}t\ensuremath{\ll}U$ (where ${k}_{B}$ is Boltzmann's constant). The conductivity then depends on spin configuration, and for both the antiferromagnetic (AF) ($T=0$ K) ground state and the equiprobable or random arrangement ($R$) (${k}_{B}T\ensuremath{\gg}\frac{{t}^{2}}{U}$) an exact expression is found. For the partially disordered regime ($T\ensuremath{\simeq}{T}_{N}$) the thermodynamical averaging is expressed in terms of a phenomenological spin-spin correlation length $l$ which interpolates between the AF and $R$ exact limits. This solution is such that for $l=\ensuremath{\infty}$ the AF line shape is recovered except at the center of the spectrum (i.e., at $\ensuremath{\omega}=U$), where the absorption always diverges logarithmically. For a ferromagnetic ($l=0$) arrangement induced by a magnetic field, the absorption then vanishes. The line shapes obtained have sharp edges at $|\ensuremath{\omega}|=U\ifmmode\pm\else\textpm\fi{}4t$ for all temperatures. At the critical value of $4t=U$ the AF ground state is found to undergo a metallic transition. A closed-form analytical expression in terms of the three elliptical integrals is given for the absorption lines. Except for the AF case, which has a square-root edge singularity, the absorption is found to vanish linearly. Finally, a comparison is made with some recent calculations.

Physical review. B, Condensed matter•Jan 15, 1978•6 citations

Charge and current in the quantum Hall matrix model

T. H. Hansson, Janik Kailasvuori, A. Karlhede

We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even in the classical model. We calculate the linear response at low momenta---this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. In the quantum theory the ordering ambiguities are more severe, but we are able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra, for the special case of $2\ifmmode\times\else\texttimes\fi{}2$ matrices. Our results support the conjecture that the quantum Hall matrix models do describe important aspects of the dynamics of electrons in the lowest Landau level.

arXiv:cond-mat/0304271•Physical review. B, Condensed matter•Jul 29, 2003•19 citations

Quantum Hall system in Tao-Thouless limit

Emil J. Bergholtz, A. Karlhede

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where ${L}_{1}$ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd $q$, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd $q$, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as ${L}_{1}$ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite ${L}_{1}$; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.

arXiv:0712.1927•Physical Review B•Apr 8, 2008•143 citations
In a recent paper by S. Moroz, C. Hoyos, and L. Radzihovsky [Phys. Rev. B 91, 195409 (2015)], it is claimed that the conductivity at low frequency \ensuremathω\ensuremath{\omega}\ensuremathω and small wave vector qqq along the edge of a quantum Hall system (that possesses Galilean invariance along the edge) contains a universal contribution of order q2{q}^{2}q2 that is determined by the orbital spin per particle in the bulk of the system, or alternatively by the shift of the ground state. (These quantities are known to be related to the Hall viscosity of the bulk.) In this Comment we calculate the real part of the conductivity, integrated over \ensuremathω\ensuremath{\omega}\ensuremathω, in this regime for the edge of a system of noninteracting electrons filling either the lowest, or the lowest \ensuremathν(\ensuremathν=1,2,...)\ensuremath{\nu} (\ensuremath{\nu}=1,\phantom{\rule{0.16em}{0ex}}2,...)\ensuremathν(\ensuremathν=1,2,...), Landau level(s), and show that the q2{q}^{2}q2 term is nonuniversal and depends on details of the confining potential at the edge. In the special case of a linear potential, a form similar to the prediction is obtained; it is possible that this corrected form of the prediction may also hold for fractional quantum Hall states in systems with special forms of interactions between electrons.
For fractional quantum Hall states on a finite disk, we study the thermoelectric transport properties under the influence of the presence of an edge and its reconstruction. In a recent study on a torus [Phys. Rev. B 101, 241101(R) (2020)], Sheng and Fu found a universal non-Fermi liquid power-law scaling of the thermoelectric conductivity ${\ensuremath{\alpha}}_{xy}\ensuremath{\propto}{T}^{\ensuremath{\eta}}$ for the gapless composite Fermi liquid state. The exponent $\ensuremath{\eta}\ensuremath{\sim}0.5$ appears an independence of the filling factors and the details of the interactions. In the presence of an edge, we find the properties of the edge spectrum dominants the low-temperature behaviors and a nonuniversal scaling behavior in low temperature is observed. To consider individually the effect of the edge states, the real space entanglement spectrum of the model wave function, which contains only the spectrum of edge excitation is employed. Its spectrum energies could be tuned by varying the radius of the subsystem. In the non-Abelian Moore-Read state, the Majorana neutral edge mode is found to have more significant effects than that of the charge mode in the low temperature.
Abstract We present a detailed microscopic study of edge excitations for n filled Landau levels. We show that the higher-level wavefunctions possess a non-trivial radial dependence that should be integrated over for properly defining the edge conformal field theory. This analysis let’s us clarify the role of the electron orbital spin s in the edge theory and to discuss its universality, thus providing a further instance of the bulk-boundary correspondence. We find that the values s i for each level, parameterize a Casimir effect or chemical potential shift that could be experimentally observed. These results are generalized to fractional and hierarchical fillings by exploiting the W-infinity symmetry of incompressible Hall fluids.
The magnetic properties of a two-dimensional interacting electron system in a strong magnetic field are considered in the quantum Hall regime, where the motion of the electrons is restricted to the lowest Landau level. We apply an approach based on the many-particle quantum-field theory which we developed recently for the spin-polarized system in the fractional quantum Hall regime, and which treats the problem of the degeneracy of a partially filled Landau level properly. Via a gauge transformation of the fermion Green's function, we achieve a dressing of the propagator with multiple collective excitations. Our theory allows a calculation of the magnetic polarization M(T)M(T)M(T) for higher temperatures TTT, where $M(T)\ensuremath{\lesssim}0.4$ is not too large. For filling factor $\ensuremath{\nu}=1$ we obtain a magnetic polarization M(T)M(T)M(T) which agrees quite well with the data of a NMR experiment by Barrett [Phys. Rev. Lett. 74, 5112 (1995)].
In this paper, the low-temperature (${k}_{B}T\ensuremath{\ll}U$) frequency-dependent $\ensuremath{\sigma}(\ensuremath{\omega})$ conductivity in a half-filled Hubbard chain is calculated to the lowest order in the parameter tU\frac{t}{U}Ut​ and in closed form to all orders in $\frac{t}{\ensuremath{\omega}}\ensuremath{-}U$. Here ttt is the nearest-neighbor hopping integral and UUU the intra-atomic Coulomb interaction. The problem is formulated in terms of the number of random walks of an electron and a hole which leave the spin configuration unchanged, at temperatures such that ${k}_{B}T\ensuremath{\ll}t\ensuremath{\ll}U$ (where kB{k}_{B}kB​ is Boltzmann's constant). The conductivity then depends on spin configuration, and for both the antiferromagnetic (AF) (T=0T=0T=0 K) ground state and the equiprobable or random arrangement (RRR) (${k}_{B}T\ensuremath{\gg}\frac{{t}^{2}}{U}$) an exact expression is found. For the partially disordered regime ($T\ensuremath{\simeq}{T}_{N}$) the thermodynamical averaging is expressed in terms of a phenomenological spin-spin correlation length lll which interpolates between the AF and RRR exact limits. This solution is such that for $l=\ensuremath{\infty}$ the AF line shape is recovered except at the center of the spectrum (i.e., at $\ensuremath{\omega}=U$), where the absorption always diverges logarithmically. For a ferromagnetic (l=0l=0l=0) arrangement induced by a magnetic field, the absorption then vanishes. The line shapes obtained have sharp edges at $|\ensuremath{\omega}|=U\ifmmode\pm\else\textpm\fi{}4t$ for all temperatures. At the critical value of 4t=U4t=U4t=U the AF ground state is found to undergo a metallic transition. A closed-form analytical expression in terms of the three elliptical integrals is given for the absorption lines. Except for the AF case, which has a square-root edge singularity, the absorption is found to vanish linearly. Finally, a comparison is made with some recent calculations.
We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even in the classical model. We calculate the linear response at low momenta---this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. In the quantum theory the ordering ambiguities are more severe, but we are able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra, for the special case of $2\ifmmode\times\else\texttimes\fi{}2$ matrices. Our results support the conjecture that the quantum Hall matrix models do describe important aspects of the dynamics of electrons in the lowest Landau level.
We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where L1{L}_{1}L1​ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd qqq, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd qqq, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1{L}_{1}L1​ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1{L}_{1}L1​; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.