Physical review. B./Physical review. B

Correlation and exchange energies of a quantum Hall ferromagnet with ν = 1 3 in the strong interaction regime

P. S. Berezhnoy, A. S. Koreev, A. B. Van'kov, Игорь Владимирович Кукушкин

Published Dec 13, 2024• 1 citation

Abstract

Collective spin excitations were utilized to probe many-particle energies of strongly correlated quantum Hall ferromagnets (FMs) at $\frac{1}{3}\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}1$. For this purpose, experiments on inelastic light scattering and polarized photoluminescence (PL) were performed on a MgZnO/ZnO heterostructure, containing a high-mobility two-dimensional electron system with the Wigner-Seitz parameter ${r}_{s}\ensuremath{\approx}11$. The dispersion of the Larmor spin exciton was utilized to extract the spin stiffness, which was found to have maxima at $\ensuremath{\nu}=\frac{1}{3}, \ensuremath{\nu}=1$, and to be strongly suppressed at intermediate filling factors. Measurements of circularly polarized PL helped to confirm a FM configuration of the system. In the close vicinity of $\ensuremath{\nu}=\frac{1}{3}$, a hybrid spin-density excitation was discovered, the energy of which enables us to probe the correlation energy of the Laughlin liquid. The measured correlation and exchange energies at $\ensuremath{\nu}=\frac{1}{3}$ and ${r}_{s}\ensuremath{\gg}1$ turned out to be much less suppressed by the Landau level mixing than the exchange energy at integer filling factors. The same answer follows from the calculations performed using the exact diagonalization method.

DOI: 10.1103/physrevb.110.235306

Related papers

Spin stiffness of a Fermi liquid in the ν=1 quantum Hall regime

A. B. Van'kov, Игорь Владимирович Кукушкин

Physical review. B./Physical review. BDec 21, 20209 citations

Variational quantum Monte Carlo study of two-dimensional Wigner crystals: Exchange, correlation, and magnetic-field effects

Xuejun Zhu, Steven G. Louie

arXiv:cond-mat/9505074Physical review. B, Condensed matterAug 15, 199558 citations

Variational Monte Carlo calculation of the spin gap in the ν=1 quantum Hall liquid

Balázs Králik, Andrew M. Rappe, Steven G. Louie

Physical review. B, Condensed matterAug 15, 19976 citations

Dispersion of collective modes in spinful fractional quantum Hall states on the sphere

Rakesh K. Dora, Ajit Coimbatore Balram

arXiv:2509.13100Physical review. B./Physical review. BFeb 3, 20261 citation

STRONGLY CORRELATED PHASES IN QUANTUM HALL SYSTEMS

Amartya Saha

Aug 10, 20210 citations

Laughlin-Jastrow-correlated Wigner crystal in a strong magnetic field

Hangmo Yi, H. A. Fertig

arXiv:cond-mat/9803098Physical review. B, Condensed matterAug 15, 199865 citations

Physical review. B./Physical review. B

Correlation and exchange energies of a quantum Hall ferromagnet with ν = 1 3 in the strong interaction regime

P. S. Berezhnoy, A. S. Koreev, A. B. Van'kov, Игорь Владимирович Кукушкин

Published Dec 13, 2024• 1 citation

Publisher

Abstract

Collective spin excitations were utilized to probe many-particle energies of strongly correlated quantum Hall ferromagnets (FMs) at

\frac{1}{3}\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}1

. For this purpose, experiments on inelastic light scattering and polarized photoluminescence (PL) were performed on a MgZnO/ZnO heterostructure, containing a high-mobility two-dimensional electron system with the Wigner-Seitz parameter

{r}_{s}\ensuremath{\approx}11

. The dispersion of the Larmor spin exciton was utilized to extract the spin stiffness, which was found to have maxima at

\ensuremath{\nu}=\frac{1}{3}, \ensuremath{\nu}=1

, and to be strongly suppressed at intermediate filling factors. Measurements of circularly polarized PL helped to confirm a FM configuration of the system. In the close vicinity of

\ensuremath{\nu}=\frac{1}{3}

, a hybrid spin-density excitation was discovered, the energy of which enables us to probe the correlation energy of the Laughlin liquid. The measured correlation and exchange energies at

\ensuremath{\nu}=\frac{1}{3}

and

{r}_{s}\ensuremath{\gg}1

turned out to be much less suppressed by the Landau level mixing than the exchange energy at integer filling factors. The same answer follows from the calculations performed using the exact diagonalization method.

DOI: 10.1103/physrevb.110.235306

Related papers

Spin stiffness of a Fermi liquid in the ν=1 quantum Hall regime

A. B. Van'kov, Игорь Владимирович Кукушкин

Anomalous behavior of spin stiffness is revealed in ZnO-based two-dimensional electron systems (2DESs) with strong Coulomb interaction and Wigner-Seitz parameter ${r}_{s}>6$ . The spin stiffness is extracted directly from the quadratic $k$ dispersion of spin excitons at $\ensuremath{\nu}=1$ probed by inelastic light scattering. The resulting values are found to be dramatically rescaled compared to the case of weakly interacting 2DESs---spin stiffness turned out to be of the order of the cyclotron energy with the effective mass of Fermi-liquid quasiparticles. This result is also confirmed by the exact diagonalization simulations.Anomalous behavior of spin stiffness is revealed in ZnO-based two-dimensional electron systems (2DESs) with strong Coulomb interaction and Wigner-Seitz parameter ${r}_{s}>6$. The spin stiffness is extracted directly from the quadratic $k$ dispersion of spin excitons at $\ensuremath{\nu}=1$ probed by inelastic light scattering. The resulting values are found to be dramatically rescaled compared to the case of weakly interacting 2DESs---spin stiffness turned out to be of the order of the cyclotron energy with the effective mass of Fermi-liquid quasiparticles. This result is also confirmed by the exact diagonalization simulations.

Physical review. B./Physical review. BDec 21, 20209 citations

Variational quantum Monte Carlo study of two-dimensional Wigner crystals: Exchange, correlation, and magnetic-field effects

Xuejun Zhu, Steven G. Louie

The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/ ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.

arXiv:cond-mat/9505074Physical review. B, Condensed matterAug 15, 199558 citations

Variational Monte Carlo calculation of the spin gap in the ν=1 quantum Hall liquid

Balázs Králik, Andrew M. Rappe, Steven G. Louie

A variational method of calculating the spin 1 excitation energies of the $\ensuremath{\nu}=1$ quantum Hall liquid is presented in detail. We are able to include Landau-level mixing and finite-thickness effects. The many-body wave functions of the states near $\ensuremath{\nu}=1$ filling are taken to be a product of a lowest-Landau-level Laughlin-type wave function and a Jastrow factor. A significant reduction in the calculated spin gap is found due both to Landau-level mixing $(\ensuremath{\sim}40%$ at ${r}_{s}=1)$ and to the effect of the finite thickness of the liquid (a further $\ensuremath{\sim}30%$ at ${r}_{s}=1)$ . We find our calculations to be consistent with recent photoluminescence experiments in $\ensuremath{\delta}$-doped samples.A variational method of calculating the spin 1 excitation energies of the $\ensuremath{\nu}=1$ quantum Hall liquid is presented in detail. We are able to include Landau-level mixing and finite-thickness effects. The many-body wave functions of the states near $\ensuremath{\nu}=1$ filling are taken to be a product of a lowest-Landau-level Laughlin-type wave function and a Jastrow factor. A significant reduction in the calculated spin gap is found due both to Landau-level mixing $(\ensuremath{\sim}40%$ at ${r}_{s}=1)$ and to the effect of the finite thickness of the liquid (a further $\ensuremath{\sim}30%$ at ${r}_{s}=1)$. We find our calculations to be consistent with recent photoluminescence experiments in $\ensuremath{\delta}$-doped samples.

Physical review. B, Condensed matterAug 15, 19976 citations

Dispersion of collective modes in spinful fractional quantum Hall states on the sphere

Rakesh K. Dora, Ajit Coimbatore Balram

Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings $ν{=}n/(2n{\pm}1)$ and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at $ν{=}n/(4n{\pm}1)$ . We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at $ν{=}2/5$ . The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings $ν{=}n/(2n{\pm}1)$ and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at $ν{=}n/(4n{\pm}1)$. We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at $ν{=}2/5$. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.

arXiv:2509.13100Physical review. B./Physical review. BFeb 3, 20261 citation

STRONGLY CORRELATED PHASES IN QUANTUM HALL SYSTEMS

Amartya Saha

Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.

Aug 10, 20210 citations

Laughlin-Jastrow-correlated Wigner crystal in a strong magnetic field

Hangmo Yi, H. A. Fertig

We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power $m$ of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power $m$ of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.

arXiv:cond-mat/9803098Physical review. B, Condensed matterAug 15, 199865 citations

Collective spin excitations were utilized to probe many-particle energies of strongly correlated quantum Hall ferromagnets (FMs) at

\frac{1}{3}\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}1

{r}_{s}\ensuremath{\approx}11

. The dispersion of the Larmor spin exciton was utilized to extract the spin stiffness, which was found to have maxima at

\ensuremath{\nu}=\frac{1}{3}, \ensuremath{\nu}=1

, and to be strongly suppressed at intermediate filling factors. Measurements of circularly polarized PL helped to confirm a FM configuration of the system. In the close vicinity of

\ensuremath{\nu}=\frac{1}{3}

, a hybrid spin-density excitation was discovered, the energy of which enables us to probe the correlation energy of the Laughlin liquid. The measured correlation and exchange energies at

\ensuremath{\nu}=\frac{1}{3}

and

{r}_{s}\ensuremath{\gg}1

Anomalous behavior of spin stiffness is revealed in ZnO-based two-dimensional electron systems (2DESs) with strong Coulomb interaction and Wigner-Seitz parameter

{r}_{s}>6

. The spin stiffness is extracted directly from the quadratic

k

dispersion of spin excitons at $\ensuremath{\nu}=1$ probed by inelastic light scattering. The resulting values are found to be dramatically rescaled compared to the case of weakly interacting 2DESs---spin stiffness turned out to be of the order of the cyclotron energy with the effective mass of Fermi-liquid quasiparticles. This result is also confirmed by the exact diagonalization simulations.

{\mathrm{Al}}_{\mathit{x}}

${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.

{r}_{s}=1)

and to the effect of the finite thickness of the liquid (a further $\ensuremath{\sim}30%$ at

{r}_{s}=1)

. We find our calculations to be consistent with recent photoluminescence experiments in $\ensuremath{\delta}$-doped samples.

ν{=}n/(2n{\pm}1)

and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at

ν{=}n/(4n{\pm}1)

. We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at

ν{=}2/5

. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.

m

of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.