A quantum Hall system is investigated by exact numerical diagonalization of the Hamiltonian. The energy, off-diagonal long-range order, crystal long-range order, and the mean occupation number \ensuremath{\rho}(j)=〈${\mathit{c}}_{\mathit{j}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{j}}$〉 of the jth Landau orbital for \ensuremath{\nu}=1/2 ground state are calculated. Our results indicate no sign of either electron pairing or phase separation, the two possible features previously suggested for the \ensuremath{\nu}=1/2 ground state. A possible quasiparticle pairing mechanism is proposed in trying to account for the numerical results and experimental observations.
A quantum Hall system is investigated by exact numerical diagonalization of the Hamiltonian. The energy, off-diagonal long-range order, crystal long-range order, and the mean occupation number \ensuremath{\rho}(j)=〈{\mathit{c}}_{\mathit{j}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{j}}〉 of the jth Landau orbital for \ensuremath{\nu}=1/2 ground state are calculated. Our results indicate no sign of either electron pairing or phase separation, the two possible features previously suggested for the \ensuremath{\nu}=1/2 ground state. A possible quasiparticle pairing mechanism is proposed in trying to account for the numerical results and experimental observations.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.
We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.
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.
We study the nature of the many-body electron states (and quantum phase transitions between them) in a double-quantum-well structure under a strong external magnetic field as a function of the materials parameters that define the two-layer system, namely, the thickness of individual layers, the separation between them, the individual well potentials confining the electrons, and the potential barrier between them. Motivated by two recent experiments, we consider two different situations, one with (almost) no quantum tunneling between the wells, and the other with substantial interwell quantum tunneling. We use the spherical system finite-size exact diagonalization technique for our calculations. By calculating the overlap of our exact small-system numerical wave functions with various (postulated) analytic wave functions, we comment on the nature of the incompressibility for various Landau-level filling factors (\ensuremath{\nu}). We investigate, in particular details, the possibility of \ensuremath{\nu}=1/2 (i.e., 1/4 average occupancy in each well), and, 1 (i.e., 1/2 occupancy in each well) incompressible states where \ensuremath{\nu} is the total filling factor for the system. We also provide results for the \ensuremath{\nu}=2/3 situation. Our conclusion, based on our use of realistic system parameters in our calculations, is that in both the recent experimental realizations of the \ensuremath{\nu}=1/2 fractional quantum Hall effect, the relevant ground state is the so-called ``331'' state which is stabilized by the competition between intrawell and interwell electron-electron correlations.We calculate for various double-quantum-well structural parameters and magnetic fields the \ensuremath{\nu}=1/2 excitation spectra and the energy gaps, and compare it with the recent experimental data, obtaining excellent quantitative agreement. We provide a detailed quantum phase diagram for the \ensuremath{\nu}=1 state as a function of the interwell Coulomb interaction and the symmetric-antisymmetric single-particle gap, and find that the \ensuremath{\nu}=1 quantum Hall effect observed in the two recent experiments, in contrast to the \ensuremath{\nu}=1/2 effect, belongs in some sense to two different universality classes. We also predict the existence of a reentrant \ensuremath{\nu}=1 quantum Hall effect in double-quantum-well systems.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 31 filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.
This paper reviews the authors' theoretical understanding for the quantum Hall anomalies recently observed at fractional fillings with even denominators, more particularly at filling 1/2, where the electron spins are still fully polarized. After a brief reminder of the basic experimental evidence dip of {rho}{sub xx}, lack of plateau in {rho}{sub xx}, temperature behavior, the author discusses the important role of electron-hole (e-h) symmetry at half-filling. In particular the two alternative possibilities of spontaneous e-h symmetry breaking or non-breaking are discussed along with their physical implications. Hartree-Fock charge density waves are presented as an explicit example of spontaneous e-h symmetry breaking. Next, several different scenarios are discussed for the ground state at half-filling. One, largely discussed in the experimental papers is that of Halperin, involving phase separation. Another is that of Fano et al. where the half-filling ground state is supposed to be incompressible and homogeneous, but deformable. Pursuing further the second scenario, numerical evidence from the recent work of Chen is presented, indicating incompressibility and deformability. This work also demonstrates a rather abrupt switching of short-range electron correlations. Just below half-filling, an electron is typically surrounded by six close-by electrons (like a triangular lattice).