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Physical review. B, Condensed matter

Quantum Hall ferromagnet in a parabolic quantum wire

康弘 都倉

Published Nov 15, 1998• 0 citations
Publisher

Abstract

The spin phase diagram of interacting electrons in a quantum wire under a high magnetic field is investigated by finite-system exact diagonalization. Various phases emerge as a function of interaction strength and aspect ratio. Stable spin-polarized states correspond to two-dimensional fractional quantum Hall states of filling fractions $1/q,$ with $q$ odd. Spin-singlet ground states for six electrons with filling fractions around $\frac{2}{5}--\frac{1}{2}$ are inspected in detail, and a real-space singlet pairing state peculiar to wire systems is found for a larger aspect ratio.

DOI
10.1103/physrevb.58.12597

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The spin phase diagram of interacting electrons in a quantum wire under a high magnetic field is investigated by finite-system exact diagonalization. Various phases emerge as a function of interaction strength and aspect ratio. Stable spin-polarized states correspond to two-dimensional fractional quantum Hall states of filling fractions 1/q,1/q,1/q, with qqq odd. Spin-singlet ground states for six electrons with filling fractions around 25−−12\frac{2}{5}--\frac{1}{2}52​−−21​ are inspected in detail, and a real-space singlet pairing state peculiar to wire systems is found for a larger aspect ratio.
A system at filling factor 2/3 could be a candidate for a quantum Hall ferromagnet at integer filling factor of composite fermions. Using exact diagonalization with electrons on a torus we study the transition from the singlet ground state to the polarized ground state at this filling and look for signatures of quantum Hall ferromagnetism. Differences between the fractional and corresponding integer systems are emphasised. Most interestingly, we find around the transition a low excited half-polarized state which might become the ground state in the thermodynamical limit. We study its structure and compare it to the singlet and polarized ground states. A new interpretation of the singlet state is suggested and comparison of the filling factors 2/3 and 2/5 is presented. Adding magnetic inhomogeneities into the system we investigate the stability of all the three involved states and the tendency to build up domains like in conventional ferromagnets.
We compare quantum Hall systems at filling factors $\ensuremath{\nu}=2$ to $\ensuremath{\nu}=\frac{2}{3}$ and 25\frac{2}{5}52​, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1P=1P=1) and paramagnetic (P=0)(P=0)(P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At $\ensuremath{\nu}=2$, we find spin domains at half-polarization (P=12)(P=\frac{1}{2})(P=21​), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0P=0P=0 and 1, as seen when, e.g., the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompressible half-polarized state is found in some models.
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.
Two-dimensional interacting electron systems become strongly correlated if the electrons are subject to a perpendicular high magnetic field. After introducing the physics of the quantum Hall regime the incompressible many-particle ground state and its excitations are studied in detail at fractional filling factors for spin-polarized electrons. The spin degree of freedom whose importance was shown in recent experiments is considered by studying the thermodynamics at filling factor one and near one.
Two-dimensional interacting electron systems become strongly correlated if the electrons are subject to a perpendicular high magnetic field. After introducing the physics of the quantum Hall regime the incompressible many-particle ground state and its excitations are studied in detail at fractional filling factors for spin-polarized electrons. The spin degree of freedom whose importance was shown in recent experiments is considered by studying the thermodynamics at filling factor one and near one.
We study the ground-state phase diagram of two-dimensional two-component (or pseudospin-12\frac{1}{2}21​) Bose gases in a high synthetic magnetic field in the space of the total filling factor and the ratio of the intercomponent coupling ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ to the intracomponent one g>0g>0g>0. Using exact diagonalization, we find that when the intercomponent coupling is attractive (${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}<0$), the product states of a pair of nearly uncorrelated quantum Hall states are remarkably robust and persist even when $|{g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}|$ is close to ggg. This contrasts with the case of an intercomponent repulsion, where a variety of spin-singlet quantum Hall states with high intercomponent entanglement emerge for ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}\ensuremath{\approx}g$. We interpret this marked dependence on the sign of ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ in light of pseudopotentials on a sphere, and also explain recent numerical results in two-component Bose gases in mutually antiparallel magnetic fields where a qualitatively opposite dependence on the sign of ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ is found. Our results thus unveil an intriguing connection between multicomponent quantum Hall systems and quantum spin Hall systems in minimal setups.