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arXiv:2201.07893 · Physical review. B./Physical review. B

Adiabatic continuity of the spinful quantum Hall states

Koji Kudo, Yasuhiro Hatsugai

Published Aug 9, 2022• 2 citations
arXivHTMLPDF

Abstract

By using the extended Hubbard model of anyons, we numerically demonstrate the adiabatic deformation of the spinful quantum Hall (QH) states by transmutation of statistical fluxes. While the ground state is always spin-polarized in a series of $\ensuremath{\nu}=1$ integer QH systems, the adiabatic continuity between the singlet QH states at $\ensuremath{\nu}=2$ and $\ensuremath{\nu}=2/5$ is confirmed. These results are consistent with the composite fermion theory with spin. The many-body Chern number of the ground-state multiplet works as an adiabatic invariant and also explains the discrete change of the topological degeneracy during the evolution. The generalized St\ifmmode \check{r}\else \v{r}\fi{}eda formula of spinful systems is justified.

DOI
10.1103/physrevb.106.075120

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By using the extended Hubbard model of anyons, we numerically demonstrate the adiabatic deformation of the spinful quantum Hall (QH) states by transmutation of statistical fluxes. While the ground state is always spin-polarized in a series of \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 integer QH systems, the adiabatic continuity between the singlet QH states at \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2 and \ensuremathν=2/5\ensuremath{\nu}=2/5\ensuremathν=2/5 is confirmed. These results are consistent with the composite fermion theory with spin. The many-body Chern number of the ground-state multiplet works as an adiabatic invariant and also explains the discrete change of the topological degeneracy during the evolution. The generalized St\ifmmode \check{r}\else \v{r}\fi{}eda formula of spinful systems is justified.
Although the adiabatic heuristic argument of the fractional quantum Hall states has been successful, continuous modification of the flux/statistics of anyons is strictly prohibited due to algebraic constrains of the braid group on a torus. We have numerically shown that the adiabatic principle for anyon is still valid even though the dimension of the Hilbert space changes irregularly. The Chern number of the ground state multiplet is the adiabatic invariant of this non-analytic evolution, while the number of the topological degeneracy behaves violently. A generalized Streda formula is proposed that explains the degeneracy pattern. Nambu-Goldston modes associated with the anyon superconductivity are also suggested numerically.
An adiabatic approach put forward by Greiter and Wilczek interpolates between the integer quantum Hall effects of electrons and composite fermions by varying the statistical flux bound to electrons continuously from zero to an even integer number of flux quanta, such that the intermediate states represent anyons in an external magnetic field with the same ``effective'' integer filling factor. We consider such anyons on a torus, and we construct representative wave functions for their ground as well as excited states. These wave functions involve higher Landau levels in general, but can be explicitly projected into the lowest Landau level for many parameters. We calculate the variational energy gap between the first excited state and the ground state, and we find that it remains open as the statistical phase is varied. Finally, we obtain from these wave functions, both analytically and numerically, various topological quantities, such as the ground-state degeneracy, the Chern number, and the Hall viscosity.
Abstract The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The effective formalism used in many-body quantum Hall theories — the Chern–Simons theory is also presented. The Chern–Simons theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall analyses, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. An introduction to multi-component quantum Hall systems and spin quantum Hall systems is sketched.
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.
The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The main formalism used in many-body quantum Hall theories -- the Chern-Simons theory is also presented. The Chern-Simons theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall theories, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy)1). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. The introduction into multi-component quantum Hall systems and spin quantum Hall systems is sketched. 1)First condensed states of composite fermion excitations have been very recently confirmed in the experiment (Pan et al. Phys. Rev. Lett. 90 (2003) 016801). a sample of this book is available at http://www.oup.co.uk/isbn/0-19-852870-1
By means of finite-size exact diagonalization we theoretically study the electronic many-body effects on the nearly flat-band structure with time-reversal symmetry in a checkerboard lattice model and identify the topological nature of two quantum phases, with ninefold and threefold degeneracy, that appear, respectively, at small and large values $\ensuremath{\lambda}$ of a nearest-neighbor spin-dependent interaction. Numerical evidence from the evolution of low-lying energy spectra and Berry phases with both spin-independent and spin-dependent twisted boundary conditions reveal that these two different ground states share the same topological spin Chern number. Quantum phase transition between these two states by tuning $\ensuremath{\lambda}$ is confirmed by evaluating the closing of energy and quasispin excitation spectra. Finally, the counting rules of spin excitation spectra are demonstrated as the fingerprints of the fractionalized quantum spin Hall states.