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1. Size dependence- and induced transformations- of fractional quantum Hall effects under tilted magnetic fields

   https://doi.org/10.1038/s41598-022-22812-x  

   Wijewardena, U. Kushan ; Nanayakkara, Tharanga R. ; Kriisa, Annika ; Reichl, Christian ; Wegscheider, Werner ; Mani, Ramesh G. ( November 2022 , Scientific Reports)

   Two-dimensional electron systems subjected to high transverse magnetic fields can exhibit Fractional Quantum Hall Effects (FQHE). In the GaAs/AlGaAs 2D electron system, a double degeneracy of Landau levels due to electron-spin, is removed by a small Zeeman spin splitting,$$g \mu _B B$$$g{\mu }_{B}B$, comparable to the correlation energy. Then, a change of the Zeeman splitting relative to the correlation energy can lead to a re-ordering between spin polarized, partially polarized, and unpolarized many body ground states at a constant filling factor. We show here that tuning the spin energy can produce fractionally quantized Hall effect transitions that include both a change in$$\nu$$$\nu$for the$$R_{xx}$$$R_{\mathrm{xx}}$minimum, e.g., from$$\nu = 11/7$$$\nu =11/7$to$$\nu = 8/5$$$\nu =8/5$, and a corresponding change in the$$R_{xy}$$$R_{\mathrm{xy}}$, e.g., from$$R_{xy}/R_{K} = (11/7)^{-1}$$$R_{\mathrm{xy}}/R_K={(11/7)}^{-1}$to$$R_{xy}/R_{K} = (8/5)^{-1}$$$R_{\mathrm{xy}}/R_K={(8/5)}^{-1}$, with increasing tilt angle. Further, we exhibit a striking size dependence in the tilt angle interval for the vanishing of the$$\nu = 4/3$$$\nu =4/3$and$$\nu = 7/5$$$\nu =7/5$resistance minima, including “avoided crossing” type lineshape characteristics, and observable shifts of$$R_{xy}$$$R_{\mathrm{xy}}$at the$$R_{xx}$$$R_{\mathrm{xx}}$minima- the latter occurring for$$\nu = 4/3, 7/5$$$\nu =4/3,7/5$and the 10/7. The results demonstrate both size dependence and the possibility, not just of competition between different spin polarized states at the same$$\nu$$$\nu$and$$R_{xy}$$$R_{\mathrm{xy}}$, but also the tilt- or Zeeman-energy-dependent- crossover between distinct FQHE associated with different Hall resistances.

    

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2. Hamiltonian reconstruction as metric for variational studies

   https://doi.org/10.21468/SciPostPhys.13.3.063  

   Zhang, Kevin ; Lederer, Samuel ; Choo, Kenny ; Neupert, Titus ; Carleo, Giuseppe ; Kim, Eun-Ah ( January 2022 , SciPost Physics)

   Variational approaches are among the most powerful techniques toapproximately solve quantum many-body problems. These encompass bothvariational states based on tensor or neural networks, and parameterizedquantum circuits in variational quantum eigensolvers. However,self-consistent evaluation of the quality of variational wavefunctionsis a notoriously hard task. Using a recently developed Hamiltonianreconstruction method, we propose a multi-faceted approach to evaluatingthe quality of neural-network based wavefunctions. Specifically, weconsider convolutional neural network (CNN) and restricted Boltzmannmachine (RBM) states trained on a square latticespin-1/2$1/2$J_1\!-\!J_2$J_1-J_2$Heisenberg model. We find that the reconstructed Hamiltonians aretypically less frustrated, and have easy-axis anisotropy near the highfrustration point. In addition, the reconstructed Hamiltonians suppressquantum fluctuations in the largeJ_2$J_2$limit. Our results highlight the critical importance of thewavefunction’s symmetry. Moreover, the multi-faceted insight from theHamiltonian reconstruction reveals that a variational wave function canfail to capture the true ground state through suppression of quantumfluctuations.

    

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3. Twisted-bilayer FeSe and the Fe-based superlattices

   https://doi.org/10.21468/SciPostPhys.15.3.081  

   Eugenio, Paul Myles ; Vafek, Oskar ( January 2023 , SciPost Physics)

   We derive BM-like continuum models for the bands of superlattice heterostructures formed out of Fe-chalcogenide monolayers: (I) a single monolayer experiencing an external periodic potential, and (II) twisted bilayers with long-range moire tunneling. A symmetry derivation for the inter-layer moire tunnelling is provided for both the\Gamma$\Gamma$andM$M$high-symmetry points. In this paper, we focus on moire bands formed from hole-band maxima centered on\Gamma$\Gamma$, and show the possibility of moire bands withC=0$C=0$or±1$\pm 1$topological quantum numbers without breaking time-reversal symmetry. In theC=0$C=0$region for\theta→0$\theta \to 0$(and similarly in the limit of large superlattice period for I), the system becomes a square lattice of 2D harmonic oscillators. We fit our model to FeSe and argue that it is a viable platform for the simulation of the square Hubbard model with tunable interaction strength.

    

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4. On Harmonic Hilbert Spaces on Compact Abelian Groups

   https://doi.org/10.1007/s00041-023-09992-4  

   Das, Suddhasattwa ; Giannakis, Dimitrios ( February 2023 , Journal of Fourier Analysis and Applications)

   Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted$$L^2$$$L^2$spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach$$^*$$${}^{\ast }$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian groupGto have the same spectrum as the$$C^*$$$C^{\ast }$-algebra of continuous functions onG. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on$$L^2(G)$$$L^2\left(G\right)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation.

    

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5. Next-generation even-denominator fractional quantum Hall states of interacting composite fermions

   https://doi.org/10.1073/pnas.2314212120  

   Wang, Chengyu ; Gupta, Adbhut ; Madathil, Pranav T. ; Singh, Siddharth K. ; Chung, Yoon Jang ; Pfeiffer, Loren N. ; Baldwin, Kirk W. ; Shayegan, Mansour ( December 2023 , Proceedings of the National Academy of Sciences)

   The discovery of the fractional quantum Hall state (FQHS) in 1982 ushered a new era of research in many-body condensed matter physics. Among the numerous FQHSs, those observed at even-denominator Landau level filling factors are of particular interest as they may host quasiparticles obeying non-Abelian statistics and be of potential use in topological quantum computing. The even-denominator FQHSs, however, are scarce and have been observed predominantly in low-disorder two-dimensional (2D) systems when an excited electron Landau level is half filled. An example is the well-studied FQHS at filling factor$\mathit{\nu }\mathbf{=}$5/2 which is believed to be a Bardeen-Cooper-Schrieffer-type, paired state of flux-particle composite fermions (CFs). Here, we report the observation of even-denominator FQHSs at$\mathit{\nu }\mathbf{=}$3/10, 3/8, and 3/4 in the lowest Landau level of an ultrahigh-quality GaAs 2D hole system, evinced by deep minima in longitudinal resistance and developing quantized Hall plateaus. Quite remarkably, these states can be interpreted as even-denominator FQHSs of CFs, emerging from pairing of higher-order CFs when a CF Landau level, rather than an electron or a hole Landau level, is half-filled. Our results affirm enhanced interaction between CFs in a hole system with significant Landau level mixing and, more generally, the pairing of CFs as a valid mechanism for even-denominator FQHSs, and suggest the realization of FQHSs with non-Abelian anyons.

    

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