Session K17: Theoretical Progress on the Fractional Quantum Hall Effect

Characterizing fractional quantum Hall states using isometric tensor networks

The simulation of strongly-correlated quantum many-body systems is a long-standing numerical challenge. Although the ground-state properties of one-dimensional systems may be efficiently distilled using the density matrix renormalization group, now understood in the framework of matrix product states, generalizing this procedure to higher dimensions is problematic, since the exact evaluation of tensor network states becomes exponentially expensive. In this talk, we remedy this by employing isometric tensor networks (isoTNS), a recently-proposed restriction of the projected entangled pair state ansatz. We evaluate isoTNS algorithms for bosons and fermions, and present current progress in applying them to characterize fractional quantum Hall states.   

Many-Body Quantum Geometric Dipole

Collective excitations of electron systems may host internal structure tied to their quantum geometric structure.  Recently this has been shown to result in electric dipole moments associated with neutral, two-body excitations, including excitons [1] and plasmons in two-dimensional [2] and quasi-one-dimensional [3] metals.  We demonstrate that these properties can be formulated without reference to two-body wavefunctions.  The approach relies on a set of momentum-labeled density matrices constructed from the many-body states, from which “hole-hosting” and “particle-hosting” single-particle states may be extracted, allowing a gauge-independent quantum geometric dipole (QGD) to be defined.  This corresponds to the electric dipole moment of the excitation.  As a concrete example, we analyze magnetoplasmon excitations above the Laughlin ground states of the fractional quantized Hall effect.  The relevant single-particle states are constructed using composite-fermions.  In the limit of very strong magnetic fields an exact result may be derived, which is equivalent to the QGD of a magnetoplasmon in the integral quantized Hall effect.   This simple result is shown to be an outcome of the state fully lying in the lowest Landau level of the original electron degrees of freedom.   

Electron tunneling into Fractional Quantum Hall Effect

The low energy excitation spectrum of the Fractional Quantum Hall Effect (FQHE) is described in terms of composite fermions, which are fundamentally distinct from electrons. When an electron is added into an FQHE state, it disintegrates into many fractionally charged composite fermions. Because of the absence of electron-like quasiparticles, the a priori expectation is that the electron tunneling spectrum will contain no sharp peaks. Surprisingly, theoretical work [1] predicted at least one sharp peak in the tunneling spectrum of FQH liquids. Recent STM experiments [2] have indeed observed one or more peaks. In this work, we show that the addition of a hole to (i.e., removal of an electron from) an FQH liquid at ν = n/(2np+1) can be exactly described in terms of a finite number of composite fermion basis functions, allowing us to predict the qualitative and quantitative features of electron tunneling spectral function.  The addition of an electron cannot be readily described in terms of composite fermions, leading to a qualitatively different behavior for the spectral function. We also study the effect of the tip potential and predict features that can arise from the insertion of a spin-reversed electron. 

  

1. J. K. Jain and M. R. Peterson, Reconstructing the Electron in a Fractionalized Quantum Fluid, Phys. Rev. Lett. 94, 186808 (2005). 

2. Y. Hu et al., High-Resolution Tunneling Spectroscopy of Fractional Quantum Hall States, http://arxiv.org/abs/2308.05789.   

Orbital and Spin Order in Graphene Multilayer Quantum Hall Systems

Graphene multilayers in a strong magnetic field have rich interaction physics.  Ordered states include broken symmetry incompressible integer and fractional Quantum Hall states, including large gap states at even denominator fractions [1]. The phase diagrams of these systems depend subtly on an interplay between internal spin, valley, and sometimes additional orbital degrees of freedom present within the anomalous Landau levels at neutrality.  We will explain why a correct description of orbital polarization requires the inclusion of exchange interactions with the negative energy sea, and why the filling factor dependence of orbital polarization is sensitive to particle-hole symmetry breaking.  These ideas will be applied to bilayer graphene, to multilayer rhombohedral graphene stacks, and to bulk graphite.

 

 

Candidate parent Hamiltonian for the 3/7 fractional quantum Hall state

The Trugman-Kivelson (TK) interaction [1] has been long known as a parent Hamiltonian for the 1/3 Laughlin wave function. When the lowest two Landau levels (LLs) are assumed to be degenerate, this interaction also produces the unprojected composite-fermion wave function at n/(2n+1) with n=2 as a unique zero-mode [2]. However, this approach encounters difficulties at n=3 (i.e. 3/7 filling) because numerous zero modes are generated by the TK interaction within the lowest three LLs. Here, we introduce a local three-body interaction to single out the 3/7 wave function from the zero-mode subspace. Our extensive diagonalization studies on finite systems provide strong support for this proposal [4].

[1] S. A. Trugman and S. Kivelson, Phys. Rev. B 31, 5280 (1985).

[2] J. K. Jain, Phys. Rev. B 41, 7653 (1990); E. H. Rezayi and A. H. MacDonald, Phys. Rev. B 44, 8395 (1991).

[3] S. Bandyopadhyay, L. Chen, M. T. Ahari, G. Ortiz, Z. Nussinov, and A. Seidel, Phys. Rev. B 98, 161118(R) (2018).

[4] K. Kudo, A. Sharma, G. J. Sreejith, and J. K. Jain, Phys. Rev. B 108, 085130 (2023).   

A realistic Hamiltonian for the bosonic Moore-Read Pfaffian state at ν =1

The fermionic fractional quantum Hall effect at filling factor 5/2, thought to support non-Abelian low-energy excitations, is experimentally challenging but sought after for the possibility of harnessing these quasi-particle excitations with fractional braiding statistics for quantum information processing applications. Alternatively, bosonic fractional quantum Hall states have been proposed in a variety of realistic physical systems. Via exact diagonalization, we numerically investigate a two-body model Hamiltonian derived from the three-body Hamiltonian that produces the Moore-Read Pfaffian as its zero-energy ground state for bosons at ν=1 in the spherical geometry (similar to the method used for the fermionic Moore-Read Pfaffian state). We fully characterize this realistic(two-body) Hamiltonian in terms of Haldane pseudopotentials and demonstrate that the ground state supports robust non-Abelian excitations and is adiabatically connected to low-energy spectrum of the three-body Hamiltonian that generates the boson Moore-Read Pfaffian state.   

The Role of Quantum Geometry in Stabilizing Fractional Chern Insulators: Coupled-Wires Approach

In the presence of strong electronic interactions, a partially filled Chern band may stabilize a fractional Chern insulator (FCI) state, the zero-field analogue of the fractional quantum Hall phase. While the FCI phase has been long hypothesized, feasible solid-state realizations only recently emerged with the rise of moir'e materials. In these systems, the quantum geometry of the electronic bands plays a critical role in stabilizing the FCI over competing correlated phases. Whereas in the limit of ``ideal'' quantum geometry this role is well understood, the role of quantum geometry far from ideality is supported only by empiric numerical evidence, without clear analytical understanding. We analyze an anisotropic model of a $left|C ight|=1$ Chern insulator. Upon partial filling of one of its bands, an FCI phase may be stabilized over a certain parameter regime. We incorporate strong electronic interaction analytically by employing the coupled-wires approach, and analyze the FCI stability, and its relation to a "chiralness" parameter which controls the quantum metric. We identify a competing anti-FCI phase benefiting from non-ideal metric, which generically does not become favorable over the FCI, however its presence hinders the FCI stabilization in favor of metallic or CDW phases. We thus establish an analytical connection between quantum geometry and FCI stability far from ideal conditions.   

Quantum mechanics of composite fermions

We present a reformulation of the theory of composite fermions based on Read's dipole picture. We show that: (i) Jain's CF wave function can be recast and derived from the dipole picture by interpreting the form as a projection into an incompressible state of vortices, which are introduced as auxiliary degrees of freedom for representing correlated holes; (ii) the state of a composite fermion in external fields is determined by a wave-equation resembling the ordinary Schrödinger equation but with drift velocity corrections, different from that assumed in the Halperin-Lee-Read theory. The reformulated theory can be easily generalized for fractional Chern insulators.   

Universality of critical exponents in the fractional quantum Hall regime

Scaling for transitions between integer quantum Hall states (IQHSs) has been studied extensively, and measurements show that the universality of the critical exponents depends on the nature of the disorder present in the samples [1-3]. The situation is much more complex when we introduce electron-electron interaction, as is the case of the fractional quantum Hall states (FQHSs). Recent theoretical calculations suggest that the critical exponents obtained for the transitions between FQHSs are identical to those of the IQHSs, provided that the FQHSs can be viewed as the IQHSs of non-interacting composite fermions [4]. However, the latest experiments on high-quality 2DESs confined to GaAs quantum wells reveal that the exponents agree with the theoretical expected value only for a limited number of FQHS transitions [5].  Possible causes for the non-universality are the strongly interacting nature of the composite fermions, and the nature of the disorder [5]. In our work, we investigate this underlying non-universality for the transitions between the FQHSs by intentionally introducing short-range impurities in the quantum well where the electrons reside. We do so by systematically changing the Al concentration (x) in a series of Al_xGa_1-xAs quantum wells with 0  x  0.016. We find that the exponents vary with the alloy fraction, suggesting the interplay of disorder and interaction in the fractional quantum Hall regime.

References:

           H. P. Wei et al., Phys. Rev. B 45, 3926 (1992).

           Wanli Li et al., Phys. Rev. Lett. 94, 206807 (2005).

           Wanli Li et al., Phys. Rev. Lett. 102, 216801 (2009).

           Songyang Pu et al., Phys. Rev. Lett. 128, 116801 (2022).

           P. T. Madathil et al., Phys. Rev. Lett. 130, 226503 (2023).   

Topological phases in twisted transition metal dichalcogenides: a DMRG study

The recent experimental observations of fractional quantum anomalous Hall (FQAH) effects in twisted bilayer MoTe2 have attracted considerable attention. We use large-scale density matrix renormalization group (DMRG) simulation to study twisted transition metal dichalcogenides (tTMD) at fractional fillings. We have identified various interesting phases of matter, including a series of fractional quantum Hall states.   

Time-Reversal Invariant Topological Moiré Flatband: A Platform for the Fractional Quantum Spin Hall Effect

Motivated by recent observation of the quantum spin Hall effect in monolayer germanene and twisted bilayer transition-metal-dichalcogenides (TMDs), we study the topological phases of moiré twisted bilayers with time-reversal symmetry and spin s_z conservation. By using a continuum model description which can be applied to both germanene and TMD bilayers, we show that at small twist angles, the emergent moiré flatbands can be topologically nontrivial due to inversion symmetry breaking. Each of these flatbands for each spin projection admits a lowest-Landau-level description in the chiral limit and at magic twist angle. This allows for the construction of a many-body Laughlin state with time-reversal symmetry which can be stabilized by a short-range pseudopotential, and therefore serves as an ideal platform for realizing the so-far elusive fractional quantum spin Hall effect with emergent spin-1/2 U(1) symmetry.   

Observation of a bubble phase of composite fermions

Non-interacting composite fermions can be used to account for the majority of fractional quantum Hall states. However, the residual interactions between composite fermions can lead to the formations of more exotic states with complex correlations. We recently observed a bubble phase of composite fermions in a GaAs two-dimensional electron gas sample near Landau level filling factor ν = 5/3. This phase with very interesting correlations was identified via a reentrant behavior of the fractional quantum Hall effect in transport measurements. Our finding reveals a new class of strongly correlated topological phases, driven by the clustering and charge ordering of emergent quasiparticles.   

Identifying Fibonacci anyons with upstream noise.

Non-Abelian phases are among the most highly-sought states of matter, with those whose anyons permit universal quantum gates constituting the ultimate prize. The most promising candidate of such a phase is the fractional quantum Hall plateau at filling factors $ u=frac{12}{5}$, which putatively facilitates Fibonacci anyons. Experimental validation of this assertion poses a major challenge and remains elusive. We present a measurement protocol that could achieve this goal with already-demonstrated experimental techniques. Interfacing the $ u=frac{12}{5}$ state with any readily-available Abelian state yields a binary outcome of upstream noise or no noise. Judicious choices of the Abelian states can produce a sequence of yes--no outcomes that fingerprint the possible non-Abelian phase by ruling out its competitors. Crucially, this identification is insensitive to the precise value of the measured noise and can uniquely identify the anyon type at filling factors $ u=frac{12}{5}$. In addition, it can distinguish any non-Abelian candidates at half-filling in graphene and semiconductor heterostructures.