arXiv subjects

J. K. Jain

Publications and source records attributed to J. K. Jain.

At least 55 records · Page 3Linked to original sources

arXiv2020

$\mathbb{Z}_{n}$ superconductivity of composite bosons and the $7/3$ fractional quantum Hall effect

The topological $p$-wave pairing of composite fermions, believed to be responsible for the 5/2 fractional quantum Hall effect (FQHE), has generated much exciting physics. Motivated by the parton theory of the FQHE, we consider the possibility of a new kind of emergent "superconductivity" in the 1/3 FQHE, which involves condensation of clusters of $n$ composite bosons. From a microscopic perspective, the state is described by the $n\bar{n}111$ parton wave function ${\cal P}_{\rm LLL} Φ_nΦ_n^*Φ_1^3$, where $Φ_n$ is the wave function of the integer quantum Hall state with $n$ filled Landau levels and ${\cal P}_{\rm LLL}$ is the lowest-Landau-level projection operator. It represents a $\mathbb{Z}_{n}$ superconductor of composite bosons, because the factor $Φ_1^3\sim \prod_{j<k}(z_j-z_k)^3$, where $z_j=x_j-iy_j$ is the coordinate of the $j$th electron, binds three vortices to electrons to convert them into composite bosons, which then condense into the $\mathbb{Z}_{n}$ superconducting state $|Φ_n|^2$. From a field theoretical perspective, this state can be understood by starting with the usual Laughlin theory and gauging a $\mathbb{Z}_n$ subgroup of the $U(1)$ charge conservation symmetry. We find from detailed quantitative calculations that the $2\bar{2}111$ and $3\bar{3}111$ states are at least as plausible as the Laughlin wave function for the exact Coulomb ground state at filling $ν=7/3$, suggesting that this physics is possibly relevant for the 7/3 FQHE. The $\mathbb{Z}_{n}$ order leads to several observable consequences, including quasiparticles with fractionally quantized charges of magnitude $e/(3n)$ and the existence of multiple neutral collective modes. It is interesting that the FQHE may be a promising venue for the realization of exotic $\mathbb{Z}_{n}$ superconductivity.

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arXiv2019

Fermi wave vector for the non-fully spin polarized composite-fermion Fermi sea

The fully spin polarized composite fermion (CF) Fermi sea at half filled lowest Landau level has a Fermi wave vector $k^*_{\rm F}=\sqrt{4πρ_e}$, where $ρ_e$ is the density of electrons or composite fermions, supporting the notion that the interaction between composite fermions can be treated perturbatively. Away from $ν=1/2$, the area is seen to be consistent with $k^*_{\rm F}=\sqrt{4πρ_e}$ for $ν<1/2$ but $k^*_{\rm F}=\sqrt{4πρ_h}$ for $ν>1/2$, where $ρ_h$ is the density of holes in the lowest Landau level. This result is consistent with particle-hole symmetry in the lowest Landau level. We investigate in this article the Fermi wave vector of the spin-singlet CF Fermi sea (CFFS) at $ν=1/2$, for which particle-hole symmetry is not a consideration. Using the microscopic CF theory, we find that for the spin-singlet CFFS the Fermi wave vectors for up and down spin CFFSs at $ν=1/2$ are consistent with $k^{*\uparrow,\downarrow}_{\rm F}=\sqrt{4πρ^{\uparrow,\downarrow}_e}$, where $ρ^{\uparrow}_e=ρ^{\downarrow}_e=ρ_e/2$, which implies that the residual interactions between composite fermions do not cause a non-perturbative correction for non-fully spin polarized CFFS either. Our results suggest the natural conjecture that for arbitrary spin polarization the CF Fermi wave vectors are given by $k^{*\uparrow}_{\rm F}=\sqrt{4πρ^{\uparrow}_e}$ and $k^{*\downarrow}_{\rm F}=\sqrt{4πρ^{\downarrow}_e}$.

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arXiv2019

Next-level composite fermions

A rich pattern of fractional quantum Hall states in graphene double layers can be naturally explained in terms of two-component composite fermions carrying both intra- and inter-layer vortices.

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arXiv2019

Kohn-Sham Theory of the Fractional Quantum Hall Effect

We formulate the Kohn-Sham equations for the fractional quantum Hall effect by mapping the original electron problem into an auxiliary problem of composite fermions that experience a density dependent effective magnetic field. Self-consistent solutions of the KS equations demonstrate that our formulation captures not only configurations with non-uniform densities but also topological properties such as fractional charge and fractional braid statistics for the quasiparticles excitations. This method should enable a realistic modeling of the edge structure, the effect of disorder, spin physics, screening, and of fractional quantum Hall effect in mesoscopic devices.

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arXiv2018

Fractional Quantum Hall Effect at $ν=2+6/13$: The Parton Paradigm for the Second Landau Level

The unexpected appearance of a fractional quantum Hall effect (FQHE) plateau at $ν=2+6/13$~ [Kumar \emph{et al.}, Phys. Rev. Lett. {\bf 105}, 246808 (2010)] offers a clue into the physical mechanism of the FQHE in the second Landau level (SLL). Here we propose a "$\bar{3}\bar{2}111$" parton wave function, which is topologically distinct from the 6/13 state in the lowest Landau level. We demonstrate the $\bar{3}\bar{2}111$ state to be a good candidate for the $ν=2+6/13$ FQHE, and make predictions for experimentally measurable properties that can reveal the nature of this state. Furthermore, we propose that the "$\bar{n}\bar{2}111$" family of parton states naturally describes many observed SLL FQHE plateaus.

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arXiv2018

Berry phase of the composite-fermion Fermi Sea: Effect of Landau-level mixing

We construct explicit lowest-Landau-level wave functions for the composite-fermion Fermi sea and its low energy excitations following a recently developed approach [Pu, Wu and Jain, Phys. Rev. B 96, 195302 (2018)] and demonstrate them to be very accurate representations of the Coulomb eigenstates. We further ask how the Berry phase associated with a closed loop around the Fermi circle, predicted to be $π$ in a Dirac composite fermion theory satisfying particle-hole symmetry [D. T. Son, Phys. Rev. X 5, 031027 (2015)], is affected by Landau level mixing. For this purpose, we consider a simple model wherein we determine the variational ground state as a function of Landau level mixing within the space spanned by two basis functions: the lowest-Landau-level projected and the unprojected composite-fermion Fermi sea wave functions. We evaluate Berry phase for a path around the Fermi circle within this model following a recent prescription, and find that it rotates rapidly as a function of Landau level mixing. We also consider the effect of a particle-hole symmetry breaking three-body interaction on the Berry phase while confining the Hilbert space to the lowest Landau level. Our study deepens the connection between the $π$ Berry phase and the exact particle-hole symmetry in the lowest Landau level.

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arXiv2018

Crystallization in the Fractional Quantum Hall Regime Induced by Landau-level Mixing

The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the non-perturbative fixed phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the $ν-κ$ plane, where $ν$ is the filling factor and $κ$ is the strength of Landau level mixing. The phase boundary is seen to exhibit a striking $ν$ dependence, with the states away from the magic filling factors $ν=n/(2pn+1)$ being much more susceptible to crystallization due to Landau level mixing than those at $ν=n/(2pn+1)$. Our results explain the qualitative difference between the experimental behaviors observed in n-doped and p-doped GaAs quantum wells, and, in particular, the existence of an insulating state for $ν<1/3$ and also for $1/3 <ν< 2/5$ in low density p-doped systems. We predict that in the vicinity of $ν=1/5$ and $ν=2/9$, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal but rather into a strongly correlated crystal of composite fermions carrying two vortices.

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arXiv2018

Inter-Landau-level Andreev Reflection at the Dirac Point in a Graphene Quantum Hall State Coupled to a NbSe2 Superconductor

Superconductivity and quantum Hall effect are distinct states of matter occurring in apparently incompatible physical conditions. Recent theoretical developments suggest that the coupling of quantum Hall effect with a superconductor can provide a fertile ground for realizing exotic topological excitations such as non-abelian Majorana fermions or Fibonacci particles. As a step toward that goal, we report observation of Andreev reflection at the junction of a quantum Hall edge state in a single layer graphene and a quasi-two dimensional niobium diselenide (NbSe2) superconductor. Our principal finding is the observation of an anomalous finite-temperature conductance peak located precisely at the Dirac point, providing a definitive evidence for inter-Landau level Andreev reflection in a quantum Hall system. Our observations are well supported by detailed numerical simulations, which offer additional insight into the role of the edge states in Andreev physics. This study paves the way for investigating analogous Andreev reflection in a fractional quantum Hall system coupled to a superconductor to realize exotic quasiparticles.

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arXiv2018

Topological superconductivity in Landau levels

The intense search for topological superconductivity is inspired by the prospect that it hosts Majorana quasiparticles. We explore in this work the optimal design for producing topological superconductivity by combining a quantum Hall state with an ordinary superconductor. To this end, we consider a microscopic model for a topologically trivial two-dimensional p-wave superconductor exposed to a magnetic field, and find that the interplay of superconductivity and Landau level physics yields a rich phase diagram of states as a function of $μ/t$ and $Δ/t$, where $μ$, $t$ and $Δ$ are the chemical potential, hopping strength, and the amplitude of the superconducting gap. In addition to quantum Hall states and topologically trivial p-wave superconductor, the phase diagram also accommodates regions of topological superconductivity. Most importantly, we find that application of a non-uniform, periodic magnetic field produced by a square or a hexagonal lattice of $h/e$ fluxoids greatly facilitates regions of topological superconductivity in the limit of $Δ/t\rightarrow 0$. In contrast, a uniform magnetic field, a hexagonal Abrikosov lattice of $h/2e$ fluxoids, or a one dimensional lattice of stripes produces topological superconductivity only for sufficiently large $Δ/t$.

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arXiv2018

Spontaneous polarization of composite fermions in the $n=1$ Landau level of graphene

Motivated by recent experiments that reveal expansive fractional quantum Hall states in the $n=1$ graphene Landau level and suggest a nontrivial role of the spin degree of freedom [Amet {\em et al.}, Nat. Common. {\bf 6}, 5838 (2014)], we perform accurate quantitative study of the the competition between fractional quantum Hall states with different spin polarizations. We find that the fractional quantum Hall effect is well described in terms of composite fermions, but the spin physics is qualitatively different from that in the $n=0$ Landau level. In particular, for the states at filling factors $n/(2n\pm 1)$, both exact diagonalization and the composite fermion theory show that the ground state is fully spin polarized and supports a robust spin wave mode even in the limit of vanishing Zeeman coupling. Thus, even though composite fermions are formed, a mean field description that treats them as weakly interacting particles breaks down, and the exchange interaction between them is strong enough to cause a qualitative change in the behavior by inducing full spin polarization. We also verify that the fully spin polarized composite fermion Fermi sea has lower energy than the paired Pfaffian state at the relevant half fillings in the $n=1$ graphene Landau level, indicating a lack of fractional quantum Hall effect at half filling in the $n=1$ graphene Landau level.

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arXiv2018

Particle-hole symmetry for composite fermions: An emergent symmetry in the fractional quantum Hall effect

The particle-hole (PH) symmetry of {\em electrons} is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. This article investigates an emergent symmetry in the fractional quantum Hall effect, namely the PH symmetry of {\em composite fermions}, which relates states at composite fermion filling factors $ν^*=n+\barν$ and $ν^*=n+1-\barν$, where the integer $n$ is the $Λ$ level index and $0\leq \barν\leq 1$. Detailed calculations using the microscopic theory of composite fermions demonstrate that for low lying $Λ$ levels (small $n$): (i) the 2-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $Λ$ level; and (ii) the 3-body interaction for composite fermions is an order of magnitude smaller than the 2-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $Λ$ levels, which is also supported by exact diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $ν={n+\barν\over 2(n+\barν)\pm 1}$ and $ν={n+1-\barν\over 2(n+1-\barν)\pm 1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $Λ$ level index, the 2-body and 3-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $ν=1/2$.

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arXiv2017

Composite Fermions on a Torus

We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the "unprojected" wave functions satisfying the correct (quasi-)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called "proper states," which includes the incompressible ground states at electron filling factors $ν=\frac{n}{2pn+ 1}$, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form $ν=\frac{ν^*}{2pν^*+ 1}$, where $n$ and $p$ are integers and $ν^*$ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors $ν=1/3$, 2/5 and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.

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arXiv2017

Density functional theory of the fractional quantum Hall effect

A conceptual difficulty in formulating the density functional theory of the fractional quantum Hall effect is that while in the standard approach the Kohn-Sham orbitals are either fully occupied or unoccupied, the physics of the fractional quantum Hall effect calls for fractionally occupied Kohn-Sham orbitals. This has necessitated averaging over an ensemble of Slater determinants to obtain meaningful results. We develop an alternative approach in which we express and minimize the grand canonical potential in terms of the composite fermion variables. This provides a natural resolution of the fractional-occupation problem because the fully occupied orbitals of composite fermions automatically correspond to fractionally occupied orbitals of electrons. We demonstrate the quantitative validity of our approach by evaluating the density profile of fractional Hall edge as a function of temperature and the distance from the delta dopant layer and showing that it reproduces edge reconstruction in the expected parameter region.

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arXiv2017

Tunnel transport and interlayer excitons in bilayer fractional quantum Hall systems

In a bilayer system consisting of a composite-fermion Fermi sea in each layer, the tunnel current is exponentially suppressed at zero bias, followed by a strong peak at a finite bias voltage $V_{\rm max}$. This behavior, which is qualitatively different from that observed for the electron Fermi sea, provides fundamental insight into the strongly correlated non-Fermi liquid nature of the CF Fermi sea and, in particular, offers a window into the short-distance high-energy physics of this state. We identify the exciton responsible for the peak current and provide a quantitative account of the value of $V_{\rm max}$. The excitonic attraction is shown to be quantitatively significant, and its variation accounts for the increase of $V_{\rm max}$ with the application of an in-plane magnetic field. We also estimate the critical Zeeman energy where transition occurs from a fully spin polarized composite fermion Fermi sea to a partially spin polarized one, carefully incorporating corrections due to finite width and Landau level mixing, and find it to be in satisfactory agreement with the Zeeman energy where a qualitative change has been observed for the onset bias voltage [Eisenstein et al., Phys. Rev. B 94, 125409 (2016)]. For fractional quantum Hall states, we predict a substantial discontinuous jump in $V_{\rm max}$ when the system undergoes a transition from a fully spin polarized state to a spin singlet or a partially spin polarized state.

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arXiv2017

The enigma of the $ν=2+\frac{3}{8}$ fractional quantum Hall effect

The fractional quantum Hall effect at $ν=2+3/8$, which has been definitively observed, is one of the last fractions for which no viable explanation has so far been demonstrated. Our detailed study suggests that it belongs to a new class of of exotic states described by the Bonderson-Slingerland wave function. Its excitations are non-Abelian anyons similar to those of the well studied Pfaffian state at 5/2, but its wave function has a more complex structure. Using the effective edge theory, we make predictions for various measurable quantities that should enable a confirmation of the underlying topological order of this state.

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arXiv2016

Landau-Level Mixing and Particle-Hole Symmetry Breaking for Spin Transitions in the Fractional Quantum Hall Effect

The spin transitions in the fractional quantum Hall effect provide a direct measure of the tiny energy differences between differently spin-polarized states, and thereby serve as an extremely sensitive test of the quantitative accuracy of the theory of the fractional quantum Hall effect, and, in particular, of the role of Landau-level mixing in lifting the particle-hole symmetry. We report on an accurate quantitative study of this physics, evaluating the effect of Landau-level mixing in a nonperturbative manner using a fixed-phase diffusion Monte Carlo method. We find excellent agreement between our calculated critical Zeeman energies and the experimentally measured values. In particular, we find, as also do experiments, that the critical Zeeman energies for fractional quantum Hall states at filling factors $ν=2-n/(2n\pm 1)$ are significantly higher than those for $ν=n/(2n\pm 1)$, a quantitative signature of the lifting of particle-hole symmetry due to Landau-level mixing.

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arXiv2016

Observation of the quantum-anomalous-Hall insulator to Anderson insulator quantum phase transition and its scaling behavior

Fundamental insight into the nature of the quantum phase transition from a superconductor to an insulator in two dimensions, or from one plateau to the next or to an insulator in quantum Hall effect, has been revealed through the study of its scaling behavior. Here, we report on the experimental observation of a quantum phase transition from a quantum-anomalous-Hall (QAH) insulator to an Anderson insulator in a magnetic topological insulator by tuning the chemical potential. Our experiment demonstrates the existence of scaling behavior from which we extract the critical exponent for this quantum phase transition. We expect that our work will motivate much further investigation of many properties of quantum phase transition in this new context.

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arXiv2016

The nature of composite fermions and the role of particle hole symmetry: A microscopic account

Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor $ν=1/2$ corresponds to an effective time reversal transformation (i.e. $\{\vec{k}_j\}$$\rightarrow$$\{-\vec{k}_j\}$) for composite fermions, and discuss how this connects to the absence of $2k_{\rm F}$ backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion $Λ$ levels (analogs of electronic Landau levels) we determine the $Λ$ level spacing and find it to be very nearly independent of the $Λ$ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a "normal form" for the unprojected wave functions of the $n/(2pn-1)$ states that correctly capture the topological properties even without lowest Landau level projection.

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