Bosonic Quantum Hall Research Articles

Collective Excitation of Bosonic Quantum Hall State

Collective Excitation of Bosonic Quantum Hall State

Crystallization of bosonic quantum Hall states in a rotating quantum gas.

The dominance of interactions over kinetic energy lies at the heart of strongly correlated quantum matter, from fractional quantum Hall liquids1, to atoms in optical lattices2 and twisted bilayer graphene3. Crystalline phases often compete with correlated quantum liquids, and transitions between them occur when the energy cost of forming a density wave approaches zero. A prime example occurs for electrons in high-strength magnetic fields, where the instability of quantum Hall liquids towards a Wigner crystal4-9 is heralded by a roton-like softening of density modulations at the magnetic length7,10-12. Remarkably, interacting bosons in a gauge field are also expected to form analogous liquid and crystalline states13-21. However, combining interactions with strong synthetic magnetic fields has been a challenge for experiments on bosonic quantum gases18,21. Here we study the purely interaction-driven dynamics of a Landau gauge Bose-Einstein condensate22 in and near the lowest Landau level. We observe a spontaneous crystallization driven by condensation of magneto-rotons7,10, excitations visible as density modulations at the magnetic length. Increasing the cloud density smoothly connects this behaviour to a quantum version of the Kelvin-Helmholtz hydrodynamic instability, driven by the sheared internal flow profile of the rapidly rotating condensate. At long times the condensate self-organizes into a persistent array of droplets separated by vortex streets, which are stabilized by a balance of interactions and effective magnetic forces.

Crystallization of Bosonic Quantum Hall States in a Rotating Quantum Gas

Crystallization of Bosonic Quantum Hall States in a Rotating Quantum Gas

Magnetic Monopoles, Landau Levels, and Bosonic Quantum Hall Effect

This item presents the abstract for a presentation at the 2021 Annual Phi Kappa Phi Student Research and Fine Arts Conference at Augusta University.

Self-dual ν=1 bosonic quantum Hall state in mixed-dimensional QED

We consider a (2+1) dimensional Wilson-Fisher boson coupled to a (3+1) dimensional U(1) gauge field. This theory possesses a strong-weak duality in terms of the coupling constant e, and is self-dual at a particular value of $e$. We derive exact relations between transport coefficients for a $\nu=1$ quantum Hall state at the self-dual point. Using boson-fermion duality, we map the $\nu=1$ bosonic quantum Hall state to a Fermi sea of the dual fermion, and observed that the exact relationships between transport coefficients at the bosonic self-dual point are reproduced by a simple random phase approximation, coupled with a Drude formula, in the fermionic theory. We explain this success of the RPA by pointing out a cancellation of a parity-breaking term in the fermion theory which occurs only at the self-dual point, resulting in the fermion self-dual theory explored previously. In addition, we argue n that the equivalence of self-dual structure can be understood in terms of electromagnetic duality or modular invariance, and these features are not inherited by the non-relativistic cousins of the present model.

Bosonic quantum Hall states in single-layer two-dimensional optical lattices

Quantum Hall (QH) states of 2D single layer optical lattices are examined using Bose-Hubbard model (BHM) in presence of artificial gauge field. We study the QH states of both the homogeneous and inhomogeneous systems. For the homogeneous case we use cluster Gutzwiller mean field (CGMF) theory with cluster sizes ranging from $2\times 2$ to $5\times 5$. We, then, consider the inhomogeneous case, which is relevant to experimental realization. In this case, we use CGMF and exact diagonalization (ED). The ED studies are using lattice sizes ranging from $3\times 3$ to $4\times 12$. Our results show that the geometry of the QH states are sensitive to the magnetic flux $\alpha$ and cluster sizes. For homogeneous system, among various combinations of $1/5\leqslant \alpha\leqslant 1/2$ and filling factor $\nu$, only the QH state of $\alpha=1/4$ with $\nu=1/2$, $1$, $3/2$ and $2$ occur as ground states. For other combinations, the competing superfluid (SF) state is the ground state and QH state is metastable. For BHM with envelope potential all the QH states observed in homogeneous system exist for box potentials, but none for the harmonic potential. The QH states also persist for very shallow Gaussian envelope potential. As a possible experimental signature we study the two point correlations of the QH and SF states.

Emergent particle-hole symmetry in spinful bosonic quantum Hall systems

When a fermionic quantum Hall system is projected into the lowest Landau level, there is an exact particle-hole symmetry between filling fractions $\nu$ and $1-\nu$. We investigate whether a similar symmetry can emerge in bosonic quantum Hall states, where it would connect states at filling fractions $\nu$ and $2-\nu$. We begin by showing that the particle-hole conjugate to a composite fermion `Jain state' is another Jain state, obtained by reverse flux attachment. We show how information such as the shift and the edge theory can be obtained for states which are particle-hole conjugates. Using the techniques of exact diagonalization and infinite density matrix renormalization group, we study a system of two-component (i.e., spinful) bosons, interacting via a $\delta$-function potential. We first obtain real-space entanglement spectra for the bosonic integer quantum Hall effect at $\nu=2$, which plays the role of a filled Landau level for the bosonic system. We then show that at $\nu=4/3$ the system is described by a Jain state which is the particle-hole conjugate of the Halperin (221) state at $\nu=2/3$. We show a similar relationship between non-singlet states at $\nu=1/2$ and $\nu=3/2$. We also study the case of $\nu=1$, providing unambiguous evidence that the ground state is a composite Fermi liquid. Taken together our results demonstrate that there is indeed an emergent particle-hole symmetry in bosonic quantum Hall systems.

Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries

Gapped quantum liquids (GQL) include both topologically ordered states (with long range entanglement) and symmetry protected topological (SPT) states (with short range entanglement). In this paper, we propose a classification of 2+1D GQL for both bosonic and fermionic systems: 2+1D bosonic/fermionic GQLs with finite on-site symmetry are classified by non-degenerate unitary braided fusion categories over a symmetric fusion category (SFC) $\cal E$, abbreviated as $\text{UMTC}_{/\cal E}$, together with their modular extensions and total chiral central charges. The SFC $\cal E$ is $\text{Rep}(G)$ for bosonic symmetry $G$, or $\text{sRep}(G^f)$ for fermionic symmetry $G^f$. As a special case of the above result, we find that the modular extensions of $\text{Rep}(G)$ classify the 2+1D bosonic SPT states of symmetry $G$, while the $c=0$ modular extensions of $\text{sRep}(G^f)$ classify the 2+1D fermionic SPT states of symmetry $G^f$. Many fermionic SPT states are studied based on the constructions from free-fermion models. But it is not clear if free-fermion constructions can produce all fermionic SPT states. Our classification does not have such a drawback. We show that, for interacting 2+1D fermionic systems, there are exactly 16 superconducting phases with no symmetry and no fractional excitations (up to $E_8$ bosonic quantum Hall states). Also, there are exactly 8 $Z_2\times Z_2^f$-SPT phases, 2 $Z_8^f$-SPT phases, and so on. Besides, we show that two topological orders with identical bulk excitations and central charge always differ by the stacking of the SPT states of the same symmetry.

Topological electromagnetic responses of bosonic quantum Hall, topological insulator, and chiral semimetal phases in all dimensions

We calculate the topological part of the electromagnetic response of Bosonic Integer Quantum Hall (BIQH) phases in odd (spacetime) dimensions, and Bosonic Topological Insulator (BTI) and Bosonic chiral semi-metal (BCSM) phases in even dimensions. To do this we use the Nonlinear Sigma Model (NLSM) description of bosonic symmetry-protected topological (SPT) phases, and the method of gauged Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in dimension $2m-1$ ($m=1,2,\dots$), the bulk response to an electromagnetic field $A_{\mu}$ is characterized by a Chern-Simons term for $A_{\mu}$ with a level quantized in integer multiples of $m!$ (factorial). We also show that BTI states (which have an extra $\mathbb{Z}_2$ symmetry) can exhibit a $\mathbb{Z}_2$ breaking Quantum Hall effect on their boundaries, with this boundary Quantum Hall effect described by a Chern-Simons term at level $\frac{m!}{2}$. We show that the factor of $m!$ can be understood by requiring gauge invariance of the exponential of the Chern-Simons term on a general Euclidean manifold, and we also use this argument to characterize the electromagnetic and gravitational responses of fermionic SPT phases with $U(1)$ symmetry in all odd dimensions. We then use our gauged boundary actions for the BIQH and BTI states to (i) construct a bosonic analogue of a chiral semi-metal (BCSM) in even dimensions, (ii) show that the boundary of the BTI state exhibits a bosonic analogue of the parity anomaly of Dirac fermions in odd dimensions, and (iii) study anomaly inflow at domain walls on the boundary of BTI states. In a series of Appendices we derive important formulas and additional results. In particular, in Appendix A we use the connection between equivariant cohomology and gauged WZ actions to give a mathematical interpretation of the actions for the BIQH and BTI boundaries constructed in this paper.

Composite Fermi liquids in the lowest Landau level

We study composite fermi liquid (CFL) states in the lowest Landau level (LLL) limit at a generic filling $\nu = \frac{1}{n}$. We begin with the old observation that, in compressible states, the composite fermion in the lowest Landau level should be viewed as a charge-neutral particle carrying vorticity. This leads to the absence of a Chern-Simons term in the effective theory of the CFL. We argue here that instead a Berry curvature should be enclosed by the fermi surface of composite fermions, with the total Berry phase fixed by the filling fraction $\phi_B=-2\pi\nu$. We illustrate this point with the CFL of fermions at filling fractions $\nu=1/2q$ and (single and two-component) bosons at $\nu=1/(2q+1)$. The Berry phase leads to sharp consequences in the transport properties including thermal and spin Hall conductances, which in the RPA approximation are distinct from the standard Halperin-Lee-Read predictions. We emphasize that these results only rely on the LLL limit, and do not require particle-hole symmetry, which is present microscopically only for fermions at $\nu=1/2$. Nevertheless, we show that the existing LLL theory of the composite fermi liquid for bosons at $\nu=1$ does have an emergent particle-hole symmetry. We interpret this particle-hole symmetry as a transformation between the empty state at $\nu=0$ and the boson integer quantum hall state at $\nu=2$. This understanding enables us to define particle-hole conjugates of various bosonic quantum Hall states which we illustrate with the bosonic Jain and Pfaffian states. The bosonic particle-hole symmetry can be realized exactly on the surface of a three-dimensional boson topological insulator. We also show that with the particle-hole and spin $SU(2)$ rotation symmetries, there is no gapped topological phase for bosons at $\nu=1$.

Bridging coupled wires and lattice Hamiltonian for two-component bosonic quantum Hall states

We investigate a model of hard-core bosons with correlated hopping on the honeycomb lattice in an external magnetic field by means of a coupled-wire approach. It has been numerically shown that this model exhibits at half filling the bosonic integer quantum Hall (BIQH) state, which is a symmetry-protected topological phase protected by the $U(1)$ particle conservation [Y.-C. He et al., Phys. Rev. Lett. 115, 116803 (2015)]. By combining the bosonization approach and a coupled-wire construction, we analytically confirm this finding and show that it even holds in the strongly anisotropic (quasi-one-dimensional) limit. We discuss the stability of the BIQH phase against tunneling that break the separate particle conservations on different sublattices down to a global particle conservation. We further argue that a phase transition between two different BIQH phases is in a deconfined quantum critical point described by two copies of the (2+1)-dimensional $O(4)$ nonlinear sigma model with the topological $\theta$ term at $\theta=\pi$. Finally we predict a possible fractional quantum Hall state, the Halperin (221) state, at 1/6 filling.

Modular matrices from universal wave-function overlaps in Gutzwiller-projected parton wave functions

We implement the universal wave function overlap (UWFO) method to extract modular $S$ and $T$ matrices for topological orders in Gutzwiller-projected parton wave functions (GPWFs). The modular $S$ and $T$ matrices generate a projective representation of $SL(2,\mathbb{Z})$ on the degenerate-ground-state Hilbert space on a torus and may fully characterize the 2+1D topological orders, i.e. the quasi-particle statistics and chiral central charge (up to $E_8$ bosonic quantum Hall states). We used the variational Monte Carlo method to computed the $S$ and $T$ matrices of the chiral spin liquid (CSL) constructed by the GPWF on the square lattice, and confirm that the CSL carries the same topological order as the $\nu=\frac{1}{2}$ bosonic Laughlin state. We find that the non-universal exponents in UWFO can be small and direct numerical computation is able to be applied on relatively large systems. We also discuss the UWFO method for GPWFs on other Bravais lattices in two and three dimensions by using the Monte Carlo method. UWFO may be a powerful method to calculate the topological order in GPWFs.

Quantum simulation architecture for lattice bosons in arbitrary, tunable, external gauge fields

I describe a lattice of asymmetrical qubit pairs in arbitrary dimension, with couplings arranged so that the motion of single-qubit excited states mimics the behavior of charged lattice bosons hopping in a magnetic field. I show, in particular, that one can choose the parameters of the many-body circuit to reach a regime where the complex hopping phase between any two elements can be tuned to any value by simply adjusting the relative phases of two applied oscillating voltage signals. I also propose specific realizations of our model using coupled three junction flux qubits or transmon qubits, in which one can reach the strongly interacting bosonic quantum Hall limit where one will find anyonic excitations. This model could also be studied in trapped ions, and the superconducting circuits could be used for topological quantum computation.

Particle entanglement spectra for quantum Hall states on lattices

We use particle entanglement spectra to characterize bosonic quantum Hall states on lattices, motivated by recent studies of bosonic atoms on optical lattices. Unlike for the related problem of fractional Chern insulators, very good trial wave functions are known for fractional quantum Hall states on lattices. We focus on the entanglement spectra for the Laughlin state at $\ensuremath{\nu}=1/2$ for the non-Abelian Moore-Read state at $\ensuremath{\nu}=1$. We undertake a comparative study of these trial states to the corresponding ground states of repulsive two-body or three-body contact interactions on the lattice. The magnitude of the entanglement gap is studied as a function of the interaction strength on the lattice, giving insight into the nature of Landau-level mixing. In addition, we compare the performance of the entanglement gap and overlaps with trial wave functions as possible indicators for the topological order in the system. We discuss how the entanglement spectra allow to detect competing phases such as a Bose-Einstein condensate.

Vortices and Dynamics in Trapped Bose-Einstein Condensates

I review the basic physics of ultracold dilute trapped atomic gases, with emphasis on Bose-Einstein condensation and quantized vortices. The hydrodynamic form of the Gross-Pitaevskii equation (a nonlinear Schr{\"o}dinger equation) illuminates the role of the density and the quantum-mechanical phase. One unique feature of these experimental systems is the opportunity to study the dynamics of vortices in real time, in contrast to typical experiments on superfluid $^4$He. I discuss three specific examples (precession of single vortices, motion of vortex dipoles, and Tkachenko oscillations of a vortex array). Other unusual features include the study of quantum turbulence and the behavior for rapid rotation, when the vortices form dense regular arrays. Ultimately, the system is predicted to make a quantum phase transition to various highly correlated many-body states (analogous to bosonic quantum Hall states) that are not superfluid and do not have condensate wave functions. At present, this transition remains elusive. Conceivably, laser-induced synthetic vector potentials can serve to reach this intriguing phase transition.

Density matrix renormalization group for bosonic quantum Hall effect

We developed a density matrix renormalization-group technique to study quantum Hall fractions of fast rotating bosons. In this paper we present a discussion of the method together with the results which we obtain in three distinct cases of the narrow channel, cylinder and spherical geometries. In the narrow channel case, which is relevant to anisotropic confining traps in the limit of extremely fast rotation, we find a series of zero-temperature phase transitions in the strongly interacting regime as a function of the interaction strength between bosons. We compute energies and density profiles for different filling fractions on a cylinder and compare the convergence rates of the method in the cylinder and a sphere geometries.

Paired states in the even integer quantum Hall effect.

We argue that a new type of quantum Hall state requiring non-perturbative Landau level mixing arises at low electron density. In these states, up and down spin electrons pair to form spinless bosons that condense into a bosonic quantum Hall state. We describe a wavefunction for a paired quantum Hall state at $\nu=2$ and argue that it is stabilized by a BCS instability arising in flux attachment calculations. Based on this state, we derive a new global phase diagram for the integral quantum Hall effect with spin. Additional experimental implications are discussed.