# Research highlights

## Stretching and squeezing the Haldane pseudopotentials: a new language for the anisotropic fractional quantum Hall effect

In a classic paper from 1983 [1], Duncan Haldane (Nobel Prize in Physics, 2016) formulated what is now known as “the Haldane pseudopotentials” to describe the fractional quantum Hall effect. The latter phenomenon, where the Hall conductance of a two-dimensional electron gas in a magnetic field is curiously quantised in terms of rational numbers like 1/3 or 2/5, had been discovered a year earlier by Tsui, Stormer and Gossard [2]. The effect, however, remained a mystery until the early 1983 when Robert Laughlin explained it as a consequence of subtle correlations between the electrons which make them form exotic kinds of quantum fluids. (Tsui, Stormer and Laughlin shared the 1998 Nobel Prize for their discovery.) One of the crucial steps in the verification and ultimate acceptance of Laughlin’s theory came from Haldane’s pseudopotentials, which allowed to write down a rotationally-invariant wave function for the Laughlin fluid and explained why such a state could describe the real system of electrons interacting via Coulomb force.

An underlying assumption of the Haldane pseudopotentials has been that the electron system is rotationally invariant – it looks the same in x and y directions. However, it is known that experimental semiconductor systems, which realise the quantum Hall effect, are not rotationally invariant: for example, semiconductors often have different effective masses along x- and y-directions. A natural question then arises: can Haldane pseudopotentials be defined for quantum Hall systems which are not invariant under x-y rotation?

In our recent paper published in Physical Review Letters and highlighted as Editors’ Suggestion [4], we have generalised the 1983 work of Haldane by formulating the Electrons in the fractional quantum Hall states bind to magnetic fluxes, thus forming composite objects with finite area (yellow). If the magnetic field is tilted from the perpendicular direction (red arrows), the shape of particle-flux composites will fluctuate. A description of such systems requires the generalised pseudopotentials developed in Ref. [4].pseudopotentials for anisotropic fractional quantum Hall systems. This work introduces a new and universal language that allows to describe a broad class of fractional quantum Hall systems without rotational symmetry, such as experiments in tilted magnetic field, or the so-called nematic quantum Hall states which spontaneously break rotational symmetry, similar to liquid crystals. The new language also illuminates a fundamental characteristic of fractional quantum Hall fluids – their geometric degree of freedom. Such degrees of freedom determine the properties of these fluids at low energies and have recently attracted much attention because of connections with models of quantum gravity in two dimensions.

**References**

[1] F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983).

[2] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).

[3] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

[4] Bo Yang, Zi-Xiang Hu, Ching-hua Lee, and Z. Papic, Phys. Rev. Lett. 118, 146403 (2017).

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## Probing the geometry of the Laughlin state

It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantum Hall states—the filling Laughlin state. We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a ‘pair amplitude’ operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. These various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.

New Journal of Physics 18, 025011 (2016)

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## Chiral spin liquid in the Haldane-Hubbard model

Motivated by recent ultracold atom experiments on Chern insulators, we study the honeycomb lattice Haldane-Hubbard Mott insulator of spin-1/2 fermions using exact diagonalization and density matrix renormalization group methods. We show that this model exhibits various chiral magnetic orders including a wide regime of triple-Q tetrahedral order. Incorporating third-neighbor hopping frustrates and quantum-melts this tetrahedral spin crystal. From analyzing the low energy spectrum, many-body Chern numbers, entanglement spectra, and modular matrices, we identify the molten state as a chiral spin liquid (CSL) with gapped semion excitations. We formulate the Chern-Simons-Higgs theory of the spin crystallization transition from the CSL to tetrahedral state.

Physical Review Letters 116, 137202 (2016).

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## Merons and deconfined criticality in the quantum Hall bilayer

Quantum Hall bilayer phase diagram with respect to interlayer distance bears a remarkable similarity with phase diagrams of strongly correlated systems as a function of doping, with magnetic ordering on the one end and Fermi-liquid-like behavior on the other. We discuss possible state of the bilayer for intemediate distances and argue there is a possibility for meron deconfinement, i.e., the deconfinement of the vortex excitations of the magnetically ordered phase.

Physical Review B 92, 195311 (2015).

## Many-body localisation transition

We propose a new approach to probing ergodicity and its breakdown in quantum many-body systems based on their response to a local perturbation. We study the distribution of matrix elements of a local operator between the system’s eigenstates, finding a qualitatively different behaviour in the many-body localized (MBL) and ergodic phases. To characterize how strongly a local perturbation modifies the eigenstates, we introduce the parameter G which represents a disorder-averaged ratio of a typical matrix element of a local operator to the energy level spacing; this parameter is reminiscent of the Thouless conductance in the single-particle localization.

Physical Review X 5, 041047 (2015).

## Fibonacci anyons

The ν = 12/5 fractional quantum Hall plateau observed in GaAs wells is a suspect in the search for non-Abelian Fibonacci anyons. We find evidence, using quantum entanglement, that this state has the topological order corresponding to Fibonacci anyons. We point out extremely close energetic competition between the Fibonacci phase and a charge-density ordered phase, which suggests that even small particle-hole symmetry breaking perturbations can explain the experimentally observed asymmetry between 12/5 and 13/5 states.

http://arxiv.org/abs/1505.02843

## How to construct parent Hamiltonians for quantum Hall states?

Many fractional quantum Hall wave functions are known to be unique and highest-density zero modes of certain “pseudopotential” Hamiltonians. Examples include the Read-Rezayi series (in particular, the Laughlin, Moore-Read and Read-Rezayi Z3 states), and more exotic non-unitary (Haldane-Rezayi, Gaffnian states) or irrational states (Haffnian state). While a systematic method to construct such Hamiltonians is available for the infinite plane or sphere geometry, its generalization to manifolds such as the cylinder or torus, where relative angular momentum is not an exact quantum number, has remained an open problem. Here we develop a geometric approach for constructing pseudopotential Hamiltonians in a universal manner that naturally applies to all geometries. Our method generalizes to the multicomponent SU(n) cases with a combination of spin or pseudospin (layer, subband, valley) degrees of freedom.

Physical Review X 5, 041003 (2015).

## A new type of Pfaffian state in the 1/3+1/3 quantum Hall bilayer

Bilayer quantum Hall systems, realized either in two separated wells or in the lowest two sub-bands of a wide quantum well, provide an experimentally realizable way to tune between competing quantum orders at the same filling fraction. Using newly developed density matrix renormalization group techniques combined with exact diagonalization, we study the problem of quantum Hall bilayers at filling 1/3 + 1/3. By slightly modifying the interlayer repulsion we find a robust non-Abelian phase which we identify as the “interlayer-Pfaffian” phase. In addition to non-Abelian statistics similar to the Moore-Read state, it exhibits a novel form of bilayer-spin charge separation.

Physical Review B 91, 205139 (2015).

## Is many-body localization possible without disorder?

Recently it has been suggested that many-body localization (MBL) can occur in translation-invariant systems, and candidate 1D models have been proposed. We find that such models, in contrast to MBL systems with quenched disorder, typically exhibit much more severe finite-size effects due to the presence of two or more vastly different energy scales. Our results suggest that MBL in translation-invariant systems with two or more very different energy scales is less robust than perturbative arguments suggest, possibly pointing to the importance of non-perturbative effects which induce delocalization in the thermodynamic limit.

Annals of Physics 362, 714 (2015).

## Tunable fractional quantum Hall effect in bilayer graphene

For a popular account of our recent work on bilayer graphene, see here.

## Solvable models for unitary and non-unitary topological states

Check out my recent paper on a zoo of solvable models for exotic animals such as Pfaffians, Gaffnians, Haffnians etc.