# Research highlights

## Quantum Many-Body Scars

Recently, researchers from Harvard and MIT succeeded in trapping a record 51 atoms and individually controlling their quantum state, realizing what is called a quantum simulator. Their experiments in this system, presented in July 2017 at a conference in Trieste, revealed completely unexpected periodic oscillations in the dynamics of the interacting atoms. Our recent paper in Nature Physics has resolved the mystery of these previously inexplicable oscillations. The theoretical explanation they proposed introduces a concept of a “quantum many-body scar” (see figure) that alters our understanding of the dynamics that are possible in many-body quantum systems.

Imagine a ball bouncing around in an oval stadium. It will bounce around chaotically, back and forth through the available space. As its motion is random, it will sooner or later visit every place in the stadium. Amidst all the chaos, however, there might be a potential for order: if the ball happens to hit the wall at a special spot and at the “correct” angle of incidence, it might end up in a *periodic orbit*, visiting the same places in the stadium over and over and not visiting the others. Such a periodic orbit is extremely unstable as the slightest perturbation will divert the ball off its track and back into chaotic pondering around the stadium.

The same idea is applicable to quantum systems, except that instead of a ball bouncing around, we are looking at a wave, and instead of a trajectory, we are observing a probability function. Classical periodic orbits can cause a quantum wave to be concentrated in its vicinity, causing a “scar”-like feature in a probability that would otherwise be uniform. Such imprints of clas

sical orbits on the probability function have been named “quantum scars”. The phenomenon, however, was only expected to happen with a single quantum particle, as the complexity of the system rises dramatically with every additional particle, making periodic orbits more and more unlikely.

In our study we explain the experimental observation with the occurrence of quantum many-body scars. We also identify the many-particle unstable periodic orbit behind the scar behavior as the coherent oscillation of atoms between the excited and ground states. Intuitively, the quantum many-body scar may be envisioned as a part of configuration space that is to some extent “shielded” from chaos, thus leading to a much slower relaxation. In other words: the system takes longer to return to chaos—the equilibrium state.

More information:

- Story on Leeds website
- Viewpoint on our paper by Vanja Dunjko and Maxim Olshanii
- Quanta magazine article https://www.quantamagazine.org/quantum-scarring-appears-to-defy-universes-push-for-disorder-20190320/

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## How to see anyons with a scanning tunneling microscope

Ordinary particles are either fermions or bosons. Bosons like to cluster together, fermions effectively repel each other – all because of a simple ±1 sign picked up by their quantum-mechanical wave function as two particles are swapped around each other. But in a solid containing many-electrons, stranger types of particles can emerge. These new particles are collective states of many-electrons and they are called anyons because their wave function can pick up an arbitrary complex phase upon the exhange, not just ±1. As a result, statistical properties of anyons are very different from bosons or fermions, for example their charge ends up being only a fraction of the electron charge.

Where could one look for anyons? Anyons are believed to be abundant in fractional quantum Hall phases, which form in two-dimensional electron gases at cryogenic temperatures and in the presence of a strong magnetic field. Two-dimensional electron gas can be fabricated in semiconductor materials like GaAs, but more recently it can also be realised in graphene, which is an atomically thin layer of carbon atoms. Theory says that anyons form because of two essential ingredients: topology and interactions. Topology is supplied by the external magnetic field, which gives electrons the Aharonov-Bohm phase as they move around, while Coulomb interactions force the electrons to surrender their identity and “fractionalise” into anyons. There is a wealth of indirect experimental evidence supporting the existence of anyons, including the observation of their fractional charge [3] and interferometric measurements of their anyonic phase [4]. But in order to unambiguously demonstrate the existence of anyons, an experiment should measure their unique “barcode”.

In our recent paper [1], we propose to use a scanning tunneling microscope (STM) to detect the unique fingerprint of anyons in the fractional quantum Hall phases in graphene (see Figure). By ejecting an electron into the strongly-correlated fractional quantum Hall liquid in graphene, the STM is sensitive to how the electron chooses to fractionalise into several anyons. The signal of the STM, resolved in energy and momentum, is predicted to display a series of sharp levels, which correspond to bound states of the anyons. The discrete sequence of these numbers is a unique fingerprint of a fractional quantum Hall phase.

Moreover, if anyons bind to an impurity lying somewhere in the sample, the local STM spectrum is predicted to change in a specific way. The theory, proposed by Haldane in 1991 [5], which governs the counting of the STM spectrum has been known as “fractional exclusion statistics”. Thus far, it has not been possible to directly test this theory because the two-dimensional electron gases in semiconductor materials were not directly accessible. With recent advances in graphene, which hosts a two-dimensional electron gas on its surface, the STM spectroscopy of anyons becomes possible. As the paper [1] explains, this enables a new approach not only to experimentally detect anyons, but also to manipulate them. The latter would be the first step towards quantum computation with anyons – a conceptually new way to store and process information which is protected from many sources of error [6].

**References**

[1] *Imaging Anyons with Scanning Tunneling Microscopy*, Zlatko Papic, Roger S. K. Mong, Ali Yazdani, and Michael P. Zaletel, Phys. Rev. X 8, 011037 (2018); see also the accompanying synopsis in Physics.

[2] *On the theory of identical particles*, J. M. Leinaas and J. Myrheim, Il Nuovo Cimento B 37, 1 (1977);* Quantum Mechanics of Fractional-Spin Particles*, F. Wilczek, Phys. Rev. Lett. 49, 957 (1982).

[3] *Direct Observation of a Fractional Charge*, R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Nature 389, 162 (1997).

[4] *Magnetic-Field-Tuned Aharonov-Bohm Oscillations and Evidence for Non-Abelian Anyons at ν=5/2*, R. L. Willett, C. Nayak, K. Shtengel, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 111, 186401 (2013).

[5] *“Fractional Statistics” in Arbitrary Dimensions: A Generalization of the Pauli Principle*, F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991).

[6] *Topological quantum computation*, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen and Zhenghan Wang, Bull. Amer. Math. Soc. 40, 31 (2003);* Introduction to Topological Quantum Computation*, Jiannis K. Pachos, Cambridge University Press (2012).

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## How complex are quantum states?

Theoretical physics aims to understand the big picture of the world. This is a very complex picture because of the myriad of emergent phenomena, such as the high-temperature superconductivity or the quantum Hall effect. If we zoom in closely on one of these, all we see are ordinary electrons; but if we look at them from far away, like in the paintings by the French pointillists, a new pattern emerges, revealing fascinating properties like the current flowing without resistance. This happens because of interactions between electrons. Interactions can make particles lose their identity and behave collectively, changing the properties of the system. The problem is – the physicists’ intuition is based almost entirely on free systems, where interactions are negligible. What about more general, interacting systems like the ones above? How do we find a free system that most closely “resembles” them?

In our recent work [1], we propose a way to crack this very fundamental question. We approach quantum many-body states as biologists who befriended computer scientists: we perform the anatomy of a quantum state by decomposing it into parts, and measure its internal correlations. From this data, we evaluate the “interaction distance” between this state and all free states. Our approach gives a new way to look at quantum states: it defines a hyperplane where all free states live, and gives us a universal “ruler” to measure the distance of any state from that plane [see Figure]. In other words, we measure the complexity of a quantum state, in the same way that the complexity of a number is specified by how many prime numbers appear in its factorisation. By factorising quantum correlations, we obtain a simple, compressed description of a quantum state, similar to modern video streaming services which provide high resolution image quality at a lower bitrate due to efficient compression.

[1] C. J. Turner, K. Meichanetzidis, Z. Papic, and J. K. Pachos, Nature Communications **8**, 14926 (2017).

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## 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)

## 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).

## 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.