# Dr Gunnar Möller

Royal Society University Research Fellow

Dr Gunnar Möller is a condensed matter theorist with an interest in strongly correlated and topologically ordered materials. Thanks to his expertise in state-of the art computer simulations, he currently holds a Royal Society University Research Fellowship, which allows him to further develop ambitious new numerical approaches to study superconductors or heavy fermion compounds. Gunnar graduated with a French Master's degree (Diplome d'Etudes Approfondies) in Theoretical Physics from the Ecole Normale Supérieure in Paris (2003). He pursued his PhD at the University of Paris XI as a member of the Laboratoire de Physique Théorique et Modèles Statistiques. During his doctoral studies, he also visited Professor Steve Simon's group at the Bell Laboratories, Murray Hill, NJ. On completion of his PhD in 2006, Gunnar moved to Cambridge, UK, to take up a postdoctoral position in the group of Professor Nigel Cooper. He developed his personal line of research on strongly correlated phases of matter as a Research Fellow at the Cavendish Laboratory thanks to the support of several prestigious fellowship awards, including a Trinity Hall Research Fellowship (2008-2011), a Leverhulme Early Career Fellowship (2011-2013), and a Royal Society University Research Fellowship (2013-2016). His international collaborations were supported by an ICAM Fellowship for a collaboration with Professor Victor Gurarie at UCO Boulder (2008-10), and by a CNRS visiting researcher position for collaboration with Dr Jerome Dubail, Université de Lorraine, Nancy, France (2015-16). Gunnar joined the faculty at the University of Kent in May 2016.

## Research interests

Dr Gunnar Möller's research revolves around the interplay of strong interactions and
topology, which can give rise to collective phases of matter with exciting new properties. Some of the most interesting topological phenomena can be found in two-dimensional quantum systems. A prominent example is fractional quantum Hall phases of electrons in strong magnetic fields, which realise fractionalised quasiparticles that carry fractions of the charge of an electron. More excitingly, they can also carry so-called non-Abelian exchange statistics which allow one to
manipulate the many-body quantum state in a well-defined manner through the controlled movement of quasiparticles, providing an ideal platform for quantum computation. In practice, Gunnar's work covers two main aspects of investigation, as outlined below.Developing high-performance numerical simulations of strongly correlated materialsGunnar develops new computer simulations of various different flavours. His approach relies on combining analytical insights into collective properties of emergent phases at low temperatures on one hand, with quantitative modelling techniques for microscopic correlations on the other hand. This combination provides powerful tools which can give insights into a wide range of strongly correlated materials, spanning topics such as correlated superconductors and frustrated magnetism.Exact diagonalisation: Gunnar is a main developer of the DiagHam library for simulations of spin systems and fractional quantum Hall physics.Variational quantum Monte Carlo: Gunnar's group has explored the physics of fractional quantum Hall states using a range of variational QMC techniques, such as energy and variance minimisation.Diagrammatic Monte Carlo: Perturbative expansions in quantum field theories can be represented graphically by Feynman diagrams. Gunnar's group uses stochastic sampling techniques in the space of Feynman graphs to analyse the properties of novel quantum phases, exploring the physics of unitary Fermi gases and the Hubbard model.Matrix / tensor product states: Insights from quantum information theory have given rise to new tools for simulating strongly interacting matter. In particular, topological phases are well suited for descriptions in terms of their local entanglement. Gunnar and his team exploit this property to develop numerical approaches capturing the physics of fractional topological insulators.Realising novel phases of matterPart of Gunnar's research focuses on realising exciting new phases by 'quantum engineering', using the tools of materials science, or cold atomic gases. Examples include the creation of systems with synthetic magnetic fields, which arise from strain or spin-orbit coupling in solid state materials, and can also be generated using light-matter coupling for cold atomic gases. As a theorist, Gunnar is most interested in using such models to explore new types of topological phases such as fractional Chern insulators, topological superfluids, as well as new types of symmetry breaking phases such as supersolid phases.

## Publications

### Article

• Caio, M., Möller, G., Cooper, N. and Bhaseen, M. (2019). Topological Marker Currents in Chern Insulators. Nature Physics [Online] 15:257-261. Available at: http://dx.doi.org/10.1038/s41567-018-0390-7.
Topological states of matter exhibit many novel properties due to the presence of robust topological invariants such as the Chern index. These global characteristics pertain to the system as a whole and are not locally defined. However, local topological markers can distinguish between topological phases, and they can vary in space. In equilibrium, we show that the topological marker can be used to extract the critical behaviour of topological phase transitions. Out of equilibrium, we show that the topological marker spreads via a flow of currents emanating from the sample boundaries, and with a bounded maximum propagation speed. We discuss the possibilities for measuring the topological marker and its flow in experiment.
• Möller, G. and Cooper, N. (2018). Synthetic Gauge Fields for Lattices with Multi-Orbital Unit Cells: Routes towards a $\pi$-flux Dice Lattice with Flat Bands. New Journal of Physics [Online] 20. Available at: https://doi.org/10.1088/1367-2630/aad134.
We propose a general strategy for generating synthetic magnetic fields in complex lattices with non-trivial connectivity based on light-matter coupling in cold atomic gases. Our approach starts from an underlying optical flux lattice in which a synthetic magnetic field is generated by coupling several internal states. Starting from a high symmetry optical flux lattice, we superpose a scalar potential with a super- or sublattice period in order to eliminate links between the original lattice sites. As an alternative to changing connectivity, the approach can also be used to create or remove lattice sites from the underlying parent lattice. To demonstrate our concept, we consider the dice lattice geometry as an explicit example, and construct a dice lattice with a flux density of half a flux quantum per plaquette, providing a pathway to flat bands with a large band gap. While the intuition for our proposal stems from the analysis of deep optical lattices, we demonstrate that the approach is robust even for shallow optical flux lattices far from the tight-binding limit.

We also provide an alternative experimental proposal to realise a synthetic gauge field in a fully frustrated dice lattice based on laser-induced hoppings along individual bonds of the lattice, again involving a superlattice potential. In this approach, atoms with a long-lived excited state are trapped using an 'anti-magic' wavelength of light, allowing the desired complex hopping elements to be induced in a specific laser coupling scheme for the dice lattice geometry.
We conclude by comparing the complexity of these alternative approaches, and advocate that complex optical flux lattices provide the more elegant and easily generalisable strategy.
• Andrews, B. and Möller, G. (2018). Stability of Fractional Chern Insulators in the Effective Continuum Limit of |C|>1 Harper-Hofstadter Bands. Physical Review B: Condensed Matter and Materials Physics [Online] 97. Available at: http://dx.doi.org/10.1103/PhysRevB.97.035159.
We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number |C|>1. We analyze the states of the composite fermion series for bosons with contact interactions and (spinless) fermions with nearest-neighbor interactions. We examine the scaling of the many-body gap as the bands are tuned to the effective continuum limit n??1/|C|. Near these points, the Hofstadter model realises large magnetic unit cells that yield bands with perfectly flat dispersion and Berry curvature. We exploit the known scaling of energies in the effective continuum limit in order to maintain a fixed square aspect ratio in finite-size calculations. Based on exact diagonalization calculations of the band-projected Hamiltonian, we show that almost all finite-size spectra yield the ground state degeneracy predicted by composite fermion theory. We confirm that states at low ranks in the composite fermion hierarchy are the most robust, and yield a clear gap in the thermodynamic limit. For bosons in |C|=2 and |C|=3 bands, our data for the composite fermion states are compatible with a finite gap in the thermodynamic limit. We also report new evidence for gapped incompressible states of fermions in |C|>1 bands, which have large entanglement gaps. For cases with a clear spectral gap, we confirm that the thermodynamic limit commutes with the effective continuum limit. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that they feature |C|2 smooth sheets. We examine two cases associated with a bosonic integer quantum Hall effect (BIQHE): for ?=2 in |C|=1 bands, we find a strong competing state with a higher ground state degeneracy, so no clear BIQHE is found in the band-projected Hofstadter model; for ?=1 in |C|=2 bands, we present additional data confirming the existence of a BIQHE state.
• Liu, Z., Möller, G. and Bergholtz, E. (2017). Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States. Physical Review Letters [Online] 119. Available at: http://dx.doi.org/10.1103/PhysRevLett.119.106801.
We investigate extrinsic wormhole-like twist defects that effectively increase the genus of space in lattice versions of multi-component fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a “proof of concept” demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.
• Sendetskyi, O., Anghinolfi, L., Scagnoli, V., Möller, G., Leo, N., Alberca, A., Kohlbrecher, J., Lüning, J., Staub, U. and Heyderman, L. (2016). Magnetic diffuse scattering in artificial kagome spin ice. Physical Review B [Online] 93:224413. Available at: http://dx.doi.org/10.1103/PhysRevB.93.224413.
The study of magnetic correlations in dipolar-coupled nanomagnet systems with synchrotron X-ray scattering provides a means to uncover emergent phenomena and exotic phases, in particular in systems with thermally active magnetic moments. From the diffuse signal of soft X-ray resonant magnetic scattering, we have measured magnetic correlations in a highly dynamic artificial kagome spin ice with sub-70 nm Permalloy nanomagnets. On comparing experimental scattering patterns with Monte Carlo simulations based on a needle-dipole model, we conclude that kagome ice I phase correlations exist in our experimental system even in the presence of moment fluctuations, which is analogous to bulk spin ice and spin liquid behavior. In addition, we describe the emergence of quasi-pinch points in the magnetic diffuse scattering in the kagome ice I phase. These quasi-pinch points bear similarities to the fully developed pinch points with singularities of a magnetic Coulomb phase, and continually evolve into the latter on lowering the temperature. The possibility to measure magnetic diffuse scattering with soft X-rays opens the way to study magnetic correlations in a variety of nanomagnetic systems.
• Jackson, T., Möller, G. and Roy, R. (2015). Geometric stability of topological lattice phases. Nature Communications [Online] 6:8629. Available at: http://dx.doi.org/10.1038/ncomms9629.
The fractional quantum Hall (FQH) effect illustrates the range of novel phenomena which can arise in a topologically ordered state in the presence of strong interactions. The possibility of realizing FQH-like phases in models with strong lattice effects has attracted intense interest as a more experimentally accessible venue for FQH phenomena which calls for more theoretical attention. Here we investigate the physical relevance of previously derived geometric conditions which quantify deviations from the Landau level physics of the FQHE. We conduct extensive numerical many-body simulations on several lattice models, obtaining new theoretical results in the process, and find remarkable correlation between these conditions and the many-body gap. These results indicate which physical factors are most relevant for the stability of FQH-like phases, a paradigm we refer to as the geometric stability hypothesis, and provide easily implementable guidelines for obtaining robust FQH-like phases in numerical or real-world experiments.
• Möller, G. and Cooper, N. (2015). Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number. Physical Review Letters [Online] 115:126401. Available at: http://dx.doi.org/10.1103/PhysRevLett.115.126401.
The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number, C.
We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C|>1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor, $\nu$, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors nu = r/(r|C| +1) for bosons, or nu = r/(2r|C| +1) for fermions. This series includes a bosonic integer quantum Hall state (bIQHE) in |C|=2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.
• Möller, G., Hormozi, L., Slingerland, J. and Simon, S. (2014). Josephson-coupled Moore-Read states. Physical Review B: Condensed Matter and Materials Physics [Online] 90:235101. Available at: http://dx.doi.org/10.1103/PhysRevB.90.235101.
We study a quantum Hall bilayer system of bosons at total filling factor nu = 1, and study the phase that results from short ranged pair-tunneling combined with short ranged interlayer interactions.
We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys. Rev. Lett. 108, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase while the other is gapless. However, on introduction of a pair tunneling term, the second system becomes gapped and develops the same topological order as the gapped Hamiltonian. Supported by the exact solution of the full zero-energy quasihole spectrum and a conformal field theory approach, we develop an intuitive picture of this system as two coupled composite fermion superconductors. In this language, pair tunneling provides a Josephson coupling of the superconducting phases of the two layers, and gaps out the Goldstone mode associated with particle transport between the layers. In particular, this implies that quasiparticles are confined between the layers. In the bulk, the resulting phase has the topological order of the Halperin 220 phase with U(1)_2 x U(1)_2 topological order, but it is realized in the symmetric/antisymmetric-basis of the layer index. Consequently, the edge spectrum at a fixed particle number reveals an unexpected U(1)_4 x U(1) structure.
• Bühler, A., Lang, N., Kraus, C., Möller, G., Huber, S. and Büchler, H. (2014). Majorana modes and p-wave superfluids for fermionic atoms in optical lattices. Nature communications [Online] 5:4504. Available at: http://dx.doi.org/10.1038/ncomms5504.
The quest for realizations of non-Abelian phases of matter, driven by their possible use in fault-tolerant
topological quantum computing, has been spearheaded by recent developments in p-wave superconductors. The chiral p_x + i p_y-wave superconductor in two-dimensions exhibiting Majorana modes provides the simplest phase supporting non-Abelian quasiparticles and can be seen as the blueprint of fractional topological order. Alternatively, Kitaev's Majorana wire has emerged as an ideal toy model to understand Majorana modes. Here, we present a way to make the transition from Kitaev's Majorana wires to two-dimensional p-wave superconductors in a system with cold atomic gases in an optical lattice. The main idea is based on an approach to generate p-wave interactions by coupling orbital degrees of freedom with strong s-wave interactions. We demonstrate how this design can induce Majorana modes at edge dislocations in the optical lattice and we provide an experimentally feasible protocol for the observation of the non-Abelian statistics.
• Scaffidi, T. and Möller, G. (2012). Adiabatic Continuation of Fractional Chern Insulators to Fractional Quantum Hall States. Physical Review Letters [Online] 109:246805. Available at: http://dx.doi.org/10.1103/PhysRevLett.109.246805.
We show how the phases of interacting particles in topological flat bands, known as fractional Chern insulators, can be adiabatically connected to incompressible fractional quantum Hall liquids in the lowest Landau-level of an externally applied magnetic field. Unlike previous evidence suggesting the similarity of these systems, our approach enables a formal proof of the equality of their topological orders, and furthermore this proof robustly extends to the thermodynamic limit. We achieve this result using the hybrid Wannier orbital basis proposed by Qi [Phys. Rev. Lett. 107, 126803 (2011)] in order to construct interpolation Hamiltonians that provide continuous deformations between the two models. We illustrate the validity of our approach for the groundstate of bosons in the half filled Chern band of the Haldane model, showing that it is adiabatically connected to the nu=1/2 Laughlin state of bosons in the continuum fractional quantum Hall problem.
• Sterdyniak, A., Regnault, N. and Möller, G. (2012). Particle entanglement spectra for quantum Hall states on lattices. Physical Review B [Online] 86:165314. Available at: http://dx.doi.org/10.1103/PhysRevB.86.165314.
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 wavefunctions are known for fractional quantum Hall states on lattices. We focus on the entanglement spectra for the Laughlin state at nu=1/2 for the non-Abelian Moore-Read state at nu=1. We undertake a comparative study of these trial states to the corresponding groundstates 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 insights into the nature of Landau-level mixing. In addition, we compare the performance of the entanglement gap and overlaps with trial wavefunctions 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.
• Hormozi, L., Möller, G. and Simon, S. (2012). Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux. Physical Review Letters [Online] 108. Available at: http://dx.doi.org/10.1103/PhysRevLett.108.256809.
We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low energy physics is mapped to a multi-species continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or \emph{pseudospin}.
We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases
can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wave functions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wave functions can be destabilized when umklapp processes become strong.
• Möller, G. and Cooper, N. (2012). Correlated Phases of Bosons in the Flat Lowest Band of the Dice Lattice. Physical Review Letters [Online] 108. Available at: http://dx.doi.org/10.1103/PhysRevLett.108.045306.
We study correlated phases occurring in the flat lowest band of the dice lattice model at flux density one half. We discuss how to realize the
dice lattice model, also referred to as the $\mathcal{T}_3$ lattice, in cold atomic gases. We construct the projection of the model to the lowest
dice band, which yields a Hubbard-Hamiltonian with interaction-assisted hopping processes. We solve this model for bosons in two limits. In the
limit of large density, we use Gross-Pitaevskii mean-field theory to reveal time-reversal symmetry breaking vortex lattice phases. At low density,
we use exact diagonalization to identify three stable phases at fractional filling factors $\nu$ of the lowest band, including a
classical crystal at $\nu=1/3$, a supersolid state at $\nu=1/2$ and a Mott insulator at $\nu=1$.
• Bonderson, P., Feiguin, A., Möller, G. and Slingerland, J. (2012). Competing Topological Orders in the ?=12/5 Quantum Hall State. Physical Review Letters [Online] 108:36806. Available at: http://dx.doi.org/10.1103/PhysRevLett.108.036806.
We provide numerical evidence that a p_x-i p_y paired Bonderson-Slingerland (BS) non-Abelian hierarchy state is a strong candidate for the observed ? =12/5 quantum Hall plateau. We confirm the existence of a gapped incompressible ?=12/5 quantum Hall state with shift S = 2 on the sphere, matching that of the BS state. The exact ground state of the Coulomb interaction at S = 2 is shown to have a large overlap with the BS trial wave function. Larger overlaps are obtained with BS-type wave functions that are hierarchical descendants of general p_x - i p_y weakly paired states at ?=12/5. We perform a finite-size scaling analysis of the ground-state energies for ?=12/5 states at shifts corresponding to the BS (S = 2) and 3-clustered Read-Rezayi (S = -2) universality classes. This analysis reveals very tight competition between these two non-Abelian topological orders.
• Wójs, A., Möller, G. and Cooper, N. (2011). Search for non-Abelian statistics in half-filled Landau levels of graphene. Journal of Physics: Conference Series [Online] 334:12048. Available at: http://dx.doi.org/10.1088/1742-6596/334/1/012048.
We have employed large scale exact numerical diagonalization in Haldane spherical geometry in a comparative analysis of the correlated many-electron states in the half-filled low Landau levels of graphene and such conventional semiconductors as GaAs, including both spin and valley (i.e., pseudospin) degrees of freedom. We present evidence that the polarized Fermi sea of essentially non-interacting composite fermions remains stable against a pairing transition in both lowest Landau levels of graphene. However, it undergoes spontaneous depolarization, which in (ideal) graphene is unprotected for the lack of a single-particle pseudospin splitting. These results point to the absence of the non-Abelian Pfaffian phase in graphene.
• Wójs, A., Sreejith, G., Möller, G., Töke, C. and Jain, J. (2011). Composite Fermion Description of the Excitations of the Paired Pfaffian Fractional Quantum Hall State. Acta Physical Polonica A [Online] 120:839-842. Available at: http://dx.doi.org/10.12693/APhysPolA.120.830.
We review the recently developed bi-partite composite fermion model, in the context of so-called Pfaffian incompressible quantum liquid with fractional and non-Abelian quasiparticle statistics, a promising model for describing the correlated many-electron ground state responsible for fractional quantum Hall effect at the Landau level filling factor ? = 5/2. We use the concept of composite fermion partitions to demonstrate the emergence of an essential ingredient of the non-Abelian braid statistics – the topological degeneracy of spatially indistinguishable configurations of multiple widely separated (non-interacting) quasiparticles.
• Möller, G. and Simon, S. (2011). Trial Wavefunctions for the Goldstone Mode in ?=1/2+1/2 Quantum Hall Bilayers. Advances in Condensed Matter Physics [Online] 2011:815169. Available at: http://dx.doi.og/10.1155/2011/815169.
Based on the known physics of the excitonic superfluid or 111 state of the quantum Hall $\nu=1/2+1/2$ bilayer, we create a simple trial wavefunction ansatz for constructing a low energy branch of (Goldstone) excitations by taking the overall ground state and boosting one layer with respect to the other. This ansatz works extremely well for any interlayer spacing. For small $d$ this is simply the physics of the Goldstone mode, whereas for large $d$ this is a reflection of composite fermion physics. We find hints that certain aspects of composite fermion physics persist to low $d$ whereas certain aspects of Goldstone mode physics persist to high $d$. Using these results we show nonmonotonic behavior of the Goldstone mode velocity as a function of $d$.
• Möller, G., Wójs, A. and Cooper, N. (2011). Neutral Fermion Excitations in the Moore-Read State at Filling Factor ?=5/2. Physical Review Letters [Online] 107:36803. Available at: http://dx.doi.org/10.1103/PhysRevLett.107.036803.
We present evidence supporting the weakly paired Moore-Read phase in the half-filled second Landau level, focusing on some of the qualitative features of its excitations. Based on numerical studies, we show that systems with odd particle number at the flux N_\phi=2N-3 can be interpreted as a neutral fermion mode of one unpaired fermion, which is gapped. The mode is found to have two distinct minima, providing a signature that could be observed by photoluminescence. In the presence of two quasiparticles the same neutral fermion excitation is shown to be gapless, confirming expectations for non-Abelian statistics of the Ising model with degenerate fusion channels 1 and \psi.
• Wójs, A., Möller, G. and Cooper, N. (2011). Composite fermion dynamics in half-filled Landau levels of graphene. Acta Physica Polonica A [Online] 119:592. Available at: http://dx.doi.org/10.12693/APhysPolA.119.592.
We report on exact-diagonalization studies of correlated many-electron states in the half-filled Landau levels of graphene, including pseudospin (valley) degeneracy. We demonstrate that the polarized Fermi sea of non-interacting composite fermions remains stable against a pairing transition in the lowest two Landau levels. However, it undergoes spontaneous depolarization, which is unprotected owing to the lack of single-particle pseudospin splitting. These results suggest the absence of the Pfaffian phase in graphene.
• Möller, G., Cooper, N. and Gurarie, V. (2011). Structure and consequences of vortex-core states in p-wave superfluids. Physical Review B: Condensed Matter and Materials Physics [Online] 83:14513. Available at: http://dx.doi.org/10.1103/PhysRevB.83.014513.
It is now well established that in two-dimensional chiral $p$-wave paired superfluids, the vortices carry zero-energy
modes which obey non-abelian exchange statistics and can potentially be used for topological quantum computation.
In such superfluids there may also exist other excitations below the bulk gap inside the cores of vortices.
We study the properties of these subgap states, and argue that their
presence affects the topological protection of the zero modes.
In conventional superconductors where the chemical potential is of the order of the Fermi energy
of a non-interacting Fermi gas, there is a large number of subgap states and the mini-gap
towards the lowest of these states is a small fraction of the Fermi energy. It is therefore difficult
to cool the system to below the mini-gap and at experimentally available temperatures, transitions
between the subgap states, including the zero modes, will occur and can alter the quantum
states of the zero-modes. Consequently, qubits defined uniquely in terms of the zero-modes
do not remain coherent.
We show that compound qubits involving the zero-modes and the parity of the occupation number
of the subgap states on each vortex are still well defined. However, practical schemes taking into
account all subgap states would nonetheless be difficult to achieve.
We propose to avoid this difficulty by working in the regime of small chemical potential $\mu$, near the transition
to a strongly paired phase, where the number of subgap states is reduced. We develop the theory to describe
this regime of strong pairing interactions and we show how the subgap states are ultimately absorbed into the bulk gap.
Since the bulk gap also vanishes as $\mu\to 0$ there is an optimum value $\mu_c$ which maximises the combined gap.
We propose cold atomic gases as candidate systems where the regime of strong interactions can be explored,
and explicitly evaluate $\mu_c$ in a Feshbach resonant $^{40}$K gas.
• Möller, G. and Cooper, N. (2010). Condensed ground states of frustrated Bose-Hubbard models. Physical Review A [Online] 82:63625. Available at: http://dx.doi.org/10.1103/PhysRevA.82.063625.
We study theoretically the ground states of two-dimensional Bose-Hubbard models which are frustrated by gauge fields. Motivated by recent proposals for the implementation of optically induced gauge potentials, we focus on the situation in which the imposed gauge fields give rise to a pattern of staggered fluxes of magnitude ? and alternating in sign along one of the principal axes. For ?=1/2 this model is equivalent to the case of uniform flux per plaquette n?=1/2, which, in the hard-core limit, realizes the “fully frustrated” spin-1/2 XY model. We show that the mean-field ground states of this frustrated Bose-Hubbard model typically break translational symmetry. Given the presence of both a non-zero superfluid fraction and translational symmetry breaking, these phases are supersolid. We introduce a general numerical technique to detect broken symmetry condensates in exact diagonalization studies. Using this technique we show that, for all cases studied, the ground state of the Bose-Hubbard model with staggered flux ? is condensed, and we obtain quantitative determinations of the condensate fraction. We discuss the experimental consequences of our results. In particular, we explain the meaning of gauge invariance in ultracold-atom systems subject to optically induced gauge potentials and show how the ability to imprint phase patterns prior to expansion can allow very useful additional information to be extracted from expansion images.
• Wójs, A., Möller, G., Simon, S. and Cooper, N. (2010). Skyrmions in the Moore-Read State at ?=5/2. Physical Review Letters [Online] 104:86801. Available at: https://doi.org/10.1103/PhysRevLett.104.086801.
We study spinful excitations in the Moore-Read state. Energetics of the skyrmion based on a spin-wave picture support the existence of skyrmion excitations in the plateau below $\nu=5/2$. This prediction is then tested numerically. We construct trial skyrmion wavefunctions for general FQHE states, and obtain significant overlaps for the predicted skyrmions of $\nu=5/2$. The case of $\nu=5/2$ is particularly interesting as skyrmions have twice the charge of quasiparticles (qp's). As the spin polarization of the system is tuned from full to none, we observe a transition between qp- and skyrmion-like behaviour of the excitation spectrum that can be interpreted as binding of qp's. Our ED results confirm that skyrmion states are energetically competitive with quasiparticles at low Zeeman coupling. Disorder and large density of quasiparticles are discussed as further mechanisms for depolarization.
• Möller, G. and Moessner, R. (2009). Magnetic multipole analysis of kagome and artificial ice dipolar arrays. Physical Review B: Condensed Matter and Materials Physics [Online] 80:140409. Available at: http://dx.doi.org/10.1103/PhysRevB.80.140409.
We analyse an array of linearly extended monodomain dipoles forming square and kagome lattices. We find that its phase diagram contains two (distinct) finite-entropy kagome ice regimes - one disordered, one algebraic - as well as a low-temperature ordered phase. In the limit of the islands almost touching, we find a staircase of corresponding entropy plateaux, which is analytically captured by a theory based on magnetic charges. For the case of a modified square ice array, we show that the charges (monopoles') are excitations experiencing two distinct Coulomb interactions: a magnetic three-dimensional' one as well as a logarithmic two dimensional' one of entropic origin.
• Möller, G. and Cooper, N. (2009). Composite Fermion Theory for Bosonic Quantum Hall States on Lattices. Physical Review Letters [Online] 103:105303. Available at: http://dx.doi.org/10.1103/PhysRevLett.103.105303.
We study the groundstates of the Bose-Hubbard model in a uniform effective magnetic field, illustrating the physics of cold atomic gases on rotating optical lattices'. Mapping the bosons to composite fermions leads to the prediction of quantum Hall fluids that have no counterpart in the continuum. We construct trial wavefunctions for these phases, and perform numerical tests of the predictions of the composite fermion model. Our results establish the existence of strongly correlated phases beyond those in the continuum limit, and provide evidence for a wider scope of the composite fermion approach beyond its application to the lowest Landau-level.
• Papi?, Z., Möller, G., Milovanovic, M., Regnault, N. and Goerbig, M. (2009). Fractional quantum Hall state at ?=(1)/(4) in a wide quantum well. Physical Review B: Condensed Matter and Materials Physics [Online] 79:245325. Available at: http://dx.doi.org/10.1103/PhysRevB.79.245325.
We investigate, with the help of Monte-Carlo and exact-diagonalization calculations in the spherical geometry, several compressible and incompressible candidate wave functions for the recently observed quantum Hall state at the filling factor $\nu=1/4$ in a wide quantum well. The quantum well is modeled as a two-component system by retaining its two lowest subbands. We make a direct connection with the phenomenological effective-bilayer model, which is commonly used in the description of a wide quantum well, and we compare our findings with the established results at $\nu=1/2$ in the lowest Landau level. At $\nu=1/4$, the overlap calculations for the Halperin (5,5,3) and (7,7,1) states, the generalized Haldane-Rezayi state and the Moore-Read Pfaffian, suggest that the incompressible state is likely to be realized in the interplay between the Halperin (5,5,3) state and the Moore-Read Pfaffian. Our numerics shows the latter to be very susceptible to changes in the interaction coefficients, thus indicating that the observed state is of multicomponent nature.
• Möller, G., Jolicoeur, T. and Regnault, N. (2009). Pairing in ultracold Fermi gases in the lowest Landau level. Physical Review A [Online] 79. Available at: http://dx.doi.org/10.1103/PhysRevA.79.033609.
We study a rapidly rotating gas of unpolarized spin-1/2 ultracold fermions in the two-dimensional regime when all atoms reside in the lowest Landau level. Due to the presence of the spin degree of freedom both s-wave and p-wave interactions are allowed at ultralow temperatures. We investigate the phase diagram of this system as a function of the filling factor in the lowest Landau level and in terms of the ratio between s- and p-wave interaction strengths. We show that the presence of attractive interactions induces a wide regime of phase separation with formation of maximally compact droplets that are either fully polarized or composed of spin-singlets. In the regime with no phase separation, we give evidence for fractional quantum Hall states. Most notably, we find two distinct singlet states at the filling nu=2/3 for different interactions. One of these states is accounted for by the composite fermion theory, while the other one is a paired state for which we identify two competing descriptions with different topological structures. This paired state may be an Abelian liquid of composite spin-singlet Bose molecules with Laughlin correlations. Alternatively, it may be a known non-Abelian paired state, indicated by good overlaps with the corresponding trial wave function. By fine tuning of the scattering lengths it is possible to create the non-Abelian critical Haldane-Rezayi state for nu = 1/2 and the permanent state of Moore and Read for nu=1. For purely repulsive interactions, we also find evidence for a gapped Halperin state at nu=2/5.
• Möller, G., Simon, S. and Rezayi, E. (2009). Trial wave functions for ?=(1)/(2)+(1)/(2) quantum Hall bilayers. Physical Review B: Condensed Matter and Materials Physics [Online] 79:125106. Available at: http://dx.doi.og/10.1103/PhysRevB.79.125106.
Quantum Hall bilayer systems at filling fractions near $\nu=\half+\half$ undergo a transition from a compressible phase with strong intralayer correlation to an incompressible phase with strong interlayer correlations as the layer separation $d$ is reduced below some critical value. Deep in the intralayer phase (large separation) the system can be interpreted as a fluid of composite fermions (CFs), whereas deep in the interlayer phase (small separation) the system can be interpreted as a fluid of composite bosons (CBs). The focus of this paper is to understand the states that occur for intermediate layer separation by using variational wavefunctions. We consider two main classes of wavefunctions. In the first class, first discussed by PRL {\bf 77}, 3009 (1996), we consider interlayer BCS pairing of two independent CF liquids. We find that these wavefunctions are exceedingly good for $d \gtrsim \ell_0$ with $\ell_0$ the magnetic length. The second class of wavefunctions naturally follows the reasoning of PRL {\bf 91}, 046803 (2003) and generalizes the idea of pairing wavefunctions by allowing the CFs also to be replaced continuously by CBs. This generalization allows us to construct exceedingly good wavefunctions for interlayer spacings of $d \lesssim \ell_0$, as well. The accuracy of the wavefunctions discussed in this work, compared with exact diagonalization, is comparable to that of the celebrated Laughlin wavefunction. We conclude that over a range of $d$ there exists a phase of interlayer BCS-paired composite fermions. At smaller $d$, we find a second order transition to a composite boson liquid, known also as the 111 phase.
• Möller, G., Simon, S. and Rezayi, E. (2008). Paired Composite Fermion Phase of Quantum Hall Bilayers at ?=(1)/(2)+(1)/(2). Physical Review Letters [Online] 101:176803. Available at: http://dx.doi.org/10.1103/PhysRevLett.101.176803.
We provide numerical evidence for composite fermion pairing in quantum Hall bilayer systems at filling nu=1/2+1/2 for intermediate spacing between the layers. We identify the phase as p_x+i p_y pairing, and construct high accuracy trial wave functions to describe the ground state on the sphere. For large distances between the layers, and for finite systems, a competing ‘‘Hund’s rule’’ state, or composite fermion liquid, prevails for certain system sizes.
• Möller, G. and Simon, S. (2008). Paired composite-fermion wave functions. Physical Review B: Condensed Matter and Materials Physics [Online] 77:75319. Available at: http://dx.doi.org/10.1103/PhysRevB.77.075319.
We construct a family of BCS paired composite fermion wavefunctions that generalize, but remain in the same topological phase as, the Moore-Read Pfaffian state for the half-filled Landau level. It is shown that for a wide range of experimentally relevant inter-electron interactions the groundstate can be very accurately represented in this form.
• Möller, G. and Cooper, N. (2007). Density Waves and Supersolidity in Rapidly Rotating Atomic Fermi Gases. Physical Review Letters: Moving Physics Forward [Online] 99:190409. Available at: http://dx.doi.org/10.1103/PhysRevLett.99.190409.
We study theoretically the low-temperature phases of a two-component atomic Fermi gas with attractive s-wave interactions under conditions of rapid rotation. We find that, in the extreme quantum limit, when all particles occupy the lowest Landau level, the normal state is unstable to the formation of “charge” density wave (CDW) order. At lower rotation rates, when many Landau levels are occupied, we show that the low-temperature phases can be supersolids, involving both CDW and superconducting order.
• Möller, G. and Moessner, R. (2006). Artificial Square Ice and Related Dipolar Nanoarrays. Physical Review Letters [Online] 96:237202. Available at: http://dx.doi.org/10.1103/PhysRevLett.96.237202.
We study a frustrated dipolar array recently manufactured lithographically by Wang et al. [Nature (London) 439, 303 (2006)] in order to realize the square ice model in an artificial structure. We discuss models for thermodynamics and dynamics of this system. We show that an ice regime can be stabilized by small changes in the array geometry; a different magnetic state, kagome ice, can similarly be constructed. At low temperatures, the square ice regime is terminated by a thermodynamic ordering transition, which can be chosen to be ferro- or antiferromagnetic. We show that the arrays do not fully equilibrate experimentally, and identify a likely dynamical bottleneck.
• Möller, G., Matveenko, S. and Ouvry, S. (2006). Dimensional Reduction on a Sphere. International Journal of Modern Physics B [Online] 20:3533-3546. Available at: http://dx.doi.org/10.1142/S0217979206035503.
The question of the dimensional reduction of two-dimensional (2d) quantum models on a sphere to one-dimensional (1d) models on a circle is adressed. A possible application is to look at a relation between the 2d anyon model and the 1d Calogero-Sutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength ? or on a circle of radius R – both parameters ? and R have to be viewed as long distance regulators – the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the Calogero-Moser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the Calogero-Sutherland model on a circle of radius R. First, the free one-body case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the N-body interacting anyon model is considered : it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus lead to define a new spherical anyon-like model deduced from the Aharonov-Bohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode.
• Möller, G. and Simon, S. (2006). Interlayer correlations versus intralayer correlations in a Quantum Hall bilayer at total filling one. Journal de Physique [Online] 131:283-284. Available at: http://dx.doi.org/10.1051/jp4:2005131072.
In Quantum Hall bilayers, at total filling factor one, a transition from a compressible phase with weak interlayer correlations to an incompressible phase with strong interlayer correlations is observed as the distance between the two layers is reduced. The transition between these two regimes can be understood using a trial wavefunction approach based on the composite particle picture.
• Möller, G. and Simon, S. (2005). Composite fermions in a negative effective magnetic field: A Monte Carlo study. Physical Review B: Condensed Matter and Materials Physics [Online] 72:45344. Available at: http://dx.doi.org/10.1103/PhysRevB.72.045344.
The method of Jain and Kamilla [PRB 55, R4895 (1997)] allows numerical generation of composite fermion trial wavefunctions for large numbers of electrons in high magnetic fields at filling fractions of the form $\nu=p/(2mp+1)$ with $m$ and $p$ positive integers. In the current paper we generalize this method to the case where the composite fermions are in an effective (mean) field with opposite sign from the actual physical field, i.e. when $p$ is negative. We examine both the ground state energies and the low energy neutral excitation spectra of these states. Using particle-hole symmetry we can confirm the correctness of our method by comparing results for the series $m=1$ with $p>0$ (previously calculated by others) to our results for the conjugate series $m=1$ with $p <0$. Finally, we present similar results for ground state energies and low energy neutral excitations for the states with $m=2$ and $p <0$ which were not previously addressable, comparing our results to the $m=1$ case and the $p > 0$, $m=2$ cases.

### Conference or workshop item

• Wójs, A., Möller, G., Simon, S. and Cooper, N. (2011). Skyrmions in a Half-Filled Second Landau Level. In: 30th International Conference on the Physics of Semiconductors. IOP Institute of Physics, pp. 631-632. Available at: http://dx.doi.org/10.1063/1.3666536.
We studied charged excitations of the ?=5/2 fractional quantum Hall state allowing for spin depolarization. It is generally accepted that the ground state is a spin?polarized incompressible quantum liquid, adiabatically connected to the Pfaffian state, whose spin?polarized quasiholes (QHs) obey non?Abelian statistics. Using numerical diagonalization and taking account of non?zero well widths we demonstrated that at a sufficiently low Zeeman energy it is energetically favorable for pairs of charge e/4 QHs to bind into charge e/2 Skyrmions. We showed that Skyrmion formation is further promoted by disorder, and argue that this can lead to a depolarized ground state in realistic experimental situations.
• Möller, G., Wójs, A. and Cooper, N. (2009). Fractional Quantum Hall States with Non-Abelian Statistics. In: XXXVIII International School and Conference on the Physics of Semiconductors “Jaszowiec”. Polish Academy of Sciences, pp. 847-848. Available at: http://przyrbwn.icm.edu.pl/APP/ABSTR/116/a116-5-22.html.
Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level. We consider filling fractions ? = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed. Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential. Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ? = 1/2 (“Pfaffian”) and ? = 2/5 (“parafermion” state).
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