**SFB 925**Light induced dynamics and control of correlated quantum systems

# Project C2

### 2018

#### Bosonic Topological Excitations from the Instability of a Quadratic Band Crossing

*G.-Q. Luo, A. Hemmerich, Z.-F. Xu*

We investigate the interaction-driven instability of a quadratic band crossing arising for ultracold bosonic atoms loaded into a two-dimensional optical lattice. We consider the case when the degenerate point becomes a local minimum of both crossing energy bands such that it can support a stable Bose–Einstein condensate. A repulsive contact interaction among the condensed bosons induces a spontaneously time-reversal-symmetry broken superfluid phase and a topological gap is opened in the excitation spectrum. We propose two concrete realizations of the desired quadratic band crossing in lattices with either fourfold or sixfold rotational symmetries via suitable tuning of the unit cell leading to reduced Brillouin zones and correspondingly folded bands. In either case, topologically protected edge excitations are found for a finite system.

#### Rotation-Symmetry-Enforced Coupling of Spin and Angular Momentum for p-Orbital Bosons

*Y. Li, J. Yuan, A. Hemmerich, X. Li*

Intrinsic spin angular-momentum coupling of an electron has a relativistic quantum origin with the coupling arising from charged orbits, which does not carry over to charge-neutral atoms. Here, we propose a mechanism of spontaneous generation of spin angular-momentum coupling with spinor atomic bosons loaded into ** p**-orbital bands of a two-dimensional optical lattice. This spin angular-momentum coupling originates from many-body correlations and spontaneous symmetry breaking in a superfluid, with the key ingredients attributed to spin-channel quantum fluctuations and an approximate rotation symmetry. The resultant spin angular-momentum intertwined superfluid has Dirac excitations. In the presence of a chemical potential difference for adjacent sites, it provides a bosonic analogue of a symmetry-protected-topological insulator. Through a dynamical mean-field calculation, this novel superfluid is found to be a generic low-temperature phase, and it gives way to Mott localization only at strong interactions and even-integer fillings. We show the temperature to reach this order is accessible with present experiments.

### 2017

#### Odd-parity topological superfluidity for fermions in a bond-centered square optical lattice

*Zhi-Fang Xu, Andreas Hemmerich, W. Vincent Liu*

We propose a physical scheme for the realization of two-dimensional topological odd-parity superfluidity in a spin-independent bond-centered square optical lattice based upon interband fermion pairing. The *D*_{4} point-group symmetry of the lattice protects a quadratic band crossing, which allows one to prepare a Fermi surface of spin-up fermions with odd parity close to the degeneracy point. In the presence of spin-down fermions with even parity populating a different energetically well-separated band, odd-parity pairing is favored. Strikingly, as a necessary prerequisite for pairing, both Fermi surfaces can be tuned to match well. As a result, topological superfluid phases emerge in the presence of merely *s*-wave interaction. Due to the *Z*_{2} symmetry of these odd-parity superfluids, we infer their topological features simply from the symmetry and the Fermi-surface topology as confirmed numerically.

### 2016

#### π-Flux Dirac Bosons and Topological Edge Excitations in a Bosonic Chiral p-Wave Superfluid

*Zhi-Fang Xu, Li You, Andreas Hemmerich, and W. Vincent Liu*

We study the topological properties of elementary excitations in a staggered *p _{x}±ip_{y}* Bose-Einstein condensate realized in recent orbital optical lattice experiments. The condensate wave function may be viewed as a configuration space variant of the famous

*p*momentum space order parameter of strontium ruthenate superconductors. We show that its elementary excitation spectrum possesses Dirac bosons with π Berry flux. Remarkably, if we induce a population imbalance between the

_{x}+ip_{y}*p*and

_{x}+ip_{y}*p*condensate components, a gap opens up in the excitation spectrum resulting in a nonzero Chern invariant and topologically protected edge excitation modes. We give a detailed description of how our proposal can be implemented with standard experimental technology.

_{x}−ip_{y}#### Orbital optical lattices with bosons

*T. Kock, C. Hippler, A. Ewerbeck, and A. Hemmerich*

This article provides a synopsis of our recent experimental work exploring Bose-Einstein condensation in metastable higher Bloch bands of optical lattices. Bipartite lattice geometries have allowed us to implement appropriate band structures, which meet three basic requirements: the existence of metastable excited states sufficiently protected from collisional band relaxation, a mechanism to excite the atoms initially prepared in the lowest band with moderate entropy increase, and the possibility of cross-dimensional tunneling dynamics, necessary to establish coherence along all lattice axes. A variety of bands can be selectively populated and a subsequent thermalisation process leads to the formation of a condensate in the lowest energy state of the chosen band. As examples the 2nd, 4th and 7th bands in a bipartite square lattice are discussed. The geometry of the 2nd and 7th band can be tuned such that two inequivalent energetically degenerate energy minima arise at the X±-points at the edge of the 1st Brillouin zone. In this case even a small interaction energy is sufficient to lock the phase between the two condensation points such that a complex-valued chiral superfluid order parameter can emerge, which breaks time reversal symmetry. In the 4th band a condensate can be formed at the Gamma-point in the center of the 1st Brillouin zone, which can be used to explore topologically protected band touching points. The new techniques to access orbital degrees of freedom in higher bands greatly extend the class of many-body scenarios that can be explored with bosons in optical lattices.

J. Phys. B: At. Mol. Opt. Phys. 49, 042001 (2016)

http://arxiv.org/abs/1601.00500

### 2015

#### Observation of chiral superfluid order by matter wave interference

* T. Kock, M. Ölschläger, A. Ewerbeck, W.-M. Huang, L. Mathey, A. Hemmerich*

The breaking of time reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool we use a matter wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.

Physical Review Letters 114, 115301 (2015)

http://arxiv.org/abs/1411.3483

### 2014

#### Controlling coherence via tuning of the population imbalance in a bipartite optical lattice

* M. Di Liberto, T. Comparin , T. Kock, M. Ölschäger, A. Hemmerich, C. Morais Smith*

The control of transport properties is a key tool at the basis of many technologically relevant effects in condensed matter. The clean and precisely controlled environment of ultracold atoms in optical lattices allows one to prepare simplified but instructive models, which can help to better understand the underlying physical mechanisms. We show that by tuning a structural deformation of the unit cell in a bipartite optical lattice, one can induce a phase transition from a superfluid into various Mott insulating phases forming a shell structure in the superimposed harmonic trap. The Mott shells are identified via characteristic features in the visibility of Bragg maxima in momentum spectra. The experimental findings are explained by Gutzwiller mean-field and quantum Monte Carlo calculations. Our system bears similarities with the loss of coherence in cuprate superconductors, known to be associated with the doping induced buckling of the oxygen octahedra surrounding the copper sites.

#### Proposed formation and dynamical signature of a chiral Bose liquid in an optical lattice

* X. Li, A. Paramekanti, A. Hemmerich, W. Vincent Liu*

Recent experiments on p-orbital atomic bosons have suggested the emergence of a spectacular ultracold superfluid with staggered orbital currents in optical lattices. This raises fundamental questions concerning the effects of thermal fluctuations as well as possible ways of directly observing such chiral order. Here we show via Monte Carlo simulations that thermal fluctuations destroy this superfluid in an unexpected two-step process, unveiling an intermediate normal phase with spontaneously broken time-reversal symmetry, dubbed a ‘chiral Bose liquid’. For integer fillings (n≥2) in the chiral Mott regime, thermal fluctuations are captured by an effective orbital Ising model, and Onsager’s powerful exact solution is adopted to determine the transition from this intermediate liquid to the para- orbital normal phase at high temperature. A lattice quench is designed to convert the staggered angular momentum, previously thought by experts difficult to directly probe, into coherent orbital oscillations, providing a time-resolved dynamical signature of chiral order.

Nature Communications 5, 3205 (2014)

http://arxiv.org/abs/1309.0523

### 2013

#### Interaction-induced chiral px ± i py superfluid order of bosons in an optical lattice

*M. Ölschläger, T. Kock, G. Wirth, A. Ewerbeck, C. Morais Smith, A. Hemmerich*

The study of superconductivity with unconventional order is complicated in condensed matter systems by their immense complexity. Optical lattices with their exceptional precision and control allow one to emulate superfluidity avoiding many of the complications of condensed matter. A promising approach to realize unconventional superfluid order is to employ orbital degrees of freedom in higher Bloch bands. In recent work, indications were found that bosons condensed in the second band of an optical chequerboard lattice might exhibit px ± i py order. Here we present experiments, which provide strong evidence for the emergence of px ± i py order driven by the interaction in the local p-orbitals. We compare our observations with a multi-band Hubbard model and find excellent quantitative agreement.

New Journal of Physics 15, 083041 (2013)

http://arxiv.org/abs/1305.1177

### 2012

#### Topologically induced avoided band crossing in an optical chequerboard lattice

* M. Ölschläger, G. Wirth, T. Kock, A. Hemmerich*

We report on the condensation of bosons in the 4th band of an optical chequerboard lattice providing a topologically induced avoided band crossing involving the second, third, and fourth bands. When the condensate is slowly tuned through the avoided crossing, accelerated band relaxation arises and the zero momentum approximately C4-invariant condensate wave function acquires finite momentum order and reduced C2 symmetry. For faster tuning Landau-Zener oscillations between different superfluid orders arise, which are used to characterize the avoided crossing.

Physical Review Letters, 108, 075302 (2012)

http://lanl.arxiv.org/abs/1110.3716

#### Topological semimetal in a fermionic optical lattice

* K. Sun, W. V. Liu, A. Hemmerich, S. Das Sarma*

Optical lattices have an important role in advancing our understanding of correlated quantum matter. The recent implementation of orbital degrees of freedom in chequerboard and hexagonal3 optical lattices opens up a new avenue towards discovering novel quantum states of matter that have no prior analogues in solid-state electronic materials. Here, we predict that an exotic topological semimetal emerges as a parity-protected gapless state in the orbital bands of a two-dimensional fermionic optical lattice. This new quantum state is characterized by a parabolic band-degeneracy point with Berry flux 2 Pi, in sharp contrast to the Pi-flux of Dirac points as in graphene.We showthat the appearance of this topological liquid is universal for all latticeswith D4 point-group symmetry, as long as orbitals with opposite parities hybridize strongly with each other and the band degeneracy is protected by odd parity. Turning on inter-particle repulsive interactions, the system undergoes a phase transition to a topological insulator whose experimental signature includes chiral gapless domain-wall modes, reminiscent of quantumHall edge states.

### 2011

#### Unconventional Superfluid Order in the F Band of a Bipartite Optical Square Lattice

* M. Ölschläger, G. Wirth, A. Hemmerich*

We report on the first observation of bosons condensed into the energy minima of an F band of a bipartite square optical lattice. Momentum spectra indicate that a truly complex-valued staggered angular momentum superfluid order is established. The corresponding wave function is composed of alternating local F_(2x^3-3x)±iF_(2y^3-3y) orbits and local S orbits residing in the deep and shallow wells of the lattice, which are arranged as the black and white areas of a checkerboard. A pattern of staggered vortical currents arises, which breaks time-reversal symmetry and the translational symmetry of the lattice potential. We have measured the populations of higher order Bragg peaks in the momentum spectra for varying relative depths of the shallow and deep lattice wells and find remarkable agreement with band calculations.

#### Evidence for orbital superfluidity in the P-band of a bipartite optical square lattice

*G. Wirth, M. Ölschläger, A. Hemmerich*

The successful emulation of the Hubbard model in optical lattices has stimulated extensive efforts to extend their scope to also capture more complex, incompletely understood scenarios of many-body physics. A promising approach is to consider higher bands, where the orbital degree of freedom gives rise to a structural diversity that is directly relevant, for example, for the physics of strongly correlated electronic matter. Here we report evidence for the formation of a superfluid in the P-band of a bipartite optical square lattice with S-orbits and P-orbits arranged in a chequerboard pattern. The observed momentum spectra feature cross-dimensional coherence with a lifetime of nearly 20 ms. Depending on the value of a small adjustable anisotropy of the lattice, our findings are explained either by real-valued striped superfluid order parameters with different orientations Px±Py, or by a complex-valued Px±iPy order parameter, which breaks time-reversal symmetry.

Nature Physics 7, 147 (2011), featured in News & Views ibid.

### 2010

#### Competing Superconducting States for Ultracold Atoms in Optical Lattices with Artificial Staggered Magnetic Field

* L.-K. Lim, A. Lazarides, A. Hemmerich, C. Morais Smith*

Abstract: We study superconductivity in an ultracold Bose-Fermi mixture loaded into a square optical lattice subjected to a staggered flux. While the bosons form a superfluid at very low temperature and weak interaction, the interacting fermions experience an additional long-ranged attractive interaction mediated by phonons in the bosonic superfluid. This leads us to consider a generalized Hubbard model with on-site and nearest-neighbor attractive interactions, which give rise to two competing superconducting channels. We use the Bardeen-Cooper-Schrieffer theory to determine the regimes where distinct superconducting ground states are stabilized, and find that the non-local pairing channel favors a superconducting ground state which breaks both the gauge and the lattice symmetries, thus realizing unconventional superconductivity. Furthermore, the particular structure of the single-particle spectrum leads to unexpected consequences, for example, a dome-shaped superconducting region in the temperature versus filing fraction phase diagram, with a normal phase that comprises much richer physics than a Fermi-liquid. Notably, the relevant temperature regime and coupling strength is readily accessible in state of the art experiments with ultracold trapped atoms.

### 2009

#### Strongly interacting two-dimensional Dirac fermions

* L.-K. Lim, A. Lazarides, A. Hemmerich, C. Morais Smith*

We show how strongly interacting two-dimensional Dirac fermions can be realized with ultracold atoms in a two-dimensional optical square lattice with an experimentally realistic, inherent gauge field, which breaks time reversal and inversion symmetries. We find remarkable phenomena in a temperature range around a tenth of the Fermi temperature, accessible with present experimental techniques: at zero chemical potential, besides a conventional s-wave superconducting phase, unconventional superconductivity with non-local bond pairing arises. In a temperature vs. doping phase diagram, the unconventional superconducting phase exhibits a dome structure, reminiscent of the phase diagram for high-temperature superconductors and heavy fermions.