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A topological ladder

The orbital degrees of freedom refer to different shapes of the wave functions with degenerate energies. In recent years, optical lattices engineered by interfering laser beams offer new means to explore interacting fermions with orbital degrees of freedom the symmetries of which differ from those found in traditional solids. We show that the orbital hopping pattern alone is sufficient for producing topologically non-trivial band structures. We unveil a topological insulator phase of fermions on a two-leg ladder of  double-well lattices, similar to those recently realized in experiments.

 

 

Topological orbital ladders, Xiaopeng Li, Erhai Zhao, W. Vincent Liu, arXiv:1205.0254

http://arxiv.org/abs/1205.0254

Phase diagram of dipolar Fermi gases


Understanding the quantum phases of interacting fermions is a fundamental, chanllenging problem in many-body physics. Broken symmetry phases, such as spin density wave order in antiferromagnetic metal Chromium, or the p-wave superfluid order in liquid Helium 3, have long been known and well understood. Motivated by recent experiments, we find theoretically that an unconventional spin-density wave phase with p-wave orbital symmetry in ultracold Fermi gases of polar molecules and magnetic atoms. It is a kind of magnetic order formed on bonds connecting the lattice sites, and can be viewed as the particle-hole analog of p-wave superconductivity.

 

Unconventional Spin Density Waves in Dipolar Fermi Gases, S. G. Bhongale, L. Mathey, Shan-Wen Tsai, Charles W. Clark, Erhai Zhao, arXiv:1209.2671

http://arxiv.org/abs/1209.2671

Ultra-cold fermionic atoms near unitarity

In recent years, atomic physics has opened a new frontier for the exploration of strongly correlated many-body systems. Atoms can be cooled to sub-nanokelvin temperatures, trapped in a small volume and placed in artificial crystalline potentials or electromagnetic fields created by lasers. Furthermore, interactions between atoms can be controlled. This enables simulations of electronic materials with more ideal properties than found in nature, and testing or developing theories of condensed matter in a new environment. Novel forms of quantum matter can also be engineered using ultra-cold atoms.

Ultra-Cold Atoms

 

In recent years, atomic physics has opened a new frontier for the exploration of strongly correlated many-body systems. Atoms can be cooled to sub-nanokelvin temperatures, trapped in a small volume and placed in artificial crystalline potentials or electromagnetic fields created by lasers. Furthermore, interactions between atoms can be controlled. This enables simulations of electronic materials with more ideal properties than found in nature, and testing or developing theories of condensed matter in a new environment. Novel forms of quantum matter can also be engineered using ultra-cold atoms. A notable example is the superfluid of fermionic atoms in the unitarity limit, which occurs in the crossover between the Bose-Einstein condensate (BEC) of diatomic molecules and the Bardeen-Cooper-Schrieffer (BCS) superfluid of Cooper pairs. Such superfluids have universal properties, independent of microscopic details like the structure of atoms and interaction potentials. Owing to universality, field-theoretical methods can be effectively used to both gain physical insight and make accurate quantitative predictions near unitarity.

Universality

The fundamental zero-temperature phase diagram of fermions with short-range attractive interactions contains two phases: superfluid and insulator. The strength of interactions can be characterized by a quantity called “detuning from the Feshbach resonance”, which is equal to the negative inverse of the atom-atom scattering length. At zero density and zero detuning, there is a quantum critical point, the zero-density Feshbach resonance. The properties of interacting fermions are universal in the vicinity of this critical point, the so called unitarity regime. For example, in the low density limit near unitarity, critical temperature is a function of only the density.

μ … chemical potential
ν … detuning (ν=-1/a)(ν>0 – BCS limit)
(ν>0 – BEC limit)

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Novel Phases by Population Imbalance

Application of a Zeeman field (h) to a fermionic superfluid introduces tendency to create a population imbalance between the two types of fermions that form Cooper pairs or molecules. The fully paired superfluid normally resists this tendency until it is destroyed in a first order transition at a critical value of the Zeeman field. Sometimes, however, unusual phases may be obtained first, in which superfluidity coexists with a Fermi sea of unpaired atoms, and potentially spontaneously breaks some symmetry. Proposals have been made for phases with nematic and smectic (FFLO) order, and ongoing research attempts to identify circumstances in which such phases might occur and be experimentally detected.

First-order phase transition (T=0), and boundaries of the hysteresis effects. The phase boundary is universal.Normal phases can have 1 or 2 Fermi seas.

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Crossover Between Band and Mott Insulators

Placing ultra-cold fermionic atoms in optical lattices (created by interfering laser beams) opens countless possibilities for simulating condensed matter systems, such as Mott insulators and high-temperature superconductors. The generic phase diagram at zero temperature contains superfluid and insulating phases at various densities (see below). We have shown that band insulators smoothly evolve into Mott insulators as the detuning is varied, similar to the smooth evolution of superfluidity from the BCS to the BEC regime. Future research includes searching for Mott insulators with arbitrary fractional density, as well as supersolid phases. Such phases, which would break lattice symmetries, might be expected near unitarity due to the extended range of effective interactions in an effective multi-band tight-binding model.

T=0 phase diagram Contours are superfluid-insulator phase boundaries for different values of detuning. The light dashed contour is at the Feshbach resonance; the contours move upwards into the BSC limit. Er is molecular recoil energy, V is amplitude of the optical lattice potential.

Papers:

Pair density wave

Pair density wave (PDW) is a quantum state of Cooper pairs that spontaneously breaks translational symmetry. The simplest PDW is a superfluid “condensed” at a finite wavevector q, a non-magnetized version of the FFLO state. A surprising result that we obtained is that the pairing instability of a generic band-insulator, caused by sufficiently strong short-range attraction, leads to a PDW superfluid, rather than the conventional superfluid that respects all lattice symmetries. This is a consequence of the intricate interplay between intra-band and inter-band pairing. From the field theory perspective, it is related to a non-analytic dependence of vertex functions on the particle crystal momenta in non-tight-binding lattice potentials. While this phenomenon is quite generic, its observation is practically possible only in appropriatelly engineered cold atom setups.

The PDW instability obtained from the mean-field approximation, and even semiclassical perturbation theory of arbitrary order, generally occurs at a wavevector that is incommensurate with the underlying lattice potential. Such a PDW would be highly frustrated, so we expect that the quantum fluctuations of topological defects (vortices) ultimately stabilize PDW ordering at a commensurate wavevector. A related natural possibility is that commensurate PDW Mott insulators can be stable adjacent to the PDW superfluid phase. Such Mott insulators would fundamentally involve multiple orbitals, such as in an orbitally ordered density wave.

T=0 PDW instability of attractively interacting fermions at fixed density of two fermions per lattice site. Contours are superfluid-insulator transitions restricted to occur at the wavevector <strong>q</strong>=(q,q,q), as a function of the inverse lattice potential amplitude 1/V, parametrized by the scattering length a (in the units of lattice spacing). Coming from the insulating state (on the right), the pairing instability always occurs at a finite wavevector (at the vertical tangents to the curves). The ordering wavevector as a function of 1/V is ploted by the thick bright line.T=0 PDW instability of attractively interacting fermions at fixed density of two fermions per lattice site. Contours are superfluid-insulator transitions restricted to occur at the wavevector <strong>q</strong>=(q,q,q), as a function of the inverse lattice potential amplitude 1/V, parametrized by the scattering length a (in the units of lattice spacing). Coming from the insulating state (on the right), the pairing instability always occurs at a finite wavevector (at the vertical tangents to the curves). The ordering wavevector as a function of 1/V is ploted by the thick bright line.

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Vortex lattices and liquids

Neutral atoms can be subjected to velocity-dependent forces that have the same effect on their motion as magnetic field on electrons. This can be accomplished by rotating an atomic cloud very fast, since the Coriolis force in the rotating frame of reference has the same mathematical description as a uniform magnetic field. More recently, transitions between internal atomic states induced by Raman scattering of laser light have been used to synthesize an artificial magnetic field, and even an SU(2) gauge field (spin-orbit interaction). The latter approach is especially promissing for achieving macroscopically entangled topological states of cold atoms, known as fractional quantum Hall states.

A fermionic superfluid allows external (artificial) magnetic field to pass through it only in the form of localized flux tubes, or vortices. A vortex is the supercurrent flow that circulates around a core depleted of atoms (singularity), and the amount of magnetic flux associated with it is quantized. Vortices interact with one another and arrange themselves into a lattice, typically honeycomb. When quantum fluctuations melt this vortex lattice, the syperfluid state is destroyed in favor of an insulating quantum vortex liquid. This is an exotic topological state of Cooper pairs, possibly related to the “pseudogap” state of high-temperature superconductors in strong magnetic fields.

A combination of strong (artificial) magnetic fields and Zeeman effect can produce a rather rich phase diagram of vortex lattices, vortex liquids and topological band insulators (integer quantum Hall states). When Zeeman effect favors a large spin magnetization in a paired superfluid state, the excess magnetic moment can be stored only in vortex cores, so that additional vortex-antivortex pairs may be nucleated in every unit cell of the vortex lattice. The result is known as a vortex Fulde-Ferrell-Larkin-Ovhinnikov (FFLO) state. The figures below illustrate the rich phase diagram of two-dimensional fermionic particles with short-range attractive interactions in magnetic field, whose dynamics is shaped both by the orbital and Zeeman effects.

Superfluid order parameter strength at zero temperature as a function of chemical potential μ and Zeeman field h. These parameters are normalized by the cyclotron energy scale 2ћω in the external artificial magnetic field (obtained from Coriolis forces in the frame of reference that rotates at the angular velocity ω). The plot on the right is for a weaker interaction between particles.

Mean-field phase diagrams of generally spin-polarized vortex lattices, liquids and integer quantum Hall states (corresponding to the first two plots above). Thick yellow line is the superfluid-insulator transition, second order along straight vertical segments and first order along curved “horisontal” segments. Dashed white lines are metal-insulator transitions of excess-spin fermions that form in the crystalline lattice of vortex cores. Thick black lines are transitions between integer quantum Hall states. Not shown in these plots are quantum vortex lattice melting transitions. They preempt all second order superfluid-insulator transitions and introduce vortex liquid phases that intervene between superfluids (SF) and integer quantum Hall states (I).

A low-resolution phase diagram of vortex lattice FFLO states in the wider range of parameters. Unusual magnetized vortex lattices whose density profiles are shown on the right are stable in larger Zeeman fields.

Papers:

Resonant scattering in lattice potentials

Lattice potentials imposed on interacting fermionic particles give rise to multiple universal regimes controlled by scattering resonances. Low-energy quasiparticle excitations of a zero-temperature band-insulator can be “particles” and “holes” that live at multiple symmetry-related wavectors in the Brillouin zone. Injected quasiparticles can resonantly scatter in Cooper and exciton channels, and form bound-state Cooper pairs or excitons respectively when interactions are strong enough. Pairs can be intra-band, inter-band, or formed between excitations at different wavevectors. This results with a variety of ordered phases, which can be superfluids, exciton condensates, charge and spin density waves, all of which can cross over between their weak-coupled (BCS) and strong-coupled (BEC) limit. Experimental realizations of tunable microscopic models in which these scattering resonances occur are possible with cold atoms in optical lattices tuned to finite-density lattice Feshbach resonances, but the universal aspects of their physics yields insight about solid state materials as well.

Whenever lattice fermions are tuned near a scattering resonance, their dynamics is universal and can be captured by a quantum field theory. The main method of calculations in field theory is the perturbative expansion, which is often plaqued by the lack of a small expansion parameter in descriptions of correlated states of condensed matter. However, focusing on a scattering resonance is mathematically very convinent because the unperturbed ground state is a band-insulator, a state barely different from vacuum by its dynamics. Many types of conventional and unconventional ordered phases of lattice fermions encountered in solid state materials can be reliably studied using field theoretical techniques by perturbing about scattering resonances. A particularly interesting insight is obtained by a renormalization group analysis about the pseudogap state of high temperature superconductors.

Transitions between ordered phases and featureles insulators in the weak-coupling limits are always of the pairing (BCS) kind. This means that the insulator is a band-insulator, and the ordered phases arise as pairing instabilities of the Fermi surface. However, the transitions in the strong-coupling limits belong to a bosonic universality class, XY or mean-field depending on whether there is particle-hole symmetry or not. This implies that the strong-coupled insulator adjacent to an ordered phase in the phase diagram is a “correlated” Mott insulator. Such a bosonic insulator of Cooper pairs or excitons is devided from the band-insulator either by a phase transition or a crossover depending on whether it breaks some symmetries. It turns out that in two spatial dimensions only bosonic transitions are possible and necessitate the existence of “pseudogap” Mott insulators when fermionic excitations are naturally gapped, as in our lattice model with Fermi energy sitting in a bandgap. Such insulators can exhibit vortex-driven transport out of equilibrium, of nature much similar to that claimed to occur in cuprate high temperature superconductors. While our model is substantionally different than cuprates, it does indicate that some phenomenology of pseudogap states may be related to quasi two-dimensional dynamics (electrons most readily move in copper-oxygen planes in underdoped cuprates) and low energy Cooper pairs that fail to superconduct due to strong quantum fluctuations.

Papers:

Dark and Bright Solitons in strongly Repulsive Bosonic Gases

Unlike weakly interacting BEC, solitons in hard core bosonic gases support both dark and bright solitons .solitary waves.These solitons survive collision and  quantum fluctuations.

  • “Quantum Dynamics of Solitons in Strongly Interacting Systems on Optical Lattices”, Chester P. Rubbo, Indubala I. Satija, William P. Reinhardt, Radha Balakrishnan,Ana Maria Rey,1 and Salvatore R. Manmana , Phys. Rev. A 85, 053617 (2012) (PDF)
  • “Particle-hole Asymmetry and Brightening of Soliton in a Strongly Repulsive BEC”, Radha Balakrishnan, Indubala Satija and Charles Clark, Phys Rev Lett, 103, 230403, 2009 (PDF)


 

Research areas

Condensed matter theory: P.Nikolic, I.Satija, E.Zhao
Condensed matter experiment: K.Vemuru
Atomic, molecular and optical physics experiment: K.Sauer, M.Tian
Materials science: Y.Mishin
High energy physics experiment: P.Rubin

  • Condensed matter theory: P.Nikolic, I.Satija, E.Zhao
  • Condensed matter experiment: K.Vemuru
  • Atomic, molecular and optical physics experiment: K.Sauer, M.Tian
  • Materials science: Y.Mishin
  • High energy physics experiment: P.Rubin

Condensed matter physics

 

Condensed matter physics is a major fundamental branch of physics that studies the collective quantum dynamics of strongly interacting particles. Unlike high-energy physics, which focuses on elementary particles and forces as the fundamental building blocks of nature, condensed matter physics views the emergent phenomena arising from correlations and entanglement among many particles as the fundamental ones. Quantum field theory, on which both branches of physics rely, makes no distinction between these fundamental views. Examples of condensed matter researched at CQS are solid-state crystals, superfluids and superconductors, magnets, topological insulators, and ultra-cold gases of trapped atoms.

CQS theorists I.Satija, E.Zhao and P.Nikolic share a common interest in topological insulators. I.Satija has been working on integer quantum Hall states in lattice potentials, with U(1) and SU(2) gauge symmetry groups, often placed in the context of ultra-cold atoms. Her collaborative work, which included the world-leading experimentalist Ian Spielman of NIST, E.Zhao, P.Nikolic and international collaborators, has resulted with the first proposals to experimentally measure Chern numbers in cold atom band-insulators, and create fermionic time-reversal-invariant topological insulators using cold atoms. She also explores novel topological quantum states that are possible only out of equilibrium, and has a long-term interest in the non-linear dynamics of solitons. E.Zhao’s research has scrutinized the transport and proximity-effect properties of interfaces between topological insulators and metals or superconductors, motivated in part by the quest for Majorana fermions. P.Nikolic has been interested in exotic strongly-correlated states of electrons in topological insulators, whose elementary particle constituents carry a quantized fraction of electron’s charge and spin.

Superconductivity is another area studied at CQS from multiple angles. E.Zhao is interested in non-equilibrium properties of superconductors, motivated by possible applications in electronic and spintronic devices, as well as quantum computers. P.Nikolic has been investigating the fundamental properties of superconductors with strong quantum fluctuations, motivated by the unending quest to understand the physics of cuprate high-temperature superconductors. His theory of vortex quantum dynamics and charge dynamics in cuprates, developed in collaboration with world-leading theorists Subir Sachdev (Harvard) and Zlatko Tesanovic (Johns Hopkins), has successfully addressed some of the key experimental observations in cuprates. P.Nikolic is also working on “topological” superconductors in large magnetic fields or topological insulators, where zero-point quantum fluctuations can melt a vortex lattice and produce “fractional” topological insulators, highly-entangled many-body quantum states amenable to quantum computation.

Other interests of the CQS theorists include the transport properties of mesoscopic to nano-scale quantum devices (E.Zhao), and exotic quantum states of localized magnetic moments found in frustrated quantum magnets (P.Nikolic).

A popular blog introduction about topological insulators and some key concepts in condensed matter physics can be found here.

Atomic, molecular and optical physics

 

Atoms are the birthplace of quantum mechanics. The historic effort to understand atoms has grown over time into modern fields such as high-energy and condensed matter physics, which explore the constituents of atoms and complex systems made of many atoms respectively. Recently, however, the fundamental interest in atoms has been reinvigorated by the recent discoveries of experimental methods to manipulate the quantum behavior of many atoms, molecules and photons (particles of light). A new playground of quantum mechanics has been opened by this new ability to create idealized simulations of electronic solid-state materials or fundamental processes, and engineer novel quantum states of matter not possible in other systems.

Atomic physics research is done at CQS both theoretically and experimentally. All three CQS theorists,I.Satija, E.Zhao and P.Nikolic, are working on ultra-cold atomic or molecular gases. I.Satija has been interested in topological band-insulating states of cold atoms, as well as soliton dynamics and aspects of noise correlations in Bose-Einstein condensates. E.Zhao has been interested in exotic modulated superfluids, the orbital ordering patterns of atoms in higher bands of periodic potentials, dipolar Fermi gases, and topological phases of cold atoms. His recent collaboration with CQS postdocS.Bhongale has discovered that interacting fermions with dipole moments form solids with periodic modulations of bonds, rather than density. The research of P.Nikolic has mainly explored the unconventional many-body quantum states of fermionic cold atoms with nearly resonant scattering (unitarity). His work has characterized the universal dynamics of imbalanced fermion gases near unitarity, and extended its field-theoretical approach to fermionic atoms in lattice potentials and artificial gauge fields. This work lead to the theoretical discovery of novel pair-density wave supersolids, and the phase-diagram maps of vortex-FFLO and quantum vortex liquid states of atoms in artificially created gauge fields.

The CQS experimentalist K.Sauer is an expert on magnetic resonance phenomena. The research done in her Magnetic Resonance Laboratory (MRL) seeks to understand and exploit spin-dynamics in such systems as nuclear quadrupole resonance and optically pumped atoms. One of the goals of this research is to push the noise in such systems to their fundamental limit, to reveal the full capability of magnetic resonance at low-fields both as an analytic tool and for the detection of contraband substances.

The CQS experimentalist Mingzhen Tian is an expert on laser atomic spectroscopy, nonlinear and quantum optics, and quantum information. Her research is currently focused on rare-earth based solid state quantum memory and quantum computation, which are the important elements in developing quantum information science and technology. The research topics also include laser spectroscopic properties of rare-earth ions trapped in inorganic crystal lattice at cryogenic temperature, the coherent and incoherent processes under the excitation of composite laser pulses, and the influence of the static electric and magnetic fields. Study of these processes provides the information needed to set up the physical systems to demonstrate quantum memory and quantum computation and analyze and optimize the performance.

Materials science

 

Materials science is an interdisciplinary field dealing with fundamental properties and characteristics of materials, and applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates elements of applied physics and chemistry. [Paraphrasing Wikipedia]

The CQS material scientist Y.Mishin is interested in the theory and atomistic modeling of materials, particularly materials interfaces, atomic diffusion, and mechanical behavior of metals and intermetallic compounds. Specific areas of interest include models of atomic interaction in materials, interfaces in materials (including grain and interphase boundaries, interfacial motion, segregation, chemical reactions and cohesion), atomistic theory and modeling of interfacial kinetics in materials, defects and diffusion in intermetallic compounds, and plastic deformation and fracture of metals and intermetallic compounds.

High energy physics

 

This fundamental branch of physics seeks to understand the fundamental building blocks and forces of our world. The information about elementary particles is revealed only at extremely high energies that are naturally obtained in the interiors of stars, accretion discs of black holes and early universe. People can nowadays accelerate particles to such extreme energies using some of the largest and most sophisticated machines ever built.

Phil Rubin conducts high-energy experiments at CERN. His current research involves experiments at accelerator facilities designed to explore the fundamental components and interactions of nature. The experiments seek evidence for rare and forbidden sub-atomic processes which might be exceptions to accepted symmetries or conservation laws.

Fractional Topological Insulators

A new class of materials with strong spin-orbit coupling, known as topological insulators (TI), are bulk insulators with edge or surface conduction channels that respect the time-reversal (TR) symmetry. In that sense they are similar to quantum Hall systems, which however are not invariant under TR due to the externally applied magnetic field. The Rashba spin-orbit coupling found in TI materials has a “dynamical” symmetry that can shape incompressible quantum liquids in the presence of strong quantum fluctuations, without an analogue in quantum Hall states. Such quantum liquids can exhibit new and not yet experimentally discovered topological orders with Abelian or non-Abelian fractional statistics.

An artistic snapshot of a vortex liquid with 1/3 fractionalized charge excitations.

 

A classical system can have multiple degrees of freedom whose properties can be measured independently and simultaneously with arbitray accuracy (limited only by the measuring device). However, quantum mechanics allows matter to exist in a “superposition” of different classical states. A quantum system in a “superposition” state will generally have properties whose measurements have random outcomes with predictable probabilities. Then, measuring different properties of the classical states that participate in the quantum superposition yields random, but correlated measurement outcomes. Such correlations are known as quantum entanglement.

A rather remarkable form of entanglement is that between a macroscopically large number of particles. The only forms of macroscopically entangled quantum matter that we have found so far in nature are superconductors and fractional quantum Hall states. The entanglement in superconductors is saddle and properly understood only when quantum fluctuations of the electromagnetic gauge field are taken into account (it is often ignored in literature). Apart from quantum Hall states, many other examples of entangled matter have been theoretically envisioned. The most notable example are spin liquids in quantum magnets, perhaps indirectly seen in a few experiments.

Quantum Hall effect and incompressible quantum liquids

 

Quantum Hall states are topological insulators without time-reversal symmetry. When electrons are placed in strong magnetic fields, their trajectories in the plane perpendicular to the field are circular “cyclotron” orbits. Since electrons moving in closed orbits cannot traverse large distances, they form an electric insulator. However, the cyclotron orbits can open up along extended obstacles, such as the crystal boundaries. They become “edge states” that propagate without dissipation along the boundary. Therefore, a crystal in a very strong magnetic field can conduct current only along its boundaries. The laws of quantum mechanics require that the edge currents exhibit a measurable quantized transverse conductivity, which is the phenomenon called quantum Hall effect.

The quantum of transverse conductivity is related to the electron charge, and therefore is a fundamental constant of nature that can be used nowadays to define the most accurate standard for conductivity. However, certain materials exhibit fractional quantum Hall effect, as if electrons were broken into pieces with fractional charge. The only other observed forms of fractionalization are quarks in atomic nuclei, and spin-charge separation in spin chains (and possibly above one dimension in some frustrated magnet materials). The observed fractionalization in quantum Hall states is incredibly rich, an entire hierarchy of fractions has been experimentally found and theoretically explored.

A new class of materials with strong spin-orbit coupling, known as topological insulators (TI), are bulk insulators with edge or surface conduction channels that respect the time-reversal (TR) symmetry. In that sense they are similar to quantum Hall systems, which however are not invariant under TR due to the externally applied magnetic field. Quantum Hall effect can be observed only because charge is conserved, and an analogous quantum spin-Hall effect would be observable if spin were conserved. It turns out that the Rashba spin-orbit coupling found in TI materials does not conserve spin, but brings about a new “dynamical” symmetry that can shape incompressible quantum liquids without an analogue in quantum Hall states. Such quantum liquids can exhibit new and not yet experimentally discovered topological orders with Abelian or non-Abelian fractional statistics. Various perturbations in materials can further lift the spin-related symmetries and scramble the link between bulk topological orders and edge states.

Incompressible quantum liquids are highly entangled forms of quantum matter, possibly applicable in quantum computation. They are said to have topological order whose manifestations are quasiparticle excitations with fractional quantum numbers and statistics, and ground-state degeneracy on non-simply connected spaces (like torus) that survives any sufficiently weak perturbation even when no symmetry is spontaneously broken.

The topological ground-state degeneracy on a torus is related to the energy cost of threading flux tubes through the torus openings.

A possible experimental realization

 

A fractional TI ground state can be stabilized only by a combination of strong interactions among electrons and a strong spin-orbit coupling. One way to accomplish this is to fabricate a heterostructure device that puts a TI quantum well in contact with a superconducting material. The superconductor induces Cooper pairing in the TI quantum well via the “proximity effect”. By applying a gate voltage, it is possible to drive a quantum phase transition in the quantum well between an insulating and a superconducting state. The TI’s spin-orbit coupling significantly modifies the character of this transition and gives rise to stable incompressible quantum liquids in the phase diagram.

The TI quantum well can host electrons in two states of spin projection and two orbital states. Inter-orbital Cooper pairs can carry spin and feel the spin-orbit coupling. This produces two helical modes, one of which has energy that decreases with momentum. That helical mode can condense at large momenta and produce a superconducting state with a TR-invariant vortex lattice of spin supercurrents. The quantum phase transition out of this vortex state, tuned by the gate voltage, is generally the first-order vortex lattice melting. The resulting vortex liquid phase is an incompressible quantum liquid, a candidate for a fractional TI.

Left: the heterostructure device with a TI quantum well. Right: the phase diagram of the quantum well (Δ is the TI’s bandgap tuned by the quantum well thickness, μ is the chemical potential tuned by the gate voltage).

 

Papers:

Effective theory

 

The topological properties of incompressible quantum liquids are hard to describe using microscopic models. Instead, it is more practical to construct an effective theory that gives up microscopic accuracy in favor of simplicity. An effective theory is not derived, but constructed according to certain requirements. It must have all the needed symmetries of the system it refers to. It must contain all low-energy degrees of freedom and reproduce their known classical equations of motion. Otherwise, its form should be the simplest one that can capture all universal aspects of dynamics (independent of the system’s microscopic details). The Standard Model of elementary particles is an effective theory in this sense, as well as any Landau-Ginzburg theory of a second order quantum phase transition. Even though effective theories are not derived, they can explain certain experimental observations at the quantitative level.

An effective theory of quantum Hall states is the well-known Chern-Simons (CS) theory. This is a gauge theory in which the physical particle densities and currents are represented by the curls of an auxiliary dynamical gauge field. The CS theory is constructed by the requirement that its equations of motion reproduce the experimentally observed relationship between the electrons’ density/current and the external magnetic/electric fields in quantum Hall states. The most general form of this theory can classify many quantum Hall states, qualitatively describe their properties and predict various features of the edge states.

The CS theory is very general, but it does formally rely on the conservation of “charge” that is coupled to an external gauge field. Electron’s electric charge is coupled to magnetic fields and conserved. Electron’s spin is similarly involved in the spin-orbit coupling which can be described by an SU(2) gauge field with a non-trivial “magnetic flux” (the SU(2) gauge fields are also used in the theory of weak nuclear interactions). If this SU(2) gauge field conserved spin, its quantum spin-Hall states could be readily described by an appropriate CS theory. However, the Rashba spin-orbit SU(2) gauge field is non-commutative and consequently does not conserve spin. A generalization of the CS theory is needed to capture all potentially existing incompressible quantum liquids shaped by spin-orbit couplings.

The generalization of CS theory is a Landau-Ginzburg theory of spinor fields enhanced by a topological term. The topological term has the SU(2) symmetry and captures topological orders of arbitrary incompressible quantum liquids in the continuum limit with that symmetry. It reduces to the CS theory when spin is conserved. The full effective theory, however, is capable of describing both conventional and topological states of quantum matter, including novel topological orders with Abelian or non-Abelian statistics that have no analogue in fractional quantum Hall states.

 

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