Frustrated Quantum Magnets

At the frontier of our efforts to understand collective phenomena is the search for unconventional phases of matter. Many ideas for exotic phases shaped by strong quantum fluctuations have been explored in recent years, motivated by the difficulty of explaining various strongly correlated materials, such as cuprates and heavy fermion systems. Perhaps the most promising platform for characterizing and observing such phases are geometrically frustrated quantum magnets.

Traditionally, understanding the microscopic constituents of physical systems has been considered fundamental. However, a relatively recent realization in physics is that another equally fundamental question exists: what kind of phenomena can emerge in physical systems as a collective effect of many interacting microscopic constituents? A variety of existing phases of matter (crystals, magnets, superconductors), and complex phenomena such as life, provide ample support for this view.

My work in the field of frustrated magnets has relied mostly on novel theoretical methods, which focused on the physical understanding of phenomena, rather than on quantitative descriptions (available only through state-of-art numerics).

Physics of frustration

Conventional many-body physical systems have a stable lowest-energy state that is directly determined by interactions between their elementary constituents. For example, antiferromagnets on bipartite lattices are long-range ordered in a pattern of alternating magnetic moments (spins), because the nearest-neighbor interactions favor oppositely oriented moments. Such ordering possesses a certain amount of robustness against thermal or quantum fluctuations. However, there are materials in which the local interactions are frustrated, so that there is no obvious long-range order that minimizes energy. Instead, a macroscopically large number of different arrangements of spins naively appear to minimize energy, leading to a massive degeneracy. Such frustrated magnets are typically based on lattices with triangular plaquettes. Even very small quantum and thermal fluctuations dramatically lift the naive degeneracy, and often lead to very unconventional phases of matter.


Neel phase of antiferromagnets		Frustration on a triangle (at least one bond is frustrated)

Perhaps the most frequent consequence of frustration is “order-by-disorder”. The massively degenerate naive low-energy states are disordered, but fluctuations entropically select some ordered state as they lift the degeneracy. The resulting true spectrum may have a ground-state with finite degeneracy (which corresponds to a spontaneously broken symmetry) and excitations that may be characterized by an energy scale much smaller than the apparent natural scale of the system. However, the most exotic possibility is the complete lifting of the degeneracy by quantum fluctuations to a unique disordered ground-state. The resulting “spin liquid” phase is characterized by topological order, which cannot be identified by any local probe.

It is believed that the physics of frustration in general is a useful way of approaching many unsolved problems in condensed matter physics. For example, the notion of “competing orders” in the “pseudo-gap” region of cuprates is essentially a matter of frustration, and there are several attempts to understand the non-Fermi liquid of the “pseudo-gap” as a spin liquid.

Kagome Lattice Antiferromagnets

The Kagome lattice is a two-dimensional lattice of corner-sharing triangles. If spins with antiferromagnetic interactions are placed on such a lattice, a very frustrated system is obtained. The three-dimensional analogue of this lattice is the pyrochlore lattice.

In collaboration with Prof. T.Senthil, I have theoretically analyzed the universal phases of the quantum Heisenberg model on the Kagome lattice, which has been also studied numerically and experimentally. This system is characterized by the absence of any observable long-range order at arbitrarily low temperatures, and the existence of a seemingly continuous band of gapless excitations that carry no magnetic moment (singlet excitations). Recent experiments also suggest the possibility that even magnetic (triplet) excitations may be gapless. Such a gapless spectrum is not easy to understand without the spontaneous breaking of a continuous symmetry.

Our analysis starts with the assumption that magnetic excitations are gapped. The low energy singlet sector below the spin gap can then be studied by an effective degenerate perturbation theory, which utilizes a Z2 gauge theory description of spin interactions. Two phases can be identified in general: 1) a fully gapped spin liquid, and 2) a valence-bond solid (VBS). The former case can be ruled out because the predicted singlet gap is large. However, the latter scenario is not inconsistent with present-day experiments and numerics, because it predicts an extremely small gap for singlet excitations. The small size of the gap is a consequence of a very large unit-cell (36 lattice sites).

The valence bond solid with a 36-site unit cell. Each short black line symbolizes a singlet pair of two fluctuating spins. A honeycomb super-structure is formed by benzene hexagons, which contain three singlet valence bonds that fluctuate between two possible local arrangements. Inside every honeycomb super-cell is a resonating loop of six singlet bonds shaped as a star of David. Excitations of these “stars” are the lowest energy excitations of this state.

This state has been also seen numerically.

It is very interesting to notice how reasonably simple microscopic models can produce very complex phases of matter, with features that resemble those of our real world. The phase depicted in the picture above can be viewed as a crystal of “atoms”, where each “star of David” can be regarded as an “atom” with its own internal structure and excitations. These “atom” constituents, and interactions between them that lead to crystalization, emerge almost mysteriously from a microscopic model whose elementary constituents (spins) are completely different. Other theories of Kagome antiferromagnets predict the emergence of gauge fields and effective electromagnetism. These interesting phenomena give rise to the idea that the world we live in is similarly emergent from a more microscopic realm that we are not aware of.

Papers:

Physics of low energy singlet states of the Kagome lattice quantum Heisenberg antiferromagnet

Other Kagome Models

In addition to the isotropic Heisenberg model, we have also explored the quantum Ising model, and various other models with easy-axis anisotropy.

Prior to our work, the quantum Ising model in a transverse field has been studied only numerically, using Monte Carlo techniques. Our theoretical analysis has provided a complete physical understanding of this model and confirmed the numerical observations. No phase transitions occur in this model as the strength of the transversal field is varied. Hence, even though some order-by-disorder, or a spin liquid, could have been naively expected for small transverse fields, none was found, and the only possible phase in the absence of extra perturbations is a conventional disordered phase. However, our model also includes a class of other perturbations which can produce a non-trivial effect. We discovered that a sufficiently strong “ring-exchange” can produce a phase with the same symmetries as the one stabilized by a longitudinal (Zeeman) field at 1/3 of the fully-saturated magnetization.

The easy-axis spin models provide a different theoretical route to understanding the isotropic Heisenberg model. Using the methods of compact U(1) gauge theory, as in the quantum Ising model analysis, I discovered a fundamental difference between theories that preserve the U(1) spin symmetry, and those that do not. Unlike the latter class, any fully disordered phase in the former class is guaranteed to possess topological order and qualify as a spin liquid. The Kagome lattice, being very frustrated, allows quantum fluctuations to overcome tendency to order. It gives rise to a genuine spin liquid in a U(1) symmetric spin model (such as XXZ) for sufficiently strong ring-exchange and other perturbations. However, in the limit of weaker perturbations to the nearest-neighbor easy-axis interactions, there is a stable long-range ordered VBS phase in the models with spin U(1) symmetry. This phase, although described using different language, is consistent with the VBS order that we discovered in the isotropic Heisenberg model (depicted above). The models without the U(1) spin symmetry on the Kagome lattice, such as the transverse field quantum Ising models, do not have such a rich phase diagram and tend to support only a conventional disordered phase.

Papers:

High-Temperature Superconductivity

High temperature superconductivity of cuprates is one of the greatest challenges in many-body quantum physics. Since their discovery in 1986, the field of condensed matter physics has been flooded with an enormous number of new ideas, theoretical and experimental techniques. While the superconducting phenomenology in cuprates has been understood very well, the origin of superconductivity is still mysterious. It is believed that the secret to high temperature superconductivity is hidden in the strong correlations that electrons experience in the so called “normal” and “pseudogap” states. Several experimental and theoretical developments suggested that certain aspects of pseudogap dynamics may result from a quantum liquid state of vortices that destroy the long-range phase coherence of robust Cooper pairs.

We have shown that vortices in high-temperature superconductors experience strong quantum fluctuations, and hence can indeed drive a phase transition to the “normal” phase. Furthermore, vortex quantum fluctuations can qualitatively explain a range of observed phenomena in cuprates that do not fit any conventional paradigm, including checkerboard density-wave patterns in the “normal” state, and the structure of vortex cores. Finally, in d-wave superconductors we found novel forces between vortices that are mediated by nodal quasiparticles. This may be important for the physics of vortex lattices and liquids.

The Importance of Vortices

After more than twenty years of extensive research, much about cuprate high-temperature superconductors is still poorly understood, especially regarding the microscopic mechanism behind the occurrence and destruction of superconductivity. There are strong indications that vortices play a crucial role in the superconducting phase transitions of cuprates. The most direct indications of this are Nernst effect measurements, where a large Nernst signal has been attributed to a thermally generated drift of vortices, even in the absence of superconductivity in the underdoped region at low temperatures. Indirect indications are ample and come from theoretical models that can qualitatively explain a range of seemingly disconnected phenomena observed in cuprates, including the Nernst effect, checkerboard patterns of the “normal” phase, and unusual structure of vortex cores seen by scanning tunneling spectroscopy (STM). These theories rely on the premise that vortices experience strong quantum fluctuations, and drive phase transitions from a superconductor to a “normal” phase, where Cooper pairs essentially survive as low-energy bosonic degrees of freedom but loose long-range phase coherence.


Phase diagram of cuprates; the unconventional “normal” phase is to the left and above of the superconducting phase		Nernst signal in cuprates (taken from Y.Wang, et.al; Phys.Rev.B 73, 024510 (2006))


STM image of the “normal” phase (taken from T.Hanaguri, et.al; Nature 430, 1001 (2004))		Vortex core structure, in a superconducting phase vortex lattice (the same checkerboard pattern)

Vortex Quantum Dynamics

The vortices of conventional superconductors are best described as large heavy objects, essentially subject to classical dynamics. This is mainly due to a large number of quasiparticle states localized inside a vortex core, which contribute to a very large vortex effective mass and friction. In d-wave superconductors the vortex core structure is fundamentally different, but the semi-classical analysis still predicts that the effective vortex mass is large, and even infra-red divergent.

My collaboration with Professor Subir Sachdev has significantly advanced our understanding of the mutual influence of quasiparticles and vortices in d-wave superconductors. In the superconducting phase, this interaction is captured well by the phenomenological d-wave Bogoliubov-de Gennes model, where a vortex is represented by the phase winding of the complex gap function. The crucial aspect of d-wave pairing is the existence of nodal directions in the momentum space along which the pairing gap vanishes. This property of the gap results in gapless nodal quasiparticles, and hence in the inability of vortices to bind quasiparticles to their cores. Therefore, the scattering of extended quasiparticles on vortices plays the main role in the vortex dynamics of cuprates. The quasiparticle quantum coherence is essential for two reasons, especially in underdoped cuprates at low temperatures. First, the vortex core size in cuprates (coherence length) measures only a few lattice spacings. Second, the low energy quasiparticles are gapless Dirac fermions whose lack of an intrinsic scale turns them into a quantum critical system and invalidates the semi-classical approach.


Vortex in conventional superconductors		Vortex in cuprates

In this picture, the virtual transitions between quasiparticle states caused by vortex motion are responsible for renormalization of the parameters that specify vortex dynamics. Taking into account the quantum coherence of gapless Dirac fermions, we calculated the effective vortex action and arrived at two essential conclusions: 1) the quasiparticle contribution to the effective vortex mass is equal to only a few electron masses; 2) in ideal circumstances at zero temperature, nodal quasiparticles give rise only to a universal super-Ohmic damping of vortex motion (cubic in frequency). In order for vortex damping to be more important than inertia, a low energy scale must be provided by finite temperature, disorder, or Zeeman fields; the Ohmic (Bardeen-Stephen-like) friction coefficient is then proportional to that energy scale squared. Therefore, d-wave vortices are much lighter and less damped than their counterparts in conventional s-wave superconductors. These conclusions support the notion that quantum fluctuations of vortices can drive the superconducting phase transitions in cuprates.

Papers:

Vortex Core Structure

Having a microscopic theory of vortex dynamics in d-wave superconductors, which supports the view of vortices as quantum objects, allows the exploration of the opposite problem: how the vortex quantum motion affects quasiparticles. The simplest situation involves a single vortex that executes zero-point quantum oscillations in a local trap created by the vortex lattice, or by a pinning impurity. We discovered that a sub-gap peak in the local density of states (LDOS), observed in several materials by STM near the vortex cores, can be naturally explained as arising from the resonant scattering of quasiparticles on a quantum fluctuating vortex. Unlike the self-consistent BCS-type calculations, our calculations correctly predict the absence of a zero-energy peak in LDOS, resulting from the vortex zero-point quantum motion. The absence of such zero-energy peaks in experimental observations indirectly confirms the quantum dynamics of vortices in underdoped cuprates.


LDOS vs. energy (distance from the core increases gradually as the plots are raised up for clarity)		STM tunneling conductance (taken from B.W.Hoogenboom, et.al; Phys.Rev.Lett. 87, 267001 (2001))

With closer scrutiny we also found additional features in the LDOS, which may be too small to resolve through noise in present-day experiments, but whose positions reflect the discrete spectrum of a trapped vortex. If these features could be resolved, we would be able to independently measure both the effective vortex mass and the vortex trapping potential in experiments.

Papers:

Vortex Lattices and Liquids

An interesting discovery deriving from my work is that there are novel forces between vortices in d-wave superconductors, which are mediated by the gapless nodal quasiparticles. Together with the well known Magnus (or Lorentz) forces, these interactions may have an important imprint on the properties of vortex lattices and the “normal” phase of cuprates.

These new interactions between vortices originate in the Doppler shifts of gapless quasiparticle spectra caused by vortex supercurrents. They are not only sensitive to the magnetic field (vortex density), but also to the spatial orientation of the interacting vortices with respect to nodal directions. Therefore, the influence of quasiparticles on the dynamics of vortex lattices can emerge in several experimentally verifiable situations. The first effect is the dependence of the effective vortex mass and other dynamical parameters on the external magnetic field. According to the model of quantum fluctuating vortices, the apparent vortex size in STM measurements reflects the spatial amplitude of vortex zero-point quantum oscillations. Therefore, the measured vortex size can be used to estimate the vortex mass, explore its dependence on the magnetic field, and compare it with calculations. The second effect is the dependence of the vortex lattice phonon modes on the orientation of the vortex lattice with respect to the substrate. A tendency of the vortex lattice to assume a specific orientation with respect to the substrate can result from quantum and thermal vortex fluctuations. Such a tendency would be a unique property of d-wave superconductors, caused by the gapless and anisotropic features of the quasiparticle spectrum.

The quasiparticle mediated forces between vortices in d-wave superconductors possess the distinctive feature that two vortices scattering off each other can dissipate some of their kinetic energy to quasiparticles. The consequences of these novel forces for vortex scattering have not yet been explored. One can expect that this kind of inter-vortex scattering would be imprinted on the transport properties of the “normal” state vortex liquid, such as the temperature and magnetic field dependence of resistivity due to quasiparticles and flux-flow.

Papers:

Effective Action for Vortex Dynamics in Clean d-wave superconductors

Ultra-cold fermionic atoms near unitarity

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)

Papers:

Renormalization group fixed points, universal phase diagram, and 1/N expansion for quantum liquids with interactions near the unitarity limit

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.

Papers:

Renormalization group fixed points, universal phase diagram, and 1/N expansion for quantum liquids with interactions near the unitarity limit

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:

Superfluid-insulator transitions of the Fermi gas with near-unitary interactions in a periodic potential

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.

Papers:

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:

Fractional Topological Insulators

vliquid2 — 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.

fancy-torus — 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.

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.