Nematic Quantum Criticality in Dirac Systems.

We investigate nematic quantum phase transitions in two different Dirac fermion models. The models feature twofold and fourfold, respectively, lattice rotational symmetries that are spontaneously broken in the ordered phase. Using negative-sign-free quantum Monte Carlo simulations and an ε-expansion renormalization group analysis, we show that both models exhibit continuous phase transitions. In contrast to generic Gross-Neveu dynamical mass generation, the quantum critical regime is characterized by large velocity anisotropies, with fixed-point values being approached very slowly. Both experimental and numerical investigations will not be representative of the infrared fixed point, but of a quasiuniversal regime where the drift of the exponents tracks the velocity anisotropy.

Bogoliubov-Fermi Surfaces in Noncentrosymmetric Multicomponent Superconductors.

We show that when the time reversal symmetry is broken in a multicomponent superconducting condensate without inversion symmetry the resulting Bogoliubov quasiparticles generically exhibit mini-Bogoliubov-Fermi (BF) surfaces, for small superconducting order parameter. The absence of inversion symmetry makes the BF surfaces stable with respect to weak perturbations. With sufficient increase of the order parameter, however, the Bogoliubov-Fermi surface may disappear through a Lifshitz transition, and the spectrum this way become fully gapped. Our demonstration is based on the computation of the effective Hamiltonian for the bands near the normal Fermi surface by the integration over high-energy states. Exceptions to the rule, and experimental consequences are briefly discussed.

Unconventional Superconductivity in Luttinger Semimetals: Theory of Complex Tensor Order and the Emergence of the Uniaxial Nematic State.

We investigate unconventional superconductivity in three-dimensional electronic systems with the chemical potential close to a quadratic band touching point in the band dispersion. Short-range interactions can lead to d-wave superconductivity, described by a complex tensor order parameter. We elucidate the general structure of the corresponding Ginzburg-Landau free energy and apply these concepts to the case of an isotropic band touching point. For a vanishing chemical potential, the ground state of the system is given by the superconductor analogue of the uniaxial nematic state, which features line nodes in the excitation spectrum of quasiparticles. In contrast to the theory of real tensor order in liquid crystals, however, the ground state is selected here by the sextic terms in the free energy. At a finite chemical potential, the nematic state has an additional instability at weak coupling and low temperatures. In particular, the one-loop coefficients in the free energy indicate that at weak coupling genuinely complex orders, which break time-reversal symmetry, are energetically favored. We relate our analysis to recent measurements in the half-Heusler compound YPtBi and discuss the role of cubic crystal symmetry.

Topological mott insulator in three-dimensional systems with quadratic band touching.

We argue that a three-dimensional electronic system with the Fermi level at the quadratic band touching point such as HgTe could be unstable with respect to the spontaneous formation of the (topological) Mott insulator at arbitrary weak long-range Coulomb interaction. The mechanism of the instability can be understood as the collision of Abrikosov's non-Fermi liquid fixed point with another, quantum critical, fixed point, which approaches it in the coupling space as the system's dimensionality d→dlow+, with the "lower critical dimension" 2

Density of states scaling at the semimetal to metal transition in three dimensional topological insulators.

The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of a disordered Z2 topological insulator as an important example, we compute the single particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent ν and the dynamical critical exponent z are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed.

Zero modes and charged Skyrmions in graphene bilayer.

We show that the electric charge of the Skyrmion in the vector order parameters that characterize the quantum anomalous spin Hall state and the layer antiferromagnet in a graphene bilayer is four and zero, respectively. The result is based on the demonstration that a vortex configuration in two broken symmetry states in bilayer graphene with the quadratic band crossing has the number of zero modes doubled relative to the single layer. The doubling can be understood as a result of Kramers's theorem implied by the "pseudo time reversal" symmetry of the vortex Hamiltonian. Disordering the quantum anomalous spin Hall state by Skyrmion condensation should produce a superconductor of an elementary charge 4e.

Topological insulator in the core of the superconducting vortex in graphene.

The core of the vortex in a general superconducting order parameter in graphene is argued to be ordered, with the possible local order parameters forming the algebra U(1) x Cl(3). A sufficiently strong Zeeman coupling of the magnetic field of the vortex to the electron spin breaks the degeneracy in the core in favor of the anomalous quantum Hall state. I consider a variety of superconducting condensates on the honeycomb lattice and demonstrate the surprising universality of this result. A way to experimentally determine the outcome of the possible competition between different types of orders in the core is proposed.

Restoration of the magnetic hc/e-periodicity in unconventional superconductors.

We consider the energy of the filled quasiparticle's Fermi sea of a macroscopic superconducting ring threaded by an hc/2e vortex, when the material of the ring is of an unconventional pairing symmetry. The energy relative to the one for the hc/e vortex configuration is finite, positive, and inversely proportional to the ring's inner radius. We argue that the existence of this energy in unconventional superconductors removes the commonly assumed degeneracy between the odd and the even vortices, with the loss of the concomitant hc/2e-periodicity in an external magnetic field as a consequence. This macroscopic quantum effect should be observable in nanosized unconventional superconductors with a small phase stiffness, such as deeply underdoped YBCO with Tc<5 K.

Coulomb interaction, ripples, and the minimal conductivity of graphene.

We argue that the unscreened Coulomb interaction in graphene provides a positive, universal, and logarithmic correction to scaling of zero-temperature conductivity with frequency. The combined effect of the disorder due to wrinkling of the graphene sheet and the long-range electron-electron interactions is a finite positive contribution to the dc conductivity. This contribution is disorder strength dependent and thus nonuniversal. The low-energy behavior of such a system is governed by the line of fixed points at which both the interaction and disorder are finite, and the density of states is exactly linear. An estimate of the typical random vector potential representing ripples in graphene brings the theoretical value of the minimal conductivity into the vicinity of 4e2/h.

Zero-energy states and fragmentation of spin in the easy-plane antiferromagnet on a honeycomb lattice.

The core of the vortex in the Néel order parameter for an easy-plane antiferromagnet on a honeycomb lattice is demonstrated to bind two zero-energy states. Remarkably, a single electron occupying this midgap band has its spin fragmented between the two sublattices: Whereas it yields a vanishing total magnetization, it shows a finite Néel order, orthogonal to the one of the assumed background. The requisite easy-plane anisotropy may be introduced by a magnetic field parallel to the graphene layer, for example. The results are relevant for spin-1/2 fermions on the graphene's or optical honeycomb lattice, in the strongly interacting regime.

Interactions and phase transitions on graphene's honeycomb lattice.

The low-energy theory of interacting electrons on graphene's two-dimensional honeycomb lattice is derived and discussed. In particular, the Hubbard model in the large-N limit is shown to have a semimetal-antiferromagnetic insulator quantum critical point in the universality class of the Gross-Neveu model. The same equivalence is conjectured to hold in the physical case N=2, and its consequences for various physical quantities are examined. The effects of the long-range Coulomb interaction and the magnetic field are discussed.

Stable Skyrmions in spinor condensates.

Globally symmetric spinor condensates in free space are argued not to support stable topological defects in either two or three dimensions. In the latter case, however, we show that a topological Skyrmion can be stabilized by forcing it to adopt certain density profiles. A sufficient condition for the existence of Skyrmion solutions in three dimensions is formulated and illustrated in simple examples. Our results pertain to Bose-Einstein condensation in 87Rb.

Effective theory of high-temperature superconductors.

The field theory of a fluctuating d-wave superconductor is constructed and proposed as an effective description of superconducting cuprates at low energies. The theory is used to resolve a puzzle posed by recent experiments on superfluid density in severely underdoped YBa2(Cu3)O(6+x). In particular, the overall temperature dependence of the superfluid density at low dopings is argued to be described well by the strongly anisotropic weakly interacting Bose gas, and thus approximately linear in temperature with an almost doping-independent slope.

Permanent confinement in the compact QED3 with fermionic matter.

We argue that the compact three dimensional electrodynamics with massless relativistic fermions is always in the confined phase, in spite of the bare interaction between the magnetic monopoles being rendered logarithmic by fermions. The effect is caused by screening by other dipoles, which transforms the logarithmic back into the Coulomb interaction at large distances. Possible implications for the chiral symmetry breaking for fermions are discussed, and the global phase diagram of the theory is proposed.

Antiferromagnetism from phase disordering of a d-wave superconductor.

The unbinding of vortex defects in the superconducting condensate with d-wave symmetry at T = 0 is shown to lead to the insulator with incommensurate spin-density-wave order. The transition is similar to the spontaneous generation of the chiral mass in the three-dimensional quantum electrodynamics. A possible relation to recent experiments on underdoped cuprates is discussed.