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.

In numerical simulations, spontaneously broken symmetry is often detected by computing twopoint correlation functions of the appropriate local order parameter. This approach, however, computes the square of the local order parameter, and so when it is small, very large system sizes at high precisions are required to obtain reliable results. Alternatively, one can pin the order by introducing a local symmetry breaking field, and then measure the induced local order parameter infinitely far from the pinning center. The method is tested here at length for the Hubbard model on honeycomb lattice, within the realm of the projective auxiliary field quantum Monte Carlo algorithm. With our enhanced resolution we find a direct and continuous quantum phase transition between the semi-metallic and the insulating antiferromagnetic states with increase of the interaction. The single particle gap in units of the Hubbard U tracks the staggered magnetization. An excellent data collapse is obtained by finite size scaling, with the values of the critical exponents in accord with the Gross-Neveu universality class of the transition.

The low-energy theory of electrons interacting via repulsive short-range interactions on graphene's honeycomb lattice at half filling is presented. The exact symmetry of the Lagrangian with local quartic terms for the Dirac field dictated by the lattice is D_2 x U_c(1) x (time reversal), where D_2 is the dihedral group, and U_c(1) is a subgroup of the SU_c(2) "chiral" group of the non-interacting Lagrangian, that represents translations in Dirac language. The Lagrangian describing spinless particles respecting this symmetry is parameterized by six independent coupling constants. We show how first imposing the rotational, then Lorentz, and finally chiral symmetry to the quartic terms, in conjunction with the Fierz transformations, eventually reduces the set of couplings to just two, in the "maximally symmetric" local interacting theory. We identify the two critical points in such a Lorentz and chirally symmetric theory as describing metal-insulator transitions into the states with either time-reversal or chiral symmetry being broken. In the site-localized limit of the interacting Hamiltonian the low-energy theory describes the continuous transitions into the insulator with either a finite Haldane's (circulating currents) or Semenoff's (staggered density) masses, both in the universality class of the Gross-Neveu model. The picture of the metal-insulator transition on a honeycomb lattice emerges at which the residue of the quasiparticle pole at the metallic and the mass-gap in the insulating phase both vanish continuously as the critical point is approached. We argue that the Fermi velocity is non-critical as a consequence of the dynamical exponent being fixed to unity by the emergent Lorentz invariance. Effects of long-range interaction and the critical behavior of specific heat and conductivity are discussed.Comment: 16 revtex pages, 4 figures; typos corrected, new and updated references; published versio

Critical phenomena is one of the most exciting areas of modern physics. This 2007 book provides a thorough but economic introduction into the principles and techniques of the theory of critical phenomena and the renormalization group, from the perspective of modern condensed matter physics. Assuming basic knowledge of quantum and statistical mechanics, the book discusses phase transitions in magnets, superfluids, superconductors, and gauge field theories. Particular attention is given to topics such as gauge field fluctuations in superconductors, the Kosterlitz-Thouless transition, duality transformations, and quantum phase transitions - all of which are at the forefront of physics research. This book contains numerous problems of varying degrees of difficulty, with solutions. These problems provide readers with a wealth of material to test their understanding of the subject. It is ideal for graduate students and more experienced researchers in the fields of condensed matter physics, statistical physics, and many-body physics.

We study the chiral Ising, the chiral XY, and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in 4 − ϵ dimensions and compute critical exponents for the GrossNeveu-Yukawa fixed points to order Oðϵ 4 Þ. Further, we provide Padé estimates for the correlation length exponent, the boson and fermion anomalous dimension, as well as the leading correction to scaling exponent in 2 þ 1 dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with N ¼ 1=4 and N ¼ 1=2 fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.

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 semi-metal -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 of the magnetic field are discussed.A graphite monolayer, or graphene, emerged recently as the new frontier in physics of electronic systems with reduced dimensionality [1]. Such two-dimensional, or quasi-two-dimensional systems have led to some of the most startling discoveries in the condensed matter physics in the recent past, the quantum Hall effects and the metal-insulator transitions in silicon-MOSFETS and Ga-As heterostructures, and the high-temperature superconductivity in cuprates being prime examples. What makes graphene qualitatively new is its semi-metallic nature with low-energy quasiparticles behaving as 'relativistic' Dirac spinors over a good portion of the conducting band. The spinor structure is a general consequence of the bipartite nature of the honeycomb lattice The relativistic spectrum and the concomitant linearly vanishing density of states at the Fermi level, similarly as in the superconducting state of cuprates, provide graphene's quasiparticles with an additional protection against the effects of interactions. Nevertheless, a sufficiently strong repulsion is expected to turn the semimetallic state into a gapped insulator, possibly breaking the translational and/or the rotational symmetry in the process. Within the simplest interacting theory defined by the Hubbard model there is convincing numerical evidence for the quantum phase transition at a large Hubbard U into an antiferromagnet (AF) [6]. On the other hand, the long-range Coulomb interaction remains unscreened in the semi-metal (SM) [7], and has been argued to favor the charge-density-wave (CDW) at strong coupling [8]. The competition between different instabilities, the universality class, or even the order, of the SM -insulator transition, and the interplay of interactions with the Landau quantization in the external magnetic field present some of the basic open problems. Although graphene in its natural state may not be near a critical point [9], one can conceive mechanical deformations that would pull it deeper into the strong-coupling regime [10]. Finally, the outcome of the competition between different interactions should have consequences for the selection of the ground state in the magnetic field, even at weak coupling [11].In the present communication some of these issues are addressed by considering the half-filled Hubbard model on a honeycomb lattice, complemented with the additional long-range Coulomb interaction between electrons. The analysis is based on a useful decomposition of Hubbard's on-site interaction on a bipartit...

We formulate the effective Gross-Neveu-Yukawa theory of the semimetal-insulator transitions on the honeycomb lattice and compute its quantum critical behavior near three (spatial) dimensions. We find that at the critical point Dirac fermions do not survive as coherent excitations and that the $\sim 1/r$ tail of the weak Coulomb interaction is an irrelevant coupling. The emergent Lorentz invariance near criticality implies a universal ratio of the low-temperature specific heats of the metallic and the rotational-symmetry-broken insulating phase.Comment: 4 pages, one figure; updated, published versio

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