Using derivative expansion applied to the Wigner transform of the two - point Green function we analyse the anomalous quantum Hall effect (AQHE), and the chiral magnetic effect (CME). The corresponding currents are proportional to the momentum space topological invariants. We reproduce the conventional expression for the Hall conductivity in $2+1$ D. In $3+1$ D our analysis allows to explain systematically the AQHE in topological insulators and Weyl semimetals. At the same time using this method it may be proved, that the equilibrium CME is absent in the wide class of solids, as well as in the properly regularized relativistic quantum field theory.Comment: Latex, 26 page

We analyse the 3 + 1 D equilibrium chiral magnetic effect (CME). We apply derivative expansion to the Wigner transform of the two -point Green function. This technique allows us to express the response of electric current to external electromagnetic field strength through the momentum space topological invariant. We consider the wide class of the lattice regularizations of quantum field theory (that includes, in particular, the regularization with Wilson fermions) and also certain lattice models of solid state physics (including those of Dirac semimetals). It appears, that in these models the mentioned topological invariant vanishes identically at nonzero chiral chemical potential. That means, that the bulk equilibrium CME is absent in those systems.

Starting from the Abelian Higgs field theory, we construct the theory of quantum Abrikosov-Nielsen-Olesen strings. It is shown that in four space -time dimensions in the limit of infinitely thin strings, the conformal anomaly is absent, and the quantum theory exists. We also study an analogue of the Aharonov-Bohm effect: the corresponding topological interaction is proportional to the linking number of the string world sheet and the particle world trajectory. The creation operators of the strings are explicitly constructed in the path integral and in the Hamiltonian formulation of the theory. We show that the Aharonov-Bohm effect gives rise to several nontrivial commutation relations.

In Ref. \cite{Horava2005} Ho\v{r}ava suggested, that the multi - fermion many-body system with topologically stable Fermi surfaces may effectively be described (in a vicinity of the Fermi surface) by the theory with coarse-grained fermions. The number of the components of these coarse-grained fermions is reduced compared to the original system. Here we consider the $3+1$ D system and concentrate on the particular case when the Fermi surface has co-dimension $p=3$, i.e. it represents the Fermi point in momentum space. First we demonstrate explicitly that in agreement with Ho\v{r}ava conjecture, in the vicinity of the Fermi point the original system is reduced to the model with two - component Weyl spinors. Next, we generalize the construction of Ho\v{r}ava to the situation, when the original $3+1$ D theory contains multi - component Majorana spinors. In this case the system is also reduced to the model of the two - component Weyl fermions in the vicinity of the topologically stable Fermi point. Those fermions experience the emergent gauge field and the gravitational field given by the emergent veirbein. Both these fields (the emergent gauge field and the emergent gravitational field) originate from certain collective excitations of the original system. We speculate, that the given construction may be relevant for the high energy physics in the paradigm, in which the Lorentz symmetry as well as the gravitational and gauge fields are the emergent phenomena, i.e. they appear dynamically in the low energy approximation of the underlined high energy theory.Comment: to appear in Nuclear Physics B. Nucl.Phys.B (2014). arXiv admin note: substantial text overlap with arXiv:1312.126

First of all, we reconsider the tight - binding model of monolayer graphene, in which the variations of the hopping parameters are allowed. We demonstrate that the emergent 2D Weitzenbock geometry as well as the emergent U(1) gauge field appear. The emergent gauge field is equal to the linear combination of the components of the zweibein. Therefore, we actually deal with the gauge fixed version of the emergent 2+1 D teleparallel gravity. In particular, we work out the case, when the variations of the hopping parameters are due to the elastic deformations, and relate the elastic deformations with the emergent zweibein. Next, we investigate the tight - binding model with the varying intralayer hopping parameters for the multilayer graphene with the ABC stacking. In this case the emergent 2D Weitzenbock geometry and the emergent U(1) gauge field appear as well, the emergent low energy effective field theory has the anisotropic scaling.Comment: details of the derivation added, Latex, 16 page

We consider the recently discovered Dirac semimetals with two Dirac points ±K. In the presence of elastic deformations each fermion propagates in a curved space, whose metric is defined by the expansion of the effective Hamiltonian near the Dirac point. Besides, there is the emergent electromagnetic field that is defined by the shift of the Dirac point. We consider the case, when the deformations are caused by the dislocations. The dislocation carries singular torsion and the quantized flux of emergent magnetic field. Both torsion singularity and emergent magnetic flux may be observed in the scattering of quasiparticles on the dislocation due to Stodolsky and Aharonov -Bohm effects. We discuss quantum anomalies in the quasiparticle currents in the presence of emergent gauge and gravitational fields and the external electromagnetic field. In particular, it is demonstrated, that in the presence of external electric field the quasiparticles/holes are pumped from vacuum along the dislocation. The appeared chiral imbalance along the dislocation drives the analogue of chiral magnetic effect, that is the appearance of electric current along the dislocation.

Higgs bosons -the amplitude modes -have been experimentally investigated in condensed matter for many years. An example is superfluid 3 He-B, where the broken symmetry leads to 4 Goldstone modes and at least 14 Higgs modes, which are characterized by angular momentum quantum number J and parity (Zeeman splitting of Higgs modes with J = 2 + and J = 2 − in magnetic field has been observed in 80's). Based on the relation E 2 J+ + E 2 J− = 4∆ 2 for the energy spectrum of these modes, Yoichiro Nambu proposed the general sum rule, which relates masses of Higgs bosons and masses of fermions. If this rule is applicable to Standard Model, one may expect that the observed Higgs boson with mass M H1 = 125 GeV has a Nambu partner -the second Higgs boson with mass M H2 = 325 GeV. Together they satisfy the Nambu relation M 2 H1 + M 2 H2 = 4M 2 top , where M top is the top quark mass. Also the properties of the Higgs modes in superfluid 3 He-A, where the symmetry breaking is similar to that of the Standard Model, suggest the possible existence of two electrically charged Higgs particles with masses M H+ = M H− ∼ 245 GeV, which together obey the Nambu rule M 2 H+ + M 2 H− = 4M 2 top . A certain excess of events at 325 GeV and at 245 GeV has been reported in 2011, though not confirmed in 2012 experiments. Besides, we consider the particular relativistic model of top -quark condensation that suggests the possibility that two twice degenerated Higgs bosons contribute to the Nambu sum rule. This gives the mass around 210 GeV for the Nambu partner of the 125 GeV Higgs boson. We also discuss the other possible lessons from the condensed matter to Standard Model, such as hidden symmetry, where light Higgs emerges as quasi Nambu-Goldstone mode, and the role of broken time reversal symmetry. 1: O.V.

We found an additional symmetry hidden in the fermion and Higgs sectors of the Standard Model. It is connected to the centers of the SU(3) and SU(2) subgroups of the gauge group. A lattice regularization of the whole Standard Model is constructed that possesses this symmetry.Comment: 6 pages, no figures. Shortened versio