arXiv subjects

Igor F. Herbut

Publications and source records attributed to Igor F. Herbut.

At least 55 records · Page 3Linked to original sources

arXiv2017

Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems

Dirac and Weyl fermions appear as quasi-particle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In this work we study the bosonized version of the standard Gross-Neveu model -- the Gross-Neveu-Yukawa theory -- at three-loop order, and compute critical exponents in $4-ε$ dimensions for general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the purely bosonic limit of the theory. We also find the exponents to satisfy the emergent super-scaling relations in the limit of a single-component fermion, order by order up to three loops. Finally, we apply the computed series for the exponents and their Padé approximants to several phase transitions of current interest: metal-insulator transitions of spin-1/2 and spinless fermions on the honeycomb lattice, emergent supersymmetric surface field theory in topological phases, as well as the disorder-induced quantum transition in Weyl semimetals. Comparison with the results of other analytical and numerical methods is discussed.

cond-mat.str-el↗

arXiv2017

Anisotropy induces non-Fermi-liquid behavior and nematic magnetic order in three-dimensional Luttinger semimetals

We illuminate the intriguing role played by spatial anisotropy in three-dimensional Luttinger semimetals featuring quadratic band touching and long-range Coulomb interactions. We observe the anisotropy to be subject to an exceptionally slow renormalization group (RG) evolution so that it can be considered approximately constant when computing the impact of quantum fluctuations on the remaining couplings of the system. Using perturbative RG we then study the competition of all local short-range interactions that are generated from the long-range interactions for fixed anisotropy. Two main effects come to light for sufficiently strong anisotropy. First, the three-dimensional system features an Abrikosov non-Fermi liquid ground state. Second, there appear qualitatively new fixed points which describe quantum phase transitions into phases with nemagnetic orders - higher-rank tensor orders that break time-reversal symmetry, and thus have both nematic and magnetic character. In real materials these phases may be realized through sufficiently strong microscopic short-range interactions. On the pyrochlore lattice, the anisotropy-induced fixed points determine the onset of all-in-all-out or spin ice ordering of local magnetic moments of the electrons.

cond-mat.str-el↗

arXiv2017

Phase diagram of electronic systems with quadratic Fermi nodes in $2<d<4$: $2+ε$ expansion, $4-ε$ expansion, and functional renormalization group

Several materials in the regime of strong spin-orbit interaction such as HgTe, the pyrochlore iridate Pr$_2$Ir$_2$O$_7$, and the half-Heusler compound LaPtBi, as well as various systems related to these three prototype materials, are believed to host a quadratic band touching point at the Fermi level. Recently, it has been proposed that such a three-dimensional gapless state is unstable to a Mott-insulating ground state at low temperatures when the number of band touching points $N$ at the Fermi level is smaller than a certain critical number $N_c$. We further substantiate and quantify this scenario by various approaches. Using $ε$ expansion near two spatial dimensions, we show that $N_c = 64/(25 ε^2) + O(1/ε)$ and demonstrate that the instability for $N < N_c$ is towards a nematic ground state that can be understood as if the system were under (dynamically generated) uniaxial strain. We also propose a truncation of the functional renormalization group equations in the dynamical bosonization scheme which we show to agree to one-loop order with the results from $ε$ expansion both near two as well as near four dimensions, and which smoothly interpolates between these two perturbatively accessible limits for general $2<d<4$. Directly in $d=3$ we therewith find $N_c = 1.86$, and thus again above the physical $N=1$. All these results are consistent with the prediction that the interacting ground state of pure, unstrained HgTe, and possibly also Pr$_2$Ir$_2$O$_7$, is a strong topological insulator with a dynamically-generated gap -- a topological Mott insulator.

cond-mat.str-el↗

arXiv2016

Note on "Quantum superconducting criticality in graphene and topological insulators"

We correct our previous conclusion regarding the fate of a charged quantum critical point across the superconducting transition for two dimensional massless Dirac fermion. Within the leading order $ε$ expansion, we now find that the requisite number of four-component Dirac fermion flavors ($N_f$) for the continuous phase transition through a charged critical point is $N_f>18.2699$. For $N_f\geq1/2$, the critical number of bosonic flavors for this transition is significantly reduced as compared to the value determined in the absence of the Dirac fermions in the theory.

cond-mat.mes-hall↗

arXiv2016

Gauge-field-assisted Kekulé quantum criticality

We study the quantum phase transition of $U(1)$ - charged Dirac fermions Yukawa-coupled to the Kekulé valence bond solid order parameter with $Z_3$ symmetry of the honeycomb lattice. The symmetry allows for the presence of the term in the action which is cubic in the Kekulé order parameter, and which is expected to prevent the quantum phase transition in question from being continuous. The Gross-Neveu-Yukawa theory for the transition is investigated using a perturbative renormalization group and within the $ε$ expansion close to four space-time dimensions. For a vanishing $U(1)$ charge we show that quantum fluctuations may render the phase transition continuous only sufficiently far away from 3+1 dimensions, where the validity of the conclusions based on the leading order $ε$ expansion appear questionable. In the presence of a fluctuating gauge field, on the other hand, we find quantum critical behavior even at weak coupling to appear close to 3+1 dimensions, that is, within the domain of validity of the perturbation theory. We also determine the renormalization group scaling of the cubic coupling at higher-loop orders and for a large number of Dirac fermions for vanishing charge.

cond-mat.str-el↗

arXiv2016

Chiral symmetry breaking in three-dimensional quantum electrodynamics as fixed point annihilation

Spontaneous chiral symmetry breaking in three dimensional ($d=3$) quantum electrodynamics is understood as annihilation of an infrared-stable fixed point that describes the large-N conformal phase by another unstable fixed point at a critical number of fermions $N=N_c$. We discuss the root of universality of $N_c$ in this picture, together with some features of the phase boundary in the $(d,N)$ plane. In particular, it is shown that as $d\rightarrow 4$, $N_c\rightarrow 0$ with a constant slope, our best estimate of which suggests that $N_c = 2.89$ in $d=3$.

hep-th↗

arXiv2016

Superconducting quantum criticality in three-dimensional Luttinger semimetals

We study a simple model of three-dimensional fermions close to a quadratic band touching point, built from the celebrated Luttinger single-particle Hamiltonian and an attractive contact interaction between the particles. Such a system displays a novel quantum critical point between the semimetallic and an s-wave superconducting phase at which the low-energy "Luttinger fermions" are inextricably coupled to the order parameter fluctuations. The quantum critical point is perturbatively accessible near four spatial dimensions, where it features nontrivial scaling with dynamical exponent 1<z<2 and emergent rotational and particle-hole symmetries. Some features of the criticality, such as oscillatory corrections to scaling and its enhanced symmetry, are discussed.

cond-mat.str-el↗

arXiv2016

Quadratic band touching with long range interactions in and out of equilibrium

Motivated by recent advances in cold atomic systems, we study the equilibrium and quench properties of two dimensional fermions with quadratic band touching at the Fermi level, in the presence of infinitely long range interactions. Unlike when only short range interactions are present, both nematic and quantum anomalous Hall (QAH) states state appear at weak interactions, separated by a narrow coexistence region, whose boundaries mark second and third order quantum phase transitions. After an interaction quench, the QAH order exhibits three distinct regions: persistent or damped oscillations and exponential decay to zero. In contrast, the nematic order always reaches a non-zero stationary value through power law damped oscillations, due to the interplay of the symmetry of the interaction and the specific topology of the quadratic band touching.

cond-mat.str-el↗

arXiv2016

Emergent Lorentz symmetry near fermionic quantum critical points in two and three dimensions

We study the renormalization group flow of the velocities in the field theory describing the coupling of the massless quasi-relativistic fermions to the bosons through the Yukawa coupling, as well as with both bosons and fermions coupled to a fluctuating $U(1)$ gauge field in two and three spatial dimensions. Different versions of this theory describe quantum critical behavior of interacting Dirac fermions in various condensed-matter systems. We perform an analysis using one-loop $ε-$expansion about three spatial dimensions, which is the upper critical dimension in the problem. In two dimensions, we find that velocities of both charged fermions and bosons ultimately flow to the velocity of light, independently of the initial conditions, the number of fermionic and bosonic flavors, and the value of the couplings at the critical point. In three dimensions, due to the analyticity of the gauge field propagator, both the $U(1)$ charge and the velocity of light flow, which leads to a richer behavior than in two dimensions. We show that all three velocities ultimately flow to a common terminal velocity, which is non-universal and different from the original velocity of light. Therefore, emergence of the Lorentz symmetry in the ultimate infrared regime seems to be a rather universal feature of this class of theories in both two and three dimensions.

hep-th↗

arXiv2016

Half vortex and fractional electrical charge in two dimensions

Despite fermion doubling, a two-dimensional quasi-relativistic spin-1/2 system can still lead to true fractionalization of electrical charge, when a massive ordered phase supports a "half-vortex". Such topological defect is possible when the order parameter in form of Dirac mass is described by two $U(1)$ angles, and each of them winds by an angle $π$ around a point. We demonstrate that such a mass configuration in an eight-dimensional Dirac Hamiltonian exhibits only a single bound zero mode, and therefore binds the charge of $e/2$. In graphene, for example, such an ordered phase is provided by the easy-plane spin-triplet Kekule valence bond solid. We argue that an application of an in-plane magnetic field can cause an excitonic instability toward such ordered phase, even for weak repulsion, when the on-site, nearest-neighbor and second neighbor components of it are of comparable strengths.

cond-mat.mes-hall↗

arXiv2016

Excitonic instability of three-dimensional gapless semiconductors: Large-N theory

Three-dimensional gapless semiconductors with quadratic band touching, such as HgTe, $α$-Sn, or Pr$_2$Ir$_2$O$_7$ are believed to display a non-Fermi-liquid ground state due to long-range electron-electron interaction. We argue that this state is inherently unstable towards spontaneous formation of a (topological) excitonic insulator. The instability can be parameterized by a critical fermion number $N_c$. For $N < N_c$ the rotational symmetry is spontaneously broken, the system develops a gap in the spectrum, and features a finite nematic order parameter. To leading order in the 1/N expansion and in the static approximation, the analogy with the problem of dynamical mass generation in (2+1)-dimensional quantum electrodynamics yields $N_c = 16/[3π(π-2)]$. Taking the important dynamical screening effects into account, we find that $N_c \geq 2.6(2)$ and therefore safely above the physical value of $N = 1$. Some experimental consequences of the nematic ground state are discussed.

cond-mat.str-el↗

arXiv2015

Competition of density waves and quantum multicritical behavior in Dirac materials from functional renormalization

We study the competition of spin- and charge-density waves and their quantum multicritical behavior for the semimetal-insulator transitions of low-dimensional Dirac fermions. Employing the effective Gross-Neveu-Yukawa theory with two order parameters as a model for graphene and a growing number of other two-dimensional Dirac materials allows us to describe the physics near the multicritical point at which the semimetallic and the spin- and charge-density-wave phases meet. With the help of a functional renormalization group approach, we are able to reveal a complex structure of fixed points, the stability properties of which decisively depend on the number of Dirac fermions $N_f$. We give estimates for the critical exponents and observe crucial quantitative corrections as compared to the previous first-order $ε$ expansion. For small $N_f$, the universal behavior near the multicritical point is determined by the chiral Heisenberg universality class supplemented by a decoupled, purely bosonic, Ising sector. At large $N_f$, a novel fixed point with nontrivial couplings between all sectors becomes stable. At intermediate $N_f$, including the graphene case ($N_f = 2$) no stable and physically admissible fixed point exists. Graphene's phase diagram in the vicinity of the intersection between the semimetal, antiferromagnetic and staggered density phases should consequently be governed by a triple point exhibiting first-order transitions.

cond-mat.str-el↗

arXiv2015

Critical $O(2)$ and $O(3)$ $ϕ^4$ theories near six dimensions

We consider $O(N)$-symmetric bosonic $ϕ^4$ field theories above four dimensions, and propose a new reformulation in terms of an irreducible tensorial field with a cubic and Yukawa terms. The $ϕ^4$ field theory so rewritten exhibits real and nontrivial IR-stable fixed points near and below six dimension, for low values of $N$ such as $N=2$ and $N=3$. The so-defined UV completions of the $O(2)$ and $O(3)$ models hence constitute precious examples of asymptotically safe quantum field theories. The possibility of an extension of our results to five dimensions is discussed.

hep-th↗

arXiv2015

Mott multicriticality of Dirac electrons in graphene

We study the multicritical behavior for the semimetal-insulator transitions on graphene's honeycomb lattice using the Gross-Neveu-Yukawa effective theory with two order parameters: the SO(3) (Heisenberg) order parameter describes the antiferromagnetic transition, and the $\mathbb{Z}_2$ (Ising) order parameter describes the transition to a staggered density state. Their coupling induces multicritical behavior which determines the structure of the phase diagram close to the multicritical point. Depending on the number of fermion flavors $N_f$ and working in the perturbative regime in vicinity of three (spatial) dimensions, we observe first order or continuous phase transitions at the multicritical point. For the graphene case of $N_f=2$ and within our low order approximation, the phase diagram displays a tetracritical structure.

cond-mat.str-el↗

arXiv2015

Nematic quantum criticality in three-dimensional Fermi system with quadratic band touching

We construct and discuss the field theory for tensorial nematic order parameter coupled to gapless four-component fermions at the quadratic band touching point in three (spatial) dimensions. Within a properly formulated epsilon-expansion this theory is found to have a quantum critical point, which describes the (presumably continuous) transition from the semimetal into a (nematic) Mott insulator. The latter phase breaks the rotational, but not the time-reversal symmetry, and may be relevant to materials such as gray tin or mercury telluride at low temperatures. The critical point represents a simple quantum analogue of the familiar classical isotropic-to-nematic transition in liquid crystals. The properties and the consequences of this quantum critical point are discussed. Its existence supports the scenario of the "fixed-point collision", according to which three-dimensional Fermi systems with quadratic band touching and long-range Coulomb interactions are unstable towards the gapped nematic ground state at low temperatures.

cond-mat.str-el↗

arXiv2015

Fermionic quantum criticality in honeycomb and $π$-flux Hubbard models: Finite-size scaling of renormalization-group-invariant observables from quantum Monte Carlo

We numerically investigate the critical behavior of the Hubbard model on the honeycomb and the $π$-flux lattice, which exhibits a direct transition from a Dirac semimetal to an antiferromagnetically ordered Mott insulator. We use projective auxiliary-field quantum Monte Carlo simulations and a careful finite-size scaling analysis that exploits approximately improved renormalization-group-invariant observables. This approach, which is successfully verified for the three-dimensional XY transition of the Kane-Mele-Hubbard model, allows us to extract estimates for the critical couplings and the critical exponents. The results confirm that the critical behavior for the semimetal to Mott insulator transition in the Hubbard model belongs to the Gross-Neveu-Heisenberg universality class on both lattices.

cond-mat.str-el↗

arXiv2014

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 $d\rightarrow d_{low} +$, with the "lower critical dimension" $2< d_{low}<4$. Arguments for the existence of the quantum critical point based on considerations in the large-$N$ limit in $d=3$, as well as close to $d=2$, are given. In the one-loop calculation we find that $d_{low} = 3.26$, and thus above, but not far from three dimensions. This translates into a temperature/energy window $(T_c, T_*)$ over which the NFL scaling should still be observable, before the Mott transition finally takes place at the critical temperature $T_c \sim T_* \exp(-z C/(d_{low} -d)^{1/2})$. We estimate $C=π/1.1$, dynamical critical exponent $z\approx 1.8$, and the temperature scale $k_B T_* \approx (4 m/ m_{el} ε^2)13.6 eV$, with $m$ as the band mass and $ε$ as the dielectric constant.

cond-mat.str-el↗

arXiv2014

Zero-modes and global antiferromagnetism in strained graphene

A novel magnetic ground state is reported for the Hubbard Hamiltonian in strained graphene. When the chemical potential lies close to the Dirac point, the ground state exhibits locally both the Néel and ferromagnetic orders, even for weak Hubbard interaction. Whereas the Néel order parameter remains of the same sign in the entire system, the magnetization at the boundary takes the opposite sign from the bulk. The total magnetization this way vanishes, and the magnetic ground state is globally only an antiferromagnet. This peculiar ordering stems from the nature of the strain-induced single particle zero-energy states, which have support on one sublattice of the honeycomb lattice in the bulk, and on the other sublattice near the boundary of a finite system. We support our claim with the self-consistent numerical calculation of the order parameters, as well as by the Monte Carlo simulations of the Hubbard model in both uniformly and non-uniformly strained honeycomb lattice. The present result is contrasted with the magnetic ground state of the same Hubbard model in the presence of a true magnetic field (and for vanishing Zeeman coupling), which is exclusively Néel ordered, with zero local magnetization everywhere in the system.

cond-mat.mes-hall↗