Perturbative Renormalization Group Research Articles

Global Phase Diagram of a Dirty Weyl Liquid and Emergent Superuniversality

Pursuing complementary field-theoretic and numerical methods, we here paint the global phase diagram of a three-dimensional dirty Weyl system. The generalized Harris criterion, augmented by a perturbative renormalization-group (RG) analysis shows that weak disorder is an irrelevant perturbation at the Weyl semimetal(WSM)-insulator quantum critical point (QCP). But, a metallic phase sets in through a quantum phase transition (QPT) at strong disorder across a multicritical point (MCP). The field theoretic predictions for the correlation length exponent $\nu=2$ and dynamic scaling exponent $z=5/4$ at this MCP are in good agreement with the ones extracted numerically, yielding $\nu=1.98 \pm 0.10$ and $z=1.26 \pm 0.05$, from the scaling of the average density of states (DOS). Deep inside the WSM phase, generic disorder is also an irrelevant perturbation, while a metallic phase appears at strong disorder through a QPT. We here demonstrate that in the presence of generic, but strong disorder the WSM-metal QPT is ultimately always characterized by the exponents $\nu=1$ and $z=3/2$ (to one-loop order), originating from intra-node or chiral symmetric (e.g., regular and axial potential) disorder. We here anchor such emergent \emph{chiral superuniversality} through complementary RG calculations, controlled via $\epsilon$-expansions, and numerical analysis of average DOS across WSM-metal QPT. In addition, we also discuss a subsequent QPT (at even stronger disorder) of a Weyl metal into an Anderson insulator by numerically computing the typical DOS at zero energy. The scaling behavior of various physical observables, such as residue of quasiparticle pole, dynamic conductivity, specific heat, Gr$\ddot{\mbox{u}}$neisen ratio, inside various phases as well as across various QPTs in the global phase diagram of a dirty Weyl liquid are discussed.

Chiral Mott insulators in frustrated Bose-Hubbard models on ladders and two-dimensional lattices: A combined perturbative and density matrix renormalization group study

We study the fully gapped chiral Mott insulator (CMI) of frustrated Bose-Hubbard models on ladders and two-dimensional lattices by perturbative strong-coupling analysis and density matrix renormalization group (DMRG). First we show the existence of a low-lying exciton state on all geometries carrying the correct quantum numbers responsible for the condensation of excitons and formation of the CMI in the intermediate interaction regime. Then we perform systematic DMRG simulations on several two-leg ladder systems with $\pi$-flux and carefully characterize the two quantum phase transitions. We discuss the possibility to extend the generally very small CMI window by including repulsive nearest-neighbour interactions or changing density and coupling ratios.

Kondo effect in a PT -symmetric non-Hermitian Hamiltonian

The combination of non-Hermitian physics and strong correlations can give rise to new effects in open quantum many-body systems with balanced gain and loss. We propose a generalized Anderson impurity model that includes non-Hermitian hopping terms between an embedded quantum dot and two wires. These non-Hermitian hopping terms respect a parity-time ($\mathcal{PT}$) symmetry. In the regime of a singly occupied localized state, we map the problem to a $\mathcal{PT}$-symmetric Kondo model and study the effects of the interactions using a perturbative renormalization group approach. We find that the Kondo effect persists if the couplings are below a critical value that corresponds to an exceptional point of the non-Hermitian Kondo interaction. On the other hand, in the regime of spontaneously broken $\mathcal{PT}$ symmetry, the Kondo effect is suppressed and the low-energy properties are governed by a local-moment fixed point with vanishing conductance.

Communication: An efficient stochastic algorithm for the perturbative density matrix renormalization group in large active spaces

We present an efficient stochastic algorithm for the recently introduced perturbative density matrix renormalization group method for large active spaces. The stochastic implementation bypasses the computational bottleneck involved in solving the first order equation in the earlier deterministic algorithm. We demonstrate the efficiency and accuracy of the algorithm on the C2 and Cr2 molecular benchmark systems.

ERG and Reggeon Field Theory

In this paper we present our recent non perturbative functional renormalization group analysis of Reggeon Field Theory to the interactions of Pomeron and Odderon fields. We establish the existence of a fixed point and its universal properties, which exhibits a novel symmetry structure in the space of Odderon-Pomeron interactions. We briefly discuss the implications of our findings for the existence of an Odderon in high energy scattering.

Multicritical Point on the de Almeida-Thouless Line in Spin Glasses in d>6 Dimensions.

The de Almeida-Thouless (AT) line in Ising spin glasses is the phase boundary in the temperature T and magnetic field h plane below which replica symmetry is broken. Using perturbative renormalization group (RG) methods, we show that, when the dimension d of space is just above six, there is a multicritical point (MCP) on the AT line, which separates a low-field regime, in which the critical exponents have mean-field values, from a high-field regime, where the RG flows run away to infinite coupling strength; as d approaches six from above, the MCP approaches the zero-field critical point exponentially in 1/(d-6). Thus, on the AT line, perturbation theory for the critical properties breaks down at a sufficiently large magnetic field even above 6 dimensions, as well as for all nonzero fields when d≤6, as was known previously. We calculate the exponents at the MCP to first order in ϵ=d-6>0. The fate of the MCP as d increases from just above six to infinity is not known.

The $$\epsilon $$ ϵ Expansion and Universality in Three Dimensions

It has been observed that the clasification into universality classes of critical behaviour, as established by perturbative renormalization group in the viscinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when purturbative renormalisation group fails in three dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when purturbative renormalisation group fails, because of the well known phenomenon of eigenvalue repulsion.

Review of Recent Developments in the Random-Field Ising Model

A lot of progress has been made recently in our understanding of the random-field Ising model thanks to large-scale numerical simulations. In particular, it has been shown that, contrary to previous statements: the critical exponents for different probability distributions of the random fields and for diluted antiferromagnets in a field are the same. Therefore, critical universality, which is a perturbative renormalization-group prediction, holds beyond the validity regime of perturbation theory. Most notably, dimensional reduction is restored at five dimensions, i.e., the exponents of the random-field Ising model at five dimensions and those of the pure Ising ferromagnet at three dimensions are the same.

Critical phenomena at the complex tensor ordering phase transition

We investigate the critical properties of the phase transition towards complex tensor order that has been proposed to occur in spin-orbit coupled superconductors. For this purpose we formulate the bosonic field theory for fluctuations of the complex irreducible second-rank tensor order parameter close to the transition. We then determine the scale dependence of the couplings of the theory by means of the perturbative Renormalization Group (RG). For the isotropic system we generically detect a fluctuation-induced first-order phase transition. The initial values for the running couplings are determined by the underlying microscopic model for the tensorial order. As an example we study three-dimensional Luttinger semimetals with electrons at a quadratic band touching point. Whereas the strong-coupling transition of the model receives substantial fluctuation corrections, the weak-coupling transition at low temperatures is rendered only weakly first-order due to the presence of a fixed point in the vicinity of the RG trajectory. If the number of fluctuating complex components of the order parameter is reduced by cubic anisotropy, the theory maps onto the field theory for frustrated magnetism.

Gapless chiral spin liquid from coupled chains on the kagome lattice

Using a perturbative renormalization group approach, we show that the extended (J_1-J_2-J_dJ1−J2−Jd) Heisenberg model on the kagome lattice with a staggered chiral interaction (J_\chiJχ) can exhibit a gapless chiral quantum spin liquid phase. Within a coupled-chain construction, this phase can be understood as a chiral sliding Luttinger liquid with algebraic decay of spin correlations along the chain directions. We calculate the low-energy properties of this gapless chiral spin liquid using the effective field theory and show that they are compatible with the predictions from parton mean-field theories with symmetry-protected line Fermi surfaces. These results may be relevant to the state observed in the kapellasite material.

Renormalization of Coulomb interactions in a system of two-dimensional tilted Dirac fermions

We investigate the effects of long-ranged Coulomb interactions in a tilted Dirac semimetal in two dimensions by using the perturbative renormalization-group method. Depending on the magnitude of the tilting parameter, the undoped system can have either Fermi points (type-I) or Fermi lines (type-II). Previous studies usually performed the renormalization-group transformations by integrating out the modes with large momenta. This is problematic when the Fermi surface is open, like type-II Dirac fermions. In this work, we study the effects of Coulomb interactions, following the spirit of Shankar\cite{Shankar}, by introducing a cutoff in the energy scale around the Fermi surface and integrating out the high-energy modes. For type-I Dirac fermions, our result is consistent with that of the previous work. On the other hand, we find that for type-II Dirac fermions, the magnitude of the tilting parameter increases monotonically with lowering energies. This implies the stability of type-II Dirac fermions in the presence of Coulomb interactions, in contrast with previous results. Furthermore, for type-II Dirac fermions, the velocities in different directions acquire different renormalization even if they have the same bare values. By taking into account the renormalization of the tilting parameter and the velocities due to the Coulomb interactions, we show that while the presence of a charged impurity leads only to charge redistribution around the impurity for type-I Dirac fermions, for type-II Dirac fermions, the impurity charge is completely screened, albeit with a very long screening length. The latter indicates that the temperature dependence of physical observables are essentially determined by the RG equations we derived. We illustrate this by calculating the temperature dependence of the compressibility and specific heat of the interacting tilted Dirac fermions.

Majorana Kramers pairs in Rashba double nanowires with interactions and disorder

We analyze the effects of electron-electron interactions and disorder on a Rashba double-nanowire setup coupled to an s-wave superconductor, which has been recently proposed as a versatile platform to generate Kramers pairs of Majorana bound states in the absence of magnetic fields. We identify the regime of parameters for which these Kramers pairs are stable against interaction and disorder effects. We use bosonization, perturbative renormalization group, and replica techniques to derive the flow equations for various parameters of the model and evaluate the corresponding phase diagram with topological and disorder-dominated phases. We confirm aforementioned results by considering a more microscopic approach which starts from the tunneling Hamiltonian between the three-dimensional s-wave superconductor and the nanowires. We find again that the interaction drives the system into the topological phase and, as the strength of the source term coming from the tunneling Hamiltonian increases, strong electron-electron interactions are required to reach the topological phase.

Renormalization group flows and fixed points for a scalar field in curved space with nonminimal F(ϕ)R coupling

Using covariant methods, we construct and explore the Wetterich equation for a non-minimal coupling $F(\phi)R$ of a quantized scalar field to the Ricci scalar of a prescribed curved space. This includes the often considered non-minimal coupling $\xi \phi^2 R$ as a special case. We consider the truncations without and with scale- and field-dependent wave function renormalization in dimensions between four and two. Thereby the main emphasis is on analytic and numerical solutions of the fixed point equations and the behavior in the vicinity of the corresponding fixed points. We determine the non-minimal coupling in the symmetric and spontaneously broken phases with vanishing and non-vanishing average fields, respectively. Using functional perturbative renormalization group methods, we discuss the leading universal contributions to the RG flow below the upper critical dimension $d=4$.

Phase diagrams for quantum Brownian motion on two-dimensional Bravais lattices

We study quantum Brownian motion (QBM) models for a particle in a dissipative environment coupled to a periodic potential. We review QBM for a particle in a one-dimensional periodic potential and extend the study to that for a particle in two-dimensional (2D) periodic potentials of four Bravais lattice types: square, rectangular, triangular (hexagonal), and centered rectangular. We perform perturbative renormalization group analyses to derive the zero temperature flow diagrams and phase boundaries for a particle in these potentials, and observe localization behavior dependent on the anisotropy of the lattice parameters.

Four-loop critical exponents for the Gross-Neveu-Yukawa models

We study the chiral Ising, the chiral XY and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in $4-\epsilon$ dimensions and compute critical exponents for the Gross-Neveu-Yukawa fixed points to order $\mathcal{O}(\epsilon^4)$. Further, we provide Pad\'e estimates for the correlation length exponent, the boson and fermion anomalous dimension as well as the leading correction to scaling exponent in 2+1 dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with $N=1/4$ and $N=1/2$ 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.

Active-to-absorbing phase transition subjected to the velocity fluctuations in the frozen limit case

The directed bond percolation process is studied in the presence of com- pressible velocity fluctuations with long-range correlations. We discuss a construction of a field theoretic action and a way of obtaining its large scale properties using the perturbative renormalization group. The most interest- ing results for the frozen velocity limit are given.

Minimally subtracted six-loop renormalization of O(n) -symmetric ϕ4 theory and critical exponents

We present the perturbative renormalization group functions of $O(n)$-symmetric $\phi^4$ theory in $4-2\varepsilon$ dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore, we perform a resummation to obtain estimates for critical exponents in three and two dimensions.

Nature of the magnetic phase transition in a Weyl semimetal

We investigate the nature of the magnetic phase transition induced by the short-ranged electron-electron interactions in a Weyl semimetal by using the perturbative renormalization-group method. We find that the critical point associated with the quantum phase transition is characterized by a Gaussian fixed point perturbed by a dangerously irrelevant operator. Although the low-energy and long-distance physics is governed by a free theory, the velocities of the fermionic quasiparticles and the magnetic excitations suffer from nontrivial renormalization effects. In particular, their ratio approaches one, which indicates an emergent Lorentz symmetry at low energies. We further investigate the stability of the fixed point in the presence of weak disorder. We show that while the fixed point is generally stable against weak disorder, among those disorders that are consistent with the emergent chiral symmetry of the clean system, a moderately strong random chemical potential and/or random vector potential may induce a quantum phase transition towards a disorder-dominated phase. We propose a global phase diagram of the Weyl semimetal in the presence of both electron-electron interactions and disorder based on our results.

How nonperturbative is the infrared regime of Landau gauge Yang-Mills correlators?

We study the Landau gauge correlators of Yang-Mills fields for infrared Euclidean momenta in the context of a massive extension of the Faddeev-Popov Lagrangian which, we argue, underlies a variety of continuum approaches. Standard (perturbative) renormalization group techniques with a specific, infrared-safe renormalization scheme produce so-called decoupling and scaling solutions for the ghost and gluon propagators, which correspond to nontrivial infrared fixed points. The decoupling fixed point is infrared stable and weakly coupled, while the scaling fixed point is unstable and generically strongly coupled except for low dimensions $d\to2$. Under the assumption that such a scaling fixed point exists beyond one-loop order, we find that the corresponding ghost and gluon scaling exponents are, respectively, $2\alpha_F=2-d$ and $2\alpha_G=d$ at all orders of perturbation theory in the present renormalization scheme. We discuss the relation between the ghost wave function renormalization, the gluon screening mass, the scale of spectral positivity violation, and the gluon mass parameter. We also show that this scaling solution does not realize the standard Becchi-Rouet-Stora-Tyutin symmetry of the Faddeev-Popov Lagrangian. Finally, we discuss our findings in relation to the results of nonperturbative continuum methods.

Estimating the boundaries of a limit cycle in a 2D dynamical system using renormalization group

While the plausibility of formation of limit cycle has been a well studied topic in context of the Poincare–Bendixson theorem, studies on estimates in regard to the possible size and shape of the limit cycle seem to be scanty in the literature. In this paper we present a pedagogical study of some aspects of the size of this limit cycle using perturbative renormalization group by doing detailed and explicit calculations upto second order for the Selkov model for glycolytic oscillations. This famous model is well known to lead to a limit cycle for certain ranges of values of the parameters involved in the problem. Within the tenets of the approximations made, reasonable agreement with the numerical plots can be achieved.