Perturbative Renormalization Group Method Research Articles

Kondo effect in twisted bilayer graphene

The emergence of flat bands in twisted bilayer graphene at the magic angle can be understood in terms of a vanishing Fermi velocity of the Dirac cone. This is associated with van Hove singularities approaching the Fermi energy and becoming higher-order. In the density of states, this is reflected by flanking logarithmic van Hove divergences pinching off the central Dirac cone in energy space. The low-energy pseudogap of the Dirac cone away from the magic angle is replaced by a power-law divergence due to the higher-order van Hove singularity at the magic angle. This plays an important role in the exotic phenomena observed in this material, such as superconductivity and magnetism, by amplifying electronic correlation effects. Here we investigate one such correlation effect---the Kondo effect due to a magnetic impurity embedded in twisted bilayer graphene. We use the Bistritzer-MacDonald model to extract the low-energy density of states of the material as a function of twist angle and study the resulting quantum impurity physics using perturbative and numerical renormalization group methods. Although at zero temperature the impurity is only Kondo screened precisely at the magic angle, we find highly nontrivial behavior at finite temperatures relevant to experiments, due to the complex interplay between Dirac, van Hove, and Kondo physics.

Nested Fermi surfaces and correlated electronic phases in hole-doped semiconductor quantum wells

We demonstrate the existence of novel interaction effects in hole-doped semiconductor quantum wells which are connected to dramatic changes in the Fermi surface geometry occurring upon variation of the doping. We present band structure calculations showing that quantum wells formed in $p$-type cubic semiconductors develop perfectly nested Fermi surfaces at a critical hole density $p\ensuremath{\sim}1/{d}^{2}$ set by the width $d$ of the quantum well. Nesting gives rise to competing superconducting and charge or spin density wave order, which we analyze using the perturbative renormalization group method. The correlated phases may be created or destroyed by tuning the hole density towards or away from the critical density. Our results establish $p$-type semiconductor quantum wells as a platform for novel correlated phases, which may be precisely controlled using electrostatic gating and external magnetic fields.

Scaling or multiscaling: Varieties of universality in a driven nonlinear model.

Physical understanding of how the interplay between symmetries and nonlinear effects can control the scaling and multiscaling properties in a coupled driven system, such as magnetohydrodynamic turbulence or turbulent binary fluid mixtures, remains elusive. To address this generic issue, we construct a conceptual nonlinear hydrodynamic model, parametrized jointly by the nonlinear coefficients, and the spatial scaling of the variances of the advecting stochastic velocity and the stochastic additive driving force, respectively. By using a perturbative one-loop dynamic renormalization group method, we calculate the multiscaling exponents of the suitably defined equal-time structure functions of the dynamical variable. We show that depending upon the control parameters the model can display a variety of universal scaling behaviors ranging from simple scaling to multiscaling.

Renormalization group study of superfluid phase transition: Effect of compressibility.

Dynamic critical behavior in superfluid systems is considered in the presence of external stirring and advecting processes. The latter are generated by means of the Gaussian random velocity ensemble with white-noise character in time variable and self-similar spatial dependence. The main focus of this work is to analyze an effect of compressible modes on the critical behavior. The model is formulated through stochastic Langevin equations, which are then recast into the Janssen-De Dominicis response formalism. Employing the field-theoretic perturbative renormalization group method we analyze large-scale properties of the model. Explicit calculations are performed to the leading one-loop approximation in the double (ɛ,y) expansion scheme, where ɛ is a deviation from the upper critical dimension d_{c}=4 and y describes a scaling property of the velocity ensemble. Altogether five distinct universality classes are expected to be macroscopically observable. In contrast to the incompressible case, we find that compressibility leads to an enhancement and stabilization of nontrivial asymptotic regimes.

One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses.

The de Almeida-Thouless (AT) line is the phase boundary in the temperature-magnetic field plane of an Ising spin glass at which a continuous (i.e., second-order) transition from a paramagnet to a replica-symmetry-breaking (RSB) phase occurs, according to mean-field theory. Here, using field-theoretic perturbative renormalization group methods on the Bray-Roberts reduced Landau-Ginzburg-type theory for a short-range Ising spin glass in space of dimension d, we show that at nonzero magnetic field the nature of the corresponding transition is modified as follows: (a) For d-6 small and positive, with increasing field on the AT line, first, the ordered phase just below the transition becomes the so-called one-step RSB, instead of the full RSB that occurs in mean-field theory; the transition on the AT line remains continuous with a diverging correlation length. Then at a higher field, a tricritical point separates the latter transition from a quasi-first-order one, that is one at which the correlation length does not diverge, and there is a jump in part of the order parameter, but no latent heat. The location of the tricritical point tends to zero as d→6^{+}. (b) For d≤6, we argue that the quasi-first-order transition could persist down to arbitrarily small nonzero fields, with a transition to full RSB still expected at lower temperature. Whenever the quasi-first-order transition occurs, it is at a higher temperature than the AT transition would be for the same field, preempting it as the temperature is lowered. These results may explain the reported absence of a diverging correlation length in the presence of a magnetic field in low-dimensional spin glasses in some simulations and in high-temperature series expansions. We also draw attention to the similarity of the "dynamically frozen" state, which occurs at temperatures just above the quasi-first-order transition, and the "metastate-average state" of the one-step RSB phase, and discuss the issue of the number of pure states in either.

Two-Species Reaction-Diffusion System: the Effect of Long-Range Spreading

We study fluctuation effects in the two-species reaction-diffusion systemA+B→ Ø andA+A→ (Ø,A). In contrast to the usually assumed ordinary short-range diffusion spreading of the reactants we consider anomalous diffusion due to microscopic long-range hops. In order to describe the latter, we employ the Lévy stochastic ensemble. The probability distribution for the Lévy flights decays inddimensions with the distanceraccording to a power-lawr−d−σ. For anomalous diffusion (including Lévy flights) the critical dimensiondc=σdepends on the control parameterσ, 0<σ ≤ 2. The model is studied in terms of the field theoretic approach based on the Feynman diagrammatic technique and perturbative renormalization group method. We demonstrate the ideas behind theBparticle density calculation.

Passive Advection in a Percolation Process: Two-Loop Approximation

The paradigmatic model of the directed percolation process is studied near its second order phase transition between an absorbing and an active state. The model is first expressed in a form of Langevin equation and later rewritten into a field-theoretic formulation. The ensuing response functional is then analyzed employing Feynman diagrammatic technique and perturbative renormalization group method. Percolation process is assumed to occur in external velocity field, which has an additional effect on spreading properties. Kraichnan rapid change ensemble is used for generation of velocity fluctuations. The structure of the fixed points structure is obtained within the two-loop approximation.

Multi-Critical Multi-Field Models: A CFT Approach to the Leading Order

We present some general results for the multi-critical multi-field models in d > 2 recently obtained using conformal field theory (CFT) and Schwinger–Dyson methods at the perturbative level without assuming any symmetry. Results in the leading non trivial order are derived consistently for several conformal data in full agreement with functional perturbative renormalization group (RG) methods. Mechanisms like emergent (possibly approximate) symmetries can be naturally investigated in this framework.

Disordered fermionic quantum critical points

We study the effect of quenched disorder on the semimetal-superconductor quantum phase transition in a model of two-dimensional Dirac semimetal with $N$ flavors of two-component Dirac fermions, using perturbative renormalization group methods at one-loop order in a double epsilon expansion. For $N\geq 2$ we find that the Harris-stable clean critical behavior gives way, past a certain critical disorder strength, to a finite-disorder critical point characterized by non-Gaussian critical exponents, a noninteger dynamic critical exponent $z>1$, and a finite Yukawa coupling between Dirac fermions and bosonic order parameter fluctuations. For $N\geq 7$ the disordered quantum critical point is described by a renormalization group fixed point of stable-focus type and exhibits oscillatory corrections to scaling.

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.

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.

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.

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 $d=4$.

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.

The nature of three dimensions: Non-local behavior in the three-dimensional (3D) Ising model

In this work, we first reveal the nature of three dimensions: the existence of non-local behavior. Taking the three-dimensional (3D) Ising model as an example, we analyze the transfer matrices as well as the partition function to discuss in details the origin of non-local behavior in three dimensions. We then point out that fail to deal with non-local behavior is one of the main disadvantages of several theoretical approaches (such as conventional low-temperature expansions, conventional high-temperature expansions, perturbative renormalization group and Monte Carlo methods, etc), all of which take local environments into account only. This work provides a deep understanding on the natural difference between 3D bulk materials and 2D materials, and shows that how to treat the non-local behavior and the interaction between spins along the third dimension is a key point for solving exactly the problems of the 3D systems with the nearest neighboring interactions. Finally, we will briefly review recent advance in the 3D Ising model and related topics.

Renormalization group analysis for an asymmetric simple exclusion process.

A perturbative renormalization group method is used to obtain steady-state density profiles of a totally asymmetric simple exclusion process with particle adsorption and evaporation. This method allows us to obtain a globally valid solution for the density profile without the asymptotic matching of bulk and boundary layer solutions. In addition, we show a nontrivial scaling of the boundary layer width with the system size close to specific phase boundaries.

Superconductivity in two-dimensional disordered Dirac semimetals

In two-dimensional Dirac semimetals, Cooper pairing instability occurs only when the attractive interaction strength $|u|$ is larger than some critical value $|u_{c}|$ because the density of states vanishes at Dirac points. Disorders enhance the low-energy density of states but meanwhile shorten the lifetime of fermions, which tend to promote and suppress superconductivity, respectively. To determine which of the two competing effects wins, we study the interplay of Cooper pairing interaction and disorder scattering by means of renormalization group method. We consider three types of disorders, including random mass, random gauge potential, and random chemical potential, and show that the first two suppress superconductivity. In particular, the critical BCS coupling $|u_{c}|$ is increased to certain larger value if the system contains only random mass or random gauge potential, which makes the onset of superconductivity more difficult. In the case of random chemical potential, the effective disorder parameter flows to the strong coupling regime, where the perturbation expansion breaks down and cannot provide a clear answer concerning the fate of superconductivity. When different types of disorder coexist in one system, their strength parameters all flow to strong couplings. In the strong coupling regime, the perturbative renormalization group method becomes invalid, and one needs to employ other methods to treat the disorder effects. We perform a simple gap equation analysis of the impact of random chemical potential on superconductivity by using the Abrikosov-Gorkov diagrammatic approach, and also briefly discuss the possible generalization of this approach.

Critical initial-slip scaling for the noisy complex Ginzburg–Landau equation

We employ the perturbative fieldtheoretic renormalization group method to investigate the universal critical behavior near the continuous non-equilibrium phase transition in the complex Ginzburg–Landau equation with additive white noise. This stochastic partial differential describes a remarkably wide range of physical systems: coupled nonlinear oscillators subject to external noise near a Hopf bifurcation instability; spontaneous structure formation in non-equilibrium systems, e.g., in cyclically competing populations; and driven-dissipative Bose–Einstein condensation, realized in open systems on the interface of quantum optics and many-body physics, such as cold atomic gases and exciton-polaritons in pumped semiconductor quantum wells in optical cavities. Our starting point is a noisy, dissipative Gross–Pitaevski or nonlinear Schrödinger equation, or equivalently purely relaxational kinetics originating from a complex-valued Landau–Ginzburg functional, which generalizes the standard equilibrium model A critical dynamics of a non-conserved complex order parameter field. We study the universal critical behavior of this system in the early stages of its relaxation from a Gaussian-weighted fully randomized initial state. In this critical aging regime, time translation invariance is broken, and the dynamics is characterized by the stationary static and dynamic critical exponents, as well as an independent ‘initial-slip’ exponent. We show that to first order in the dimensional expansion about the upper critical dimension, this initial-slip exponent in the complex Ginzburg–Landau equation is identical to its equilibrium model A counterpart. We furthermore employ the renormalization group flow equations as well as construct a suitable complex spherical model extension to argue that this conclusion likely remains true to all orders in the perturbation expansion.

Spin-triplet superconductivity in a weak-coupling Hubbard model for the quasi-one-dimensional compoundLi0.9Mo6O17

The purple bronze Li$_{0.9}$Mo$_6$O$_{17}$ is of interest due to its quasi-one-dimensional electronic structure and the possible Luttinger liquid behavior resulting from it. For sufficiently low temperatures, it is a superconductor with a pairing symmetry that is still to be determined. To shed light on this issue, we analyze a minimal Hubbard model for this material involving four molybdenum orbitals per unit cell near quarter filling, using asymptotically exact perturbative renormalization group methods. We find that spin triplet odd-parity superconductivity is the dominant instability. Approximate nesting properties of the two quasi-one-dimensional Fermi surfaces enhance certain second-order processes, which play crucial roles in determining the structure of the pairing gap. Notably, we find that the gap has accidental nodes, i.e. it has more sign changes than required by the point-group symmetry.

Strong interaction effects and criticality of bosons in shaken optical lattices.

We study the quantum phase transitions and identify a tricritical point between a normal Bose superfluid, a superfluid that breaks additional Z(2) Ising symmetry, and a Mott insulator in a recent shaken optical lattice experiment. We show that near the transition between normal and Z(2) symmetry breaking superfluids, bosons can condense into a momentum state with high or even locally maximum kinetic energies due to the interaction effect. We present a general low-energy effective field theory that treats both the superfluid transition and the Ising transition in a uniform framework. Using the perturbative renormalization group method, we find that the critical behavior of the quantum phase transition belongs to a universality class different from that of a dilute Bose gas.