Functional Renormalization Group Scheme Research Articles

Review of recent developments of the functional renormalization group for systems out of equilibrium

We recapitulate recent developments of the functional renormalization group (FRG) approach to the steady state of systems out of thermal equilibrium. In particular, we discuss second-order truncation schemes which account for the frequency-dependence of the two particle vertex and which incorporate inelastic processes. Our focus is on two different types of one-dimensional fermion chains: (i) infinite, open systems which feature a translation symmetry, and (ii) finite systems coupled to left and right reservoirs. In addition to giving a detailed and unified review of the technical derivation of the FRG schemes, we briefly summarize some of the key physical results. In particular, we compute the non-equilibrium phase diagram and analyze the fate of the Berezinskii–Kosterlitz–Thouless transition in the infinite, open system.Graphic abstract

Second-order functional renormalization group approach to quantum wires out of equilibrium

The functional renormalization group (FRG) provides a flexible tool to study correlations in low-dimensional electronic systems. In this paper, we present a novel FRG approach to the steady-state of quantum wires out of thermal equilibrium. Our method is correct up to second order in the two-particle interaction and accounts for inelastic scattering. We combine semi-analytic solutions of the flow equations with MPI parallelization techniques, which allows us to treat systems of up to 60 lattice sites. The equilibrium limit is well-understood and serves as a benchmark. We compute effective distribution functions, the local density of states, and the steady-state current and demonstrate that all of these quantities depend strongly on the choice of the cutoff employed within the FRG. Non-equilibrium is plagued by the lack of physical arguments in favor of a certain cutoff as well as by the appearance of secular higher-order terms which are only partly included in our approach. This demonstrates the inadequacy of a straightforward second-order FRG scheme to study interacting quantum wires out of equilibrium in the absence of a natural cutoff choice.

Functional renormalization group for fermion lattice models in three dimensions: Application to the Hubbard model on the cubic lattice

The channel-decomposed functional renormalization group (FRG) approach, most recently in the variant of truncated-unity-(TU-)FRG, has so far been used for various two-dimensional model systems. Yet, for many interesting material systems the third spatial dimension is of clear relevance. Therefore FRG schemes working in three spatial dimensions (3D) are definitely on the wishlist. Here we demonstrate that a 3D TUFRG scheme can be set up in straightforward extension of previous 2D codes and gives physically sensible results with affordable numerical effort, both regarding the qualitative as well as the quantitative description. The computed phase diagram of the three-dimensional Hubbard model at half filling or perfect nesting shows a phase transition to a \((\pi,\pi,\pi)\)-ordered antiferromagnetic ground state for repulsive interactions at an energy scale that compares well with other numerical approaches in the literature. Furthermore, the method allowed us to detect a \(d\)-wave pairing and a concurring \((\pi,\pi,0)\) antiferromagnetic ground state in the hole doped Hubbard model.

High-frequency asymptotics of the vertex function: Diagrammatic parametrization and algorithmic implementation

Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet-equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and non-local static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.

Phase transition of the frustrated antiferromagntic J1-J2-J3 spin-1/2 Heisenberg model on a simple cubic lattice

We have comprehensively investigated the frustrated J1-J2-J3 Heisenberg model on a simple cubic lattice. This model allows three regimes of magnetic order, viz., (π, π, π), (0, π, π) and (0, 0, π,), denoted as AF1, AF2, and AF3, respectively. The effects of the interplay of neighboring couplings on the model are studied in the entire temperature range. The zero temperature magnetic properties of this model are discussed utilizing the linear spin wave (LSW) theory, nonlinear spin wave (NLSW) theory, and Green’s function (GF) method. The zero temperature phase diagrams evaluated by the LSW and NLSW methods are illustrated, and are observed to exhibit different parameter boundaries. In certain regions and along the parameter boundaries, the possible phase transformations driven by the parameters are discussed. The results obtained using the LSW, NLSW, and GF methods are compared with those obtained using the series expansion (SE) method, and are observed to be in good agreement when the value of J2 is not close to the parameter boundaries. The ground state energies obtained using the LSW and NLSW methods are close to that obtained using the SE method. At finite temperatures, only the GF method is employed to evaluate the magnetic properties, and the calculated phase diagram is observed to be identical to the classical phase diagram. The results indicate that at the parameter boundaries, a temperature-driven first-order phase transition between AF1 and AF2 may occur along the boundary line. Along the AF1-AF3 and AF2-AF3 boundary lines, AF3 is less stable than AF1 and AF2. Our calculated critical temperature agrees with that obtained using Monte Carlo simulations and pseudofermion functional renormalization group scheme.

Nonperturbative renormalization group treatment of amplitude fluctuations for |φ|4 topological phase transitions

The study of the Berezinskii-Kosterlitz-Thouless transition in two-dimensional $|\varphi|^4$ models can be performed in several representations, and the amplitude-phase (AP) Madelung parametrization is a natural way to study the contribution of density fluctuations to nonuniversal quantities. We introduce a functional renormalization group scheme in AP representation where amplitude fluctuations are integrated first to yield an effective sine-Gordon model with renormalized superfluid stiffness. By a mapping between the lattice XY and continuum $|\varphi|^4$ models, our method applies to both on equal footing. Our approach correctly reproduces the existence of a line of fixed points and of universal thermodynamics and it allows to estimate universal and nonuniversal quantities of the two models, finding good agreement with available Monte Carlo results. The presented approach is flexible enough to treat parameter ranges of experimental relevance.

Adiabatically deformed ensemble: Engineering nonthermal states of matter

We propose a route towards engineering non-thermal states of matter, which show largely unexplored physics. The main idea relies on the adiabatic passage of a thermal ensemble under slow variations of the system Hamiltonian. If the temperature of the initial thermal ensemble is either zero or infinite the ensemble after the passage is a simple thermal one with the same vanishing or infinite temperature. However, for any finite non-zero temperature intriguing non-thermal ensembles can be achieved. We exemplify this in: (a) a single oscillator (b) a dimerized interacting one dimensional chain of spinless fermions, (c) a BCS-type superconductor and (d) the topological Kitaev chain. We solve these models with a combination of methods; either exactly, numerically using the density matrix renormalization group (DMRG) or within an approximate functional renormalization group (FRG) scheme. The designed states show strongly non-thermal behavior in each of the considered models. For example, for the chain of spinless fermions we exemplify how long ranged non-thermal power-law correlations can be stabilized and for the Kitaev chain we elucidate how the non-thermal ensemble can largely alter the transition temperature separating topological and trivial phases.

Competing electronic instabilities of extended Hubbard models on the honeycomb lattice: A functional renormalization group calculation with high-wave-vector resolution

We investigate the quantum many-body instabilities for electrons on the honeycomb lattice at half-filling with extended interactions, motivated by a description of graphene and related materials. We employ a recently developed fermionic functional Renormalization Group scheme which allows for highly resolved calculations of wavevector dependences in the low-energy effective interactions. We encounter the expected anti-ferromagnetic spin density wave for a dominant on-site repulsion between electrons, and charge order with different modulations for dominant pure $n$-th nearest neighbor repulsive interactions. Novel instabilities towards incommensurate charge density waves take place when non-local density interactions among several bond distances are included simultaneously. Moreover, for more realistic Coulomb potentials in graphene including enough non-local terms there is a suppression of charge order due to competition effects between the different charge ordering tendencies, and if the on-site term fails to dominate, the semi-metallic state is rendered stable. The possibility of a topological Mott insulator being the favored tendency for dominating second nearest neighbor interactions is not realized in our results with high momentum resolution.

Universal short-time dynamics: Boundary functional renormalization group for a temperature quench

We present a method to calculate short-time non-equilibrium universal exponents within the functional renormalization-group scheme. As an example, we consider the classical critical dynamics of the relaxational model A after a quench of the temperature of the system and calculate the initial slip exponent which characterizes the non-equilibrium universal short-time behaviour of both the order parameter and correlation functions. The value of this exponent is found to be consistent with the result of a perturbative dimensional expansion and of Monte Carlo simulations in three spatial dimensions.

Functional renormalization group for three-dimensional quantum magnetism

We formulate a pseudofermion functional renormalization group (PFFRG) scheme to address frustrated quantum magnetism in three dimensions. In a scenario where many numerical approaches fail due to sign problem or small system size, three-dimensional (3D) PFFRG allows for a quantitative investigation of the quantum spin problem and its observables. We illustrate 3D PFFRG for the simple cubic ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ quantum Heisenberg antiferromagnet, and benchmark it against other approaches, if available.

Renormalization group flow of the Luttinger-Ward functional: Conserving approximations and application to the Anderson impurity model

We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle irreducible vertex functions, given a cut-off in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to non-conserving approximations introduced in an earlier publication ctuation exchange approximation as well as to nonconserving approximations introducen [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle irreducible functional renormalization group schemes. However, we identify a simple static two-particle irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position.

Functional and local renormalization groups

We discuss the relation between functional renormalization group (FRG) and local renormalization group (LRG), focussing on the two dimensional case as an example. We show that away from criticality the Wess-Zumino action is described by a derivative expansion with coefficients naturally related to RG quantities. We then demonstrate that the Weyl consistency conditions derived in the LRG approach are equivalent to the RG equation for the $c$-function available in the FRG scheme. This allows us to give an explicit FRG representation of the Zamolodchikov-Osborn metric, which in principle can be used for computations.

Two-particle irreducible functional renormalization group schemes—a comparative study

We derive functional renormalization group (fRG) schemes for Fermi systems which are based on the two-particle irreducible approach to the quantum many-body problem. In a first step, the cutoff is introduced in the non-interacting propagator as it is commonly done in fRG based on one-particle irreducible vertex functions. The most natural truncation of the resulting infinite hierarchy of flow equations is shown to be fully equivalent to self-consistent perturbation theory. An earlier suggested alternative truncation strategy is considered as well. In a second step, the cutoff is introduced in the two-particle interaction. Again two truncation procedures are investigated, one of which was derived before. In the latter, the mean-field solution of the many-body problem is considered as the starting point of the renormalization group flow. We compare the performance and the required numerical resources for solving the coupled flow equations for all the approximate schemes by applying them to the problem of the quantum anharmonic oscillator. In a functional integral representation, this model has a formal similarity to the quantum many-body problem. The perspectives for applying the derived two-particle irreducible fRG approaches to zero- and one-dimensional systems of correlated fermions are discussed.

Griffiths wings associated with electronic nematic transition

A spatial xy anisotropy generated by a uniaxial pressure and strain can control a phase transition. In particular, the xy anisotropy couples directly to electronic nematic order and thus is expected to play a crucial role in a system exhibiting nematicity. Here we study the phase diagram of an electronic nematic phase transition by introducing the xy anisotropy. In a mean-field theory, we find that a first order transition occurs near van Hove filling and its phase boundary forms a wing structure, which we term a Griffiths wing. The anisotropy of the electronic system exhibits a discontinuous change by crossing the wing, leading to a meta-nematic transition, i.e., the analog to a meta-magnetic transition observed in ferromagnetic systems such as UGe2 and UCoAl. However, in contrast to ferromagnetic systems, the upper edge of the wing shows a non-monotonic temperature dependence. The Griffiths wing interpolates between a two-dimensional system and a nearly one-dimensional system by changing the xy anisotropy. We also study the effect of nematic order-parameter fluctuations in a functional renormalization-group scheme. We find that the Griffiths wing is very sensitive to the fluctuations and is easily broken into two pieces, leading to a phase diagram topologically different from the mean-field result. We discuss that the Griffiths wing associated with nematicity can be related to various materials such as high-Tc cuprates, bilayer ruthenates, quasi-one-dimensional metals, and cold atom systems.

Criticality of the net-baryon number probability distribution at finite density

We compute the probability distribution P(N) of the net-baryon number at finite temperature and quark-chemical potential, μ, at a physical value of the pion mass in the quark-meson model within the functional renormalization group scheme. For μ/T<1, the model exhibits the chiral crossover transition which belongs to the universality class of the O(4) spin system in three dimensions. We explore the influence of the chiral crossover transition on the properties of the net baryon number probability distribution, P(N). By considering ratios of P(N) to the Skellam function, with the same mean and variance, we unravel the characteristic features of the distribution that are related to O(4) criticality at the chiral crossover transition. We explore the corresponding ratios for data obtained at RHIC by the STAR Collaboration and discuss their implications. We also examine O(4) criticality in the context of binomial and negative-binomial distributions for the net proton number.

Cluster functional renormalization group

Functional renormalization group (FRG) has become a diverse and powerful tool to derive effective low-energy scattering vertices of interacting many-body systems. Starting from a free expansion point of the action, the flow of the RG parameter $\ensuremath{\Lambda}$ allows us to trace the evolution of the effective one- and two-particle vertices towards low energies by taking into account the vertex corrections between all parquet channels in an unbiased fashion. In this work, we generalize the expansion point at which the diagrammatic resummation procedure is initiated from a free UV limit to a cluster product state. We formulate a cluster FRG scheme where the noninteracting building blocks (i.e., decoupled spin clusters) are treated exactly, and the intercluster couplings are addressed via RG. As a benchmark study, we apply our cluster FRG scheme to the spin-$\frac{1}{2}$ bilayer Heisenberg model (BHM) on a square lattice where the neighboring sites in the two layers form the individual two-site clusters. Comparing with existing numerical evidence for the BHM, we obtain reasonable findings for the spin susceptibility, the spin-triplet excitation energy, and quasiparticle weight even in coupling regimes close to antiferromagnetic order. The concept of cluster FRG promises applications to a large class of interacting electron systems.

Two-particle correlations in a functional renormalization group scheme using a dynamical mean-field theory approach

We apply a recently introduced hybridization flow functional renormalization group scheme for Anderson-like impurity models as an impurity solver in a dynamical mean-field theory (DMFT) approach to lattice Hubbard models. We present how this scheme is capable of reproducing metallic and insulating solutions of the lattice model. Our setup also offers a numerically rather inexpensive method to calculate two-particle correlation functions. For the paramagnetic Hubbard model on the Bethe lattice in infinite dimensions we calculate the local two-particle vertex for the metallic and the insulating phase. Then we go to a two-site cluster DMFT scheme for the two-dimensional Hubbard model that includes short-range antiferromagnetic fluctuations and obtain the local and nonlocal two-particle vertex functions. We discuss the rich frequency structures of these vertices and compare with the vertex in the single-site solution.

Strong Quantum Interference in Strongly Disordered Bosonic Insulators

We study the variable-range hopping of bosons in an array of sites with short-range interactions and a large characteristic coordination number. The latter leads to strong quantum interference phenomena yet allows for their analytical study. We develop a functional renormalization-group scheme that repeatedly eliminates high-energy sites properly renormalizing the tunneling between the low-energy ones. Using this approach we determine the temperature and magnetic field dependence of the hopping conductivity and find a large positive magnetoresistance. With increasing magnetic field the behavior of the conductivity crossovers from the Mott's law to an activational behavior with the activation gap proportional to the magnetic field.

Critical scales in anisotropic spin systems from functional renormalization

We apply a recently developed functional renormalization group (fRG) scheme for quantum spin systems to the spin-1/2 antiferromagnetic XXZ model on a two-dimensional square lattice. Based on an auxiliary fermion representation we derive flow equations which allow a resummation of the perturbation series in the spin-spin interactions. Spin susceptibilities are calculated for different values of the anisotropy parameter. The phase transition between planar and axial ordering at the isotropic point is reproduced correctly. The results for the critical scales from the fRG as quantitative measures for the ordering temperatures are in good agreement with the exact solution in the Ising limit. On the easy-plane side, the deviations from critical temperatures obtained with quantum Monte Carlo are larger but still acceptable. However, at the isotropic point the Mermin-Wagner theorem is violated such that a precise description of the behavior in the vicinity of the phase transition is not possible. We discuss possible reasons for these discrepanies.

Effective interactions in multiband systems from constrained summations

In correlated electron materials, the application of many-body techniques for the study of interaction effects or unconventional superconductivity often requires the formulation of an effective low-energy model that contains only the relevant bands near the Fermi level. However, the bands away from the Fermi level are known to renormalize the low-energy interactions substantially. Here we compare different schemes to derive low-energy effective theories for interacting electrons in solids. The frequently used constrained random phase approximation (cRPA) is identified as a particular resummation of higher-order interaction terms that includes important virtual corrections. We then propose an adapted functional renormalization group scheme that includes the cPRA, but also allows one to go beyond the cRPA approximation. We study a simple two-band model in order to explore the differences between the approximations.