Nonlinear responses in condensed matter are intensively studied because they provide rich information about materials and hold the possibility of being applied in diodes or high-frequency optical devices. While nonlinear responses in noninteracting models have been explored widely, the effect of strong correlations on the nonlinear response is still poorly understood, even though it has been suggested that correlations can enhance the nonlinear response. In this work, we first give an analytical derivation of nonlinear responses using the Green's function methods at finite temperature. Then, we discuss the difficulties of considering dissipation using conventional methods, such as the reduced density matrix method. We reveal that the relaxation time approximation leads to severe limitations when considering optical responses. Finally, we demonstrate that correlation effects, such as the renormalization of the band structure and different lifetime in orbitals or sublattices, can significantly enhance nonlinear responses and can even change the sign of the nonlinear conductivity.
We investigate the Kondo effect in an open quantum system, motivated by recent experiments with ultracold alkaline-earth(-like) atoms. Because of inelastic collisions and the associated atom losses, this system is described by a complex-valued Kondo interaction and provides a non-Hermitian extension of the Kondo problem. We show that the non-Hermiticity induces anomalous reversion of renormalization-group flows which violate the $g$-theorem due to non-unitarity and produce a quantum phase transition unique to non-Hermiticity. Furthermore, we exactly solve the non-Hermitian Kondo Hamiltonian us…
We provide a prescription for constructing Hamiltonians representing the low energy physics of correlated electron materials with dynamically screened Coulomb interactions. The key feature is a renormalization of the hopping and hybridization parameters by the processes that lead to the dynamical screening. The renormalization is shown to be non-negligible for various classes of correlated electron materials. The bandwidth reduction effect is necessary for connecting models to materials behavior and for making quantitative predictions for low-energy properties of solids.
The purpose of this paper is (i) to present a generic and fully functional implementation of the density-matrix renormalization group (DMRG) algorithm, and (ii) to describe how to write additional strongly-correlated electron models and geometries by using templated classes. Besides considering general models and geometries, the code implements Hamiltonian symmetries in a generic way and parallelization over symmetry-related matrix blocks.
We studied the correlated quasi-one-dimensional systems by one-loop renormalization group techniques in weak coupling. In contrast to conventional g-ology approach, we formulate the theory in terms of bilinear currents and obtain all possible interaction vertices. Furthermore, the one-loop renormalization group equations are derived by operator product expansions of these currents at short length scale. It is rather remarkable that these coupled non-linear equations, after appropriate rescaling, can be casted into potential flows. The existence of what we nicknamed "RG potential" provides a na…
The Numerical Renormalization Group method (NRG) has been developed by Wilson in the 1970's to investigate the Kondo problem. The NRG allows the non-perturbative calculation of static and dynamic properties for a variety of impurity models. In addition, this method has been recently generalized to lattice models within the Dynamical Mean Field Theory. This paper gives a brief historical overview of the development of the NRG and discusses its application to the Hubbard model; in particular the results for the Mott metal-insulator transition at low temperatures.
We derive an efficient method for treating renormalization contributions at two-loop level within the functional renormalization group in the one-particle irreducible formalism for fermions. It is based on a decomposition of the two-particle vertex in charge, magnetic and pairing channels. The method treats single-particle and collective excitations in all channels on equal footing, allows for the description of symmetry-breaking and captures collective mode fluctuation physics in the infrared. As a first application, we study the superfluid ground state of the two-dimensional attractive Hubba…
We demonstrate the utility of effective Hamilonians for studying strongly correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical Contractor Renormalization (CORE) method is applied in two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cut-off using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to …
We study the one dimensional t-t'-J model for generic couplings using two complementary theories, the extremely correlated Fermi liquid theory and time-dependent density matrix renormalization group over a broad energy scale. The two methods provide a unique insight into the strong momentum dependence of the self-energy of this prototypical non-Fermi liquid, described at low energies as a Tomonaga-Luttinger liquid. We also demonstrate its intimate relationship to spin-charge separation, i.e. the splitting of Landau quasiparticles of higher dimensions into two constituents, driven by strong qua…
The transmission of correlated electrons through a domain wall in a ferromagnetic one dimensional system is studied theoretically in the limit of a domain wall width smaller or comparable to the electron Fermi wavelength. The domain wall gives rise to both potential and spin dependent scattering of the charge carriers. Using a poor man's renormalization group approach for the electron-electron interactions, we obtain the low temperature behavior of the reflection and transmission coefficients. The results show that the low-temperature conductance is governed by the electron correlations, which…
The introduction of non-Hermiticity has greatly enriched the research field of traditional condensed matter physics, and eventually led to a series of discoveries of exotic phenomena. We investigate the effect of non-Hermitian imaginary hoppings on the attractive Hubbard model. The exact bound-pair solution shows that the electron-electron correlation suppresses the non-Hermiticity, resulting in off-diagonal long-range order (ODLRO) ground state. In a large negative $U $ limit, the ODLRO ground state corresponds to $η$-spin ferromagnetic states. We also study the system with mixed hopping conf…
We present a general frame to extend functional renormalization group (fRG) based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a non-perturbative inclusion of correlations. Introducing auxiliary fermionic fields by means of a Hubbard-Stratonovich transformation, we derive the flow equations for the auxiliary fields and determine the relation to the conventional weak-coupling truncation of the hierarchy of flow equations. As a specific example we consider the dynamic…
Within the time-dependent Gutzwiller approximation (TDGA) applied to Holstein- and SSH-Hubbard models we study the influence of electron correlations on the phonon self-energy. For the local Holstein coupling we find that the phonon frequency renormalization gets weakened upon increasing the onsite interaction $U$ for all momenta. In contrast, correlations can enhance the phonon frequency shift for small wave-vectors in the SSH-Hubbard model. Moreover the TDGA applied to the latter model provides a mechanism which leads to phonon frequency corrections at intermediate momenta due to the couplin…
Emergence of exceptional points in two dimensions is one of the remarkable phenomena in non-Hermitian systems. We here elucidate the impacts of symmetry on the non-Hermitian physics. Specifically, we analyze chiral symmetric correlated systems in equilibrium where the non-Hermitian phenomena are induced by the finite lifetime of quasi-particles. Intriguingly, our analysis reveals that the combination of symmetry and non-Hermiticity results in novel topological degeneracies of energy bands which we call symmetry-protected exceptional rings (SPERs). We observe the emergence of SPERs by analyzing…
Single-band Hubbard model at criticality of the metal-insulator transition is studied using approximations derived from parquet theory. It is argued that only the electron-hole and interaction two-particle channels in the parquet algebra are relevant. A scheme is proposed how to reduce the parquet equations to a manageable form the complexity of which is comparable with single-channel approximations such as the renormalized RPA. The newly derived approximation, however, contains dynamical vertex corrections, remains self-consistent at the two-particle level and allows only for integrable singu…
We introduce a real time version of the functional renormalization group which allows to study correlation effects on nonequilibrium transport through quantum dots. Our method is equally capable to address (i) the relaxation out of a nonequilibrium initial state into a (potentially) steady state driven by a bias voltage and (ii) the dynamics governed by an explicitly time-dependent Hamiltonian. All time regimes from transient to asymptotic can be tackled; the only approximation is the consistent truncation of the flow equations at a given order. As an application we investigate the relaxation …
Several density-matrix renormalization group methods have been proposed to compute the momentum- and frequency-resolved dynamical correlation functions of low-dimensional strongly correlated systems. The most relevant approaches are discussed in this contribution. Their applications in various studies of quasi-one-dimensional strongly correlated systems (spin chains, itinerant electron systems, electron-phonon systems) are reviewed.
Metallic behavior of band-insulator/ Mott-insulator interfaces was observed in artificial perovskite superlattices such as in nanoscale SrTiO3/LaTiO3 multilayers. Applying a semiclassical perspective to the parallel electronic transport we identify two major ingredients relevant for such systems: i) the quantum confinement of the conduction electrons (superlattice modulation) leads to a complex, quasi-two dimensional subband structure with both hole- and electron-like Fermi surfaces. ii) strong electron-electron interaction requires a substantial renormalization of the quasi-particle dispersio…
The low temperature thermodynamics of correlated 1D fermionic models with spin and charge degrees of freedom is obtained by exact diagonalization (ED) of small systems and followed by density matrix renormalization group (DMRG) calculations that target the lowest hundreds of states $\{E(N)\}$ at system size $N$ instead of the ground state. Progressively larger $N$ reaches $T < 0.05t$ in correlated models with electron transfer $t$ between first neighbors and bandwidth $4t$. The size dependence of the many-fermion basis is explicitly included for arbitrary interactions by scaling the partition …
The crossover from fluctuating atomic constituents to a collective state as one lowers temperature or energy is at the heart of the dynamical mean-field theory description of the solid state. We demonstrate that the numerical renormalization group is a viable tool to monitor this crossover in a real-materials setting. The renormalization group flow from high to arbitrarily small energy scales clearly reveals the emergence of the Fermi-liquid state of Sr$_2$RuO$_4$. We find a two-stage screening process, where orbital fluctuations are screened at much higher energies than spin fluctuations, and…
Calculations using the (exact) fermionic functional renormalization group are usually truncated at the second order of the corresponding hierarchy of coupled ordinary differential equations. We present a method for the systematic determination of higher order vertex functions. This method is applied to a study of transport properties of various correlated quantum dot systems. It is shown that for large Coulomb correlations higher order vertex functions cannot be neglected, and a static approximation is insufficient.
We study the weakly interacting Hubbard model on the square lattice using a one-loop renormalization group approach. The transition temperature T_c between the metallic and (nearly) ordered states is found. In the parquet regime, (T_c >> |mu|), the dominant correlations at temperatures below T_c are antiferromagnetic while in the BCS regime (T_c << |mu|) at T_c the d-wave singlet pairing susceptibility is most divergent.
Linear response theory for the density matrix renormalization group (DMRG-LRT) was first presented in terms of the DMRG renormalization projectors [Dorando et al., J. Chem. Phys. 130, 184111 (2009)]. Later, with an understanding of the manifold structure of the matrix product state (MPS) ansatz, which lies at the basis of the DMRG algorithm, a way was found to construct the linear response space for general choices of the MPS gauge in terms of the tangent space vectors [Haegeman et al., Phys. Rev. Lett. 107, 070601 (2011)]. These two developments led to the formulation of the Tamm-Dancoff and …
We derive an efficient and unbiased method for computing order parameters in correlated electron systems with competing instabilities. Charge, magnetic and pairing fluctuations above the energy scale of spontaneous symmetry breaking are taken into account by a functional renormalization group flow, while the formation of order below that scale is treated in mean-field theory. The method captures fluctuation driven instabilities such as $d$-wave superconductivity. As a first application we study the competition between antiferromagnetism and superconductivity in the ground state of the two-dime…
We present an alternative functional renormalization group (fRG) approach to the single-impurity Anderson model at finite temperatures. Starting with the exact self-energy and interaction vertex of a small system ('core') containing a correlated site, we switch on the hybridization with a non-interacting bath in the fRG-flow and calculate spectra of the correlated site. Different truncations of the RG-flow-equations and choices of the core are compared and discussed. Furthermore we calculate the linear conductance and the magnetic susceptibility as functions of temperature and interaction stre…
Functional renormalization group without functional integrals: implementing Hilbert space projections for strongly correlated electrons via Hubbard X-operators
We improve the density-matrix renormalization group (DMRG) evaluation of the Kubo formula for the zero-temperature linear conductance of one-dimensional correlated systems.The dynamical DMRG is used to compute the linear response of a finite system to an applied AC source-drain voltage, then the low-frequency finite-system response is extrapolated to the thermodynamic limit to obtain the DC conductance of an infinite system. The method is demonstrated on the one-dimensional spinless fermion model at half filling. Our method is able to replicate several predictions of the Luttinger liquid theor…
We present a technique for calculating non-equilibrium Green functions for impurity systems with local interactions. We use an analogy to the calculation of response functions in the x-ray problem.The initial state and the final state problems, which correspond to the situations before and after the disturbance (an electric or magnetic field, for example) is suddenly switched on, are solved with the aid of Wilson's momentum shell renormalization group. The method is illustrated by calculating the non-equilibrium dynamics of the ohmic two-state problem.
A new scheme of first-principles computation for strongly correlated electron systems is proposed. This scheme starts from the local-density approximation (LDA) at high-energy band structure, while the low-energy effective Hamiltonian is constructed by a downfolding procedure using combinations of the constrained LDA and the GW method. Thus obtained low-energy Hamiltonian is solved by the path-integral renormalization-group method, where spatial and dynamical fluctuations are fully considered. An application to Sr$_2$VO$_4$ shows that the scheme is powerful in agreement with experimental resul…
We propose new approach for treatment of local and non-local interactions in correlated electronic systems, which uses self-energy and the two-particle irreducible vertices, obtained from (extended) dynamical mean-field theory, as an input of two-particle irreducible functional renormalization-group (2PI-fRG) approach. Using 2PI-fRG approach allows us to treat both, local and non-local interactions. In case of purely local interaction the corresponding equations have similar (although not identical) structure to the earlier developed DMF$^2$RG approach. In a simplest truncation, neglecting sca…
This thesis comprises two parts centered around the functional renormalization-group framework: in the first part, I study the role of symmetries and conservation laws in approximate solutions, while in the second part I analyze Friedel oscillations and transport in Luttinger liquids with impurities. The functional renormalization group (fRG) has been developed as a new computational tool in the theory of interacting Fermi systems. The effective behavior of a given microscopic model is calculated by solving coupled differential flow equations for the Green functions with an energy scale as the…
A numerical algorithm for studying strongly correlated electron systems is proposed. The groundstate wavefunction is projected out after numerical renormalization procedure in the path integral formalism. The wavefunction is expressed from the optimized linear combination of retained states in the truncated Hilbert space with numerically chosen basis. This algorithm does not suffer from the negative sign problem and can be applied to any type of Hamiltonian in any dimension. The efficiency is tested in examples of the Hubbard model where the basis of Slater determinants is numerically optimize…
The projective construction (the slave-particle approach) has played an very important role in understanding strongly correlated systems, such as the emergence of fermions, anyons, and gauge theory in quantum spin liquids and quantum Hall states. Recently, fermionic Projected Entangled Pair States (fPEPS) have been introduced to effciently represent many-body fermionic states. In this paper, we show that the strongly correlated bosonic/fermionic states obtained both from the projective construction and the fPEPS approach can be represented systematically as Grassmann tensor product states. Thi…
Numerical approaches are an important tool to study strongly correlated quantum systems. However, their fragility with respect to rounding errors is not well studied and numerically verified enclosures of the results are not available. In this work we apply interval arithmetic to the well established numerical renormalization group scheme. This extension enables us to provide a numerically verified NRG excitation spectrum.
In this work we investigate charge transport in one-dimensional (1D) insulators via semi-classical and perturbative renormalization group (RG) methods. We consider the problem of electron-electron, electron-phonon and electron-two-level system interactions. We show that non-linear collective modes such as polarons and solitons are reponsible for transport. We find a new excitation in the Mott insulator: the polaronic soliton. We discuss the differences between band and Mott insulators in terms of their spin spectrum and obtain the charge and spin gaps in each one of these systems. We show that…
We study Friedel oscillations and screening effects of the impurity potential in the Hubbard model. Electronic correlations are accounted for by solving the real-space dynamical mean-field theory equations using the continuous time quantum Monte-Carlo simulations at finite temperatures and using a homogeneous self-energy approximation with the numerical renormalization group at zero temperature. We find that in the Fermi liquid phase both the amplitudes of Friedel oscillations and the screening charge decrease with increasing the interaction and follow the behavior of the Fermi liquid renormal…
The large-degeneracy expansion for dilute magnetic alloys is extended to account for conduction electrons interactions. Particular attention is paid to the renormalization of the hybridization vertex which affects the low-energy excitations. As a first example, we calculate the enhanced characteristic energy $k_BT_0$ in the limit of weakly correlated conduction electrons. The metallic regime with strongly correlated electrons is discussed.
We formulate the exact Wilsonian renormalization group for a system of interacting fermions on a lattice. The flow equations for all vertices of the Wilson effective action are expressed in form of the Polchinski equation. We apply this method to the Hubbard model on a square lattice using both zero- and finite- temperature methods. Truncating the effective action at the sixth term in fermionic variables we obtain the one-loop functional renormalization equations for the effective interaction. We find the temperature of the instability Tc^{RG} as function of doping. We calculate furthermore th…
We re-adapt a spectral renormalization method, introduced in nonlinear optics, to solve the Kohn-Sham (KS) equations of density functional theory (DFT), with a focus on functionals based on the strictly-correlated electrons (SCE) regime, which are particularly challenging to converge. Important aspects of the method are: (i) the eigenvalues and the density are computed simultaneously; (ii) it converges using randomized initial guesses; (iii) easy to implement. Using this method we could converge for the first time the Kohn-Sham equations with functionals that include the next leading term in t…
A spin-1/2 magnetic impurity coupled to a one-dimensional correlated electron system have been studied by applying the density renormalization group method. The Kondo temperature is substantially enhanced by strong repulsive interactions in the chain, but changes non-monotonically in the case of electron attraction. The magnetization of the impurity at zero- temperature shows local Fermi liquid behavior.
We present a recently-developed renormalization group scheme, the functional renormalization group (fRG), as a many-particle method suited to account for the two-particle interactions between the electrons in complex quantum dot geometries. A detailed derivation of a truncated set of RG flow equations that requires only basic knowledge of the functional integral approach to many-particle physics is given. Using fRG we study linear-response transport properties of a variety of quantum dots in the zero-temperature limit. For parallel geometries, we observe a generic crossover from mesoscopic to …
Long-range effects induce some interesting behavior and considered as a gateway to understand the non-local behavior in the quantum systems. Especially, the long-range topological models became a platform for the realization of new quasi-particles, which are believed to be potential candidates for the topological qubits. In this work, we consider non-Hermitian Su-Schriffer-Heeger (SSH) model and discuss the interplay of non-Hermiticity and long-range effects. We use the approach of momentum space characterization, critical exponents and curvature renormalization group (CRG) method to understan…
Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems, (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals, (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and non-equilibrium transport through quantum wires…
A self-consistent theory for two-particle fluctuations with renormalized irreducible vertices is proposed. Using the Parquet formalism, we construct the fully antisymmetric full vertex in terms of the two-particle fluctuations in the charge, the spin and the particle-particle channels on an equal footing to satisfy the Pauli principle. The fluctuations are determined self-consistently, which are reflected into the one-particle self-energy via the Schwinger-Dyson equation. We demonstrate the application of the present theory to the impurity Anderson model and the Hubbard model on a square latti…
We present a consistent fusion of functional renormalization group and mean-field theory which explicitly introduces a bosonic field via a Hubbard-Stratonovich transformation at the critical scale, at which the order sets in. We show that a minimal truncation of the flow equations, that neglects order parameter fluctuations, is integrable and fulfills fundamental constraints as the Goldstone theorem and the Ward identity connected with the broken global symmetry. To introduce the bosonic field, we present a technique to factorize the most singular part of the vertex, even when the full depende…
We calculate the electronic structure of Sr2RuO4, treating correlations in the framework of dynamical mean-field theory. The approach successfully reproduces several experimental results and explains the key properties of this material: the anisotropic mass renormalization of quasiparticles and the crossover into an incoherent regime at a low temperature. While the orbital differentiation originates from the proximity of the van Hove singularity, strong correlations are caused by the Hund's coupling. The generality of this mechanism for other correlated materials is pointed out.
According to standard theory the magnetoresistance magnitude in ferromagnetic manganites crucially depends on the electron-phonon coupling strength. We showed that in La0.7Sr0.3MnO3 the phonon renormalization is strong, despite its relatively small magnetoresistance. Here, we report results of a similar inelastic neutron scattering investigation of a closely related compound, La0.8Sr0.2MnO3, where the magnetoresistance is enhanced. We find similar phonon renormalization and dynamic CE-type polaron correlations as in La0.7Sr0.3MnO3. However, quantitative comparison of the results for the two sa…
Mott's metal-insulator transition at an interface due to band bending is studied by the density matrix renormalization group (DMRG). We show that the result can be recovered by a simple modification of the conventional Poisson's equation approach used in semi-conductor heterojunctions. A novel mechanism of colossal electroresistance is proposed, which incorporates the hysteretic behavior of the transition in higher dimensions.
The renormalization of the Wiedemann-Franz (WF) ratio in strongly correlated electron systems is analyzed within the Landau quasiparticle picture. We demonstrate that the WF law is violated: (i) at the quantum critical point, where the effective mass diverges, and (ii) beyond a point of fermion condensation, where the single-particle spectrum $ε(p)$ becomes flat. Results of the analysis are compared with available experimental data.
We analyse the dynamical density fluctuation spectra for cuprates starting from the $t-J$ model in a slave-boson 1/N representation. The results obtained are consistent with diagonalization studies and show novel low-energy structure on the energy scale $J+δt$ due to the correlated motion of holes in a RVB spin liquid. The low-energy response implies an anomalous renormalization of several phonon modes. Here we discuss the renormalization of the highest breathing phonons in La$_{2-x}$Sr$_x$CuO$_4$.
The time-dependent numerical renormalization-group approach (TD-NRG), originally devised for tracking the real-time dynamics of quantum-impurity systems following a single quantum quench, is extended to multiple switching events. This generalization of the TD-NRG encompasses the possibility of periodic switching, allowing for coherent control of strongly correlated systems by an external time-dependent field. To this end, we have embedded the TD-NRG in a hybrid framework that combines the outstanding capabilities of the numerical renormalization group to systematically construct the effective …
We study a model of a covalent band insulator with on-site Coulomb repulsion at half-filling using dynamical mean-field theory. Upon increasing the interaction strength the system undergoes a discontinuous transition from a correlated band insulator to a Mott insulator with hysteretic behavior at low temperatures. Increasing the temperature in the band insulator close to the insulator-insulator transition we find a crossover to a Mott insulator at elevated temperatures. Remarkably, correlations decrease the energy gap in the correlated band insulator. The gap renormalization can be traced to t…
We present a novel scheme for an unbiased and non-perturbative treatment of strongly correlated fermions. The proposed approach combines two of the most successful many-body methods, i.e., the dynamical mean field theory (DMFT) and the functional renormalization group (fRG). Physically, this allows for a systematic inclusion of non-local correlations via the flow equations of the fRG, after the local correlations are taken into account non-perturbatively by the DMFT. To demonstrate the feasibility of the approach, we present numerical results for the two-dimensional Hubbard model at half-filli…
