Tracking the Footprints of Spin Fluctuations: A Multi-Method, Multi-Messenger Study of the Two-Dimensional Hubbard Model
TL;DR: In this article, a comparative study of state-of-the-art quantum many-body methods using the half-filled Hubbard model at weak coupling was performed using several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo) serving as a benchmark.
Abstract: The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state of the art quantum many-body methods Upon cooling into its insulating antiferromagnetic ground-state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate temperature metallic regime and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo) serving as a benchmark By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes Based on this analysis, we explain how quasiparticles can coexist with increasingly long-range antiferromagnetic correlations, and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime We also critically discuss whether imaginary time methods are able to capture the non-Fermi liquid singularities of this fully nested system
Citations
More filters
TL;DR: In this article, the authors study thermodynamic properties of the SU$N$ Fermi-Hubbard model in 2D square lattices and discover simple scaling laws of the energy, the number of on-site pairs, and the kinetic energy with respect to $N$ when the temperature is above the superexchange energy.
Abstract: The authors study thermodynamic properties of the SU($N$) Fermi-Hubbard model in 2D square lattices and discover simple scaling laws of the energy, the number of on-site pairs, and the kinetic energy with respect to $N$ when the temperature is above the superexchange energy. The results establish a limit to the cooling for SU($N$) fermions in 2D, in contrast to the arbitrarily low temperatures that could potentially be achieved in 1D.
28 citations
TL;DR: A comprehensive overview of the different approaches which have been recently developed and applied to identify the dominant twoparticle scattering processes controlling the shape of the one-particle spectral functions and, in some cases, of the physical response of the system is presented in this paper.
Abstract: While calculations and measurements of single-particle spectral properties often offer the most direct route to study correlated electron systems, the underlying physics may remain quite elusive, if information at higher particle levels is not explicitly included. Here, we present a comprehensive overview of the different approaches which have been recently developed and applied to identify the dominant two-particle scattering processes controlling the shape of the one-particle spectral functions and, in some cases, of the physical response of the system. In particular, we will discuss the underlying general idea, the common threads and the specific peculiarities of all the proposed approaches. While all of them rely on a selective analysis of the Schwinger-Dyson (or the Bethe-Salpeter) equation, the methodological differences originate from the specific two-particle vertex functions to be computed and decomposed. Finally, we illustrate the potential strength of these methodologies by means of their applications the two-dimensional Hubbard model, and we provide an outlook over the future perspective and developments of this route for understanding the physics of correlated electrons.
22 citations
TL;DR: In this article, the authors present a comprehensive analysis of collective electronic fluctuations and their effect on single-particle properties of the Hubbard model, based on a standard dual fermion and boson scheme with the interaction truncated at the twoparticle level, and compare various approximations that differ in the set of diagrams (ladder vs exact diagrammatic Monte Carlo), and in the form of the four-point interaction vertex (exact vs partially bosonized).
Abstract: In this work we present a comprehensive analysis of collective electronic fluctuations and their effect on single-particle properties of the Hubbard model Our approach is based on a standard dual fermion and boson scheme with the interaction truncated at the two-particle level Within this framework we compare various approximations that differ in the set of diagrams (ladder vs exact diagrammatic Monte Carlo), and/or in the form of the four-point interaction vertex (exact vs partially bosonized) This allows to evaluate the effect of all components of the four-point vertex function on the electronic self-energy In particular, we observe that contributions that are not accounted for by the partially bosonized approximation for the vertex have only a minor effect on electronic degrees of freedom in a broad range of model parameters In addition, we find that in the regime, where the ladder dual fermion approximation provides an accurate solution of the problem, the leading contribution to the self-energy is given by the longitudinal bosonic modes This can be explained by the fact that contributions of transverse particle-hole and particle-particle modes partially cancel each other Our results justify the applicability of the recently introduced dual triply irreducible local expansion (D-TRILEX) method that represents one of the simplest consistent diagrammatic extensions of the dynamical mean-field theory We find that the self-consistent D-TRILEX approach is reasonably accurate also in challenging regimes of the Hubbard model, even where the dynamical mean-field theory does not provide the optimal local reference point (impurity problem) for the diagrammatic expansion
19 citations
TL;DR: Shinaoka et al. as discussed by the authors implemented a fluctuation exchange (FLEX) approach to spin fluctuations and superconductivity using the ''intermediate representation basis'' (IR) for Matsubara Green functions, which is numerically very efficient and enables them to reach temperatures on the order of ${10}^{\ensuremath{-}4}$ in units of the electronic bandwidth in multiorbital systems.
Abstract: Superconductivity arises mostly at energy and temperature scales that are much smaller than the typical bare electronic energies. Since the computational effort of diagrammatic many-body techniques increases with the number of required Matsubara frequencies and thus with the inverse temperature, phase transitions that occur at low temperatures are typically hard to address numerically. In this work, we implement a fluctuation exchange (FLEX) approach to spin fluctuations and superconductivity using the ``intermediate representation basis'' (IR) [Shinaoka et al., Phys. Rev. B 96, 035147 (2017)] for Matsubara Green functions. This $\mathrm{FLEX}+\mathrm{IR}$ approach is numerically very efficient and enables us to reach temperatures on the order of ${10}^{\ensuremath{-}4}$ in units of the electronic bandwidth in multiorbital systems. After benchmarking the method in the doped repulsive Hubbard model on the square lattice, we study the possibility of spin-fluctuation-mediated superconductivity in the hydrated sodium cobalt material ${\mathrm{Na}}_{x}{\mathrm{CoO}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}{y\mathrm{H}}_{2}\mathrm{O}$ reaching the scale of the experimental transition temperature ${T}_{\mathrm{c}}=4.5\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and below.
14 citations
TL;DR: In this article, the authors developed an algorithm that solves the Bethe-Salpeter equation in O(L^8)$ time with O (L^4) memory, where $L$ grows only logarithmically with inverse temperature, bandwidth, and desired accuracy.
Abstract: The Bethe-Salpeter equation plays a crucial role in understanding the physics of correlated fermions, relating to optical excitations in solids as well as resonances in high-energy physics. Yet, it is notoriously difficult to control numerically, typically requiring an effort that scales polynomially with energy scales and accuracy. This puts many interesting systems out of computational reach. Using the intermediate representation and sparse modelling for two-particle objects on the Matsubara axis, we develop an algorithm that solves the Bethe-Salpeter equation in $O(L^8)$ time with $O(L^4)$ memory, where $L$ grows only logarithmically with inverse temperature, bandwidth, and desired accuracy, This opens the door for computations in hitherto inaccessible regimes. We benchmark the method on the Hubbard atom and on the multi-orbital weak-coupling limit, where we observe the expected exponential convergence to the analytical results. We then showcase the method for a realistic impurity problem.
13 citations
References
More filters
TL;DR: In this paper, it is rigorously proved that at any nonzero temperature, a one- or two-dimensional isotropic spin-S$ Heisenberg model with finite-range exchange interaction can be neither ferromagnetic nor antiferromagnetic.
Abstract: It is rigorously proved that at any nonzero temperature, a one- or two-dimensional isotropic spin-$S$ Heisenberg model with finite-range exchange interaction can be neither ferromagnetic nor antiferromagnetic. The method of proof is capable of excluding a variety of types of ordering in one and two dimensions.
6,236 citations
TL;DR: The dynamical mean field theory of strongly correlated electron systems is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition.
Abstract: We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.
5,230 citations
TL;DR: In this paper, the Hartree-Fock approximation of the correlation problem for the d-and f-bands was applied to a simple, approximate model for the interaction of electrons in narrow energy bands.
Abstract: It is pointed out that one of the main effects of correlation phenomena in d- and f-bands is to give rise to behaviour characteristic of the atomic or Heitler-London model. To investigate this situation a simple, approximate model for the interaction of electrons in narrow energy bands is introduced. The results of applying the Hartree-Fock approximation to this model are examined. Using a Green function technique an approximate solution of the correlation problem for this model is obtained. This solution has the property of reducing to the exact atomic solution in the appropriate limit and to the ordinary uncorrelated band picture in the opposite limit. The condition for ferromagnetism of this solution is discussed. To clarify the physical meaning of the solution a two-electron example is examined.
5,151 citations
TL;DR: The universe itself is thought to have passed through several phase transitions as the high-temperature plasma formed by the big bang cooled to form the world as we know it today as mentioned in this paper.
Abstract: Nature abounds with phase transitions. The boiling and freezing of water are everyday examples of phase transitions, as are more exotic processes such as superconductivity and superfluidity. The universe itself is thought to have passed through several phase transitions as the high-temperature plasma formed by the big bang cooled to form the world as we know it today.
3,749 citations
TL;DR: In this paper, a review of the physics of high-temperature superconductors from the point of view of the doping of a Mott insulator is presented, with the goal of putting the resonating valence bond idea on a more formal footing.
Abstract: This article reviews the physics of high-temperature superconductors from the point of view of the doping of a Mott insulator. The basic electronic structure of cuprates is reviewed, emphasizing the physics of strong correlation and establishing the model of a doped Mott insulator as a starting point. A variety of experiments are discussed, focusing on the region of the phase diagram close to the Mott insulator (the underdoped region) where the behavior is most anomalous. The normal state in this region exhibits pseudogap phenomenon. In contrast, the quasiparticles in the superconducting state are well defined and behave according to theory. This review introduces Anderson's idea of the resonating valence bond and argues that it gives a qualitative account of the data. The importance of phase fluctuations is discussed, leading to a theory of the transition temperature, which is driven by phase fluctuations and the thermal excitation of quasiparticles. However, an argument is made that phase fluctuations can only explain pseudogap phenomenology over a limited temperature range, and some additional physics is needed to explain the onset of singlet formation at very high temperatures. A description of the numerical method of the projected wave function is presented, which turns out to be a very useful technique for implementing the strong correlation constraint and leads to a number of predictions which are in agreement with experiments. The remainder of the paper deals with an analytic treatment of the $t\text{\ensuremath{-}}J$ model, with the goal of putting the resonating valence bond idea on a more formal footing. The slave boson is introduced to enforce the constraint againt double occupation and it is shown that the implementation of this local constraint leads naturally to gauge theories. This review follows the historical order by first examining the U(1) formulation of the gauge theory. Some inadequacies of this formulation for underdoping are discussed, leading to the SU(2) formulation. Here follows a rather thorough discussion of the role of gauge theory in describing the spin-liquid phase of the undoped Mott insulator. The difference between the high-energy gauge group in the formulation of the problem versus the low-energy gauge group, which is an emergent phenomenon, is emphasized. Several possible routes to deconfinement based on different emergent gauge groups are discussed, which leads to the physics of fractionalization and spin-charge separation. Next the extension of the SU(2) formulation to nonzero doping is described with a focus on a part of the mean-field phase diagram called the staggered flux liquid phase. It will be shown that inclusion of the gauge fluctuation provides a reasonable description of the pseudogap phase. It is emphasized that $d$-wave superconductivity can be considered as evolving from a stable U(1) spin liquid. These ideas are applied to the high-${T}_{c}$ cuprates, and their implications for the vortex structure and the phase diagram are discussed. A possible test of the topological structure of the pseudogap phase is described.
3,246 citations