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arXiv:cond-mat/0301254 · Physical Review B

Nonperturbative approach to full Mott behavior

Tudor D. Stanescu, Philip Phillips

Published Jun 15, 2004• 38 citations
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Abstract

Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale ${t}^{2}/U)$ is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along $x=\ifmmode\pm\else\textpm\fi{}y$ that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level $x<1/3.$ The sign change is tied to a termination of strong correlation physics in the doped Mott state.

DOI
10.1103/physrevb.69.245104

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arXiv:cond-mat/0104478•Physical review. B, Condensed matter•Nov 28, 2001•17 citations
Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale t2/U){t}^{2}/U)t2/U) is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along x=\ifmmode±\else\textpm\fiyx=\ifmmode\pm\else\textpm\fi{}yx=\ifmmode±\else\textpm\fiy that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level x<1/3.x<1/3.x<1/3. The sign change is tied to a termination of strong correlation physics in the doped Mott state.
Abstract We present a T = 0 K renormalization group (RG) phase diagram for the electronic Hubbard model in two dimensions on the square lattice at half filling. The RG procedure treats quantum fluctuations in the single particle occupation number nonperturbatively via the unitarily decoupling of one electronic state at every RG step. The resulting phase diagram thus possesses the quantum fluctuation energy scale (ω) as one of its axes. A relation is derived between ω and the effective temperature scale upto which gapless, as well as emergent gapped phases can be obtained. We find that the normal and insulating phases of the half-filled Hubbard model correspond, for any on-site repulsion, to a marginal Fermi liquid normal phase and a topologically-ordered gapped Mott insulating liquid respectively. The marginal Fermi liquid is found to arise from singular forward scattering in directions normal to the nested Fermi surface, while singular backscattering events lead to Mott liquid state. The transition between these two phases involves passage through a pseudogapped phase bookended by Fermi surface topology-changing Lifshitz transitions. The pseudogap phase is observed to arise from the electronic differentiation encoded within the nested Fermi surface, and involves the gradual gapping of the Fermi surface from antinodes to nodes via charge and spin excitations that are mutually entangled. We obtain effective Hamiltonians for various phases, as well as wavefunctions for the low-energy many-body eigenstates of the Mott liquid. Benchmarking of the ground-state energy per particle and the double-occupancy fraction for the Mott liquid against existing numerical results yields excellent agreement. Presence of a Néel ordering symmetry-breaking perturbation in the RG leads to an antiferromagnetic spin-ordered charge insulating Mott state. Our results thus offer novel insights on a variety of aspects of the Mott–Hubbard problem, and can be extended to the doped system.
We introduce a slave-fermion formulation in which to study the charge dynamics of the half-filled Hubbard model on the square lattice. In this description, the charge degrees of freedom are represented by fermionic holons and doublons and the Mott-insulating characteristics of the ground state are the consequence of holon–doublon bound-state formation. The bosonic spin degrees of freedom are described by the antiferromagnetic Heisenberg model, yielding long-ranged (Néel) magnetic order at zero temperature. Within this framework and in the self-consistent Born approximation, we perform systematic calculations of the average double occupancy, the electronic density of states, the spectral function and the optical conductivity. Qualitatively, our method reproduces the lower and upper Hubbard bands, the spectral-weight transfer into a coherent quasiparticle band at their lower edges and the renormalisation of the Mott gap, which is associated with holon–doublon binding, due to the interactions of both quasiparticle species with the magnons. The zeros of the Green function at the chemical potential give the Luttinger volume, the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalised hopping parameter and the optical gap is directly related to the Mott gap. Quantitatively, the square-lattice Hubbard model is one of the best-characterised problems in correlated condensed matter and many numerical calculations, all with different strengths and weaknesses, exist with which to benchmark our approach. From the semi-quantitative accuracy of our results for all but the weakest interaction strengths, we conclude that a self-consistent treatment of the spin-fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott–Hubbard insulator. We remark in addition that an analytical approximation with these properties serves a vital function in developing a full understanding of the fundamental physics of the Mott state, both in the antiferromagnetic insulator and at finite temperatures and dopings.
Abstract We present a T = 0 K renormalization group (RG) phase diagram for the hole-doped 2D Hubbard model on the square lattice. The RG method employed is nonperturbative in treating quantum fluctuations of the single-particle occupation number via the unitarily decoupling of one electronic state at every RG step. As a result, the RG phase diagram possesses the quantum fluctuation energy scale (ω) as one of its axes. Using effective Hamiltonians and wavefunctions for the low-energy many-body eigenstates for the doped Mott liquid obtained from the stable fixed point of the RG flows, we demonstrate the collapse of the pseudogap for charge excitations (Mottness) at a quantum critical point (QCP) possessing a nodal non-Fermi liquid with superconducting fluctuations, and spin-pseudogapping near the antinodes. The QCP is characterised using both thermodynamic and quantum information-theoretic measures. d-wave superconducting order is shown to arise from this quantum critical state of matter. The pseudogap phase possesses a variety of fluctuations that lead to several symmetry-broken phases at low-energies. Benchmarking of the ground state energy per particle and the double-occupancy fraction obtained from a finite-size scaling analysis against existing numerical results yields excellent agreement. We present detailed insight into the T = 0 origin of several experimentally observed findings in the cuprates, including Homes law and Planckian dissipation. We also establish that the heirarchy of temperature scales for the pseudogap (T PG), onset temperature for pairing (T ons), formation of the Mott liquid (T ML) and superconductivity (T C) obtained from our analysis is quantitatively consistent with that observed experimentally for some members of the cuprates. Our results offer insight on the ubiquitous origin of superconductivity in doped Mott insulating states, and pave the way towards a systematic search for higher superconducting transition temperatures in such systems.
Since the discovery of half-filled cuprate to be a Mott insulator, the excitation spectra above the chemical potential for the unoccupied states has attracted much research attention. There were many theoretical works using different numerical techniques to study this problem, but many have reached different conclusions. One of the reasons is the lack of very detailed high-resolution experimental results for the theories to be compared with. Recently, the scanning tunneling spectroscopy [P. Cai et al., Nat. Phys. 12, 1047 (2016); C. Ye et al., Nat. Commun. 4, 1365 (2013)] on lightly doped Mott insulator with an antiferromagnetic order found the presence of in-gap states with energy of order half an eV above the chemical potential. The measured spectral properties with doping are not quite consistent with earlier theoretical works. Although the experiment has disorder and localization effect, but for the energy scale we will study here, a model without disorder is sufficed to illustrate the underlying physics. We perform a diagonalization method on top of the variational Monte Carlo calculation to study the evolution of antiferromagnetic Mott state with doped hole concentration in the Hubbard model. Our results found in-gap states that behave similarly with ones reported by STS. These in-gap states acquire a substantial amount of dynamical spectral weight transferred from the upper Hubbard band. The in-gap states move toward chemical potential with increasing spectral weight as doping increases. Our result also provides information about the energy scale of these in-gap states in relation with the Coulomb coupling strength UUU.
The Fermi liquid theory is a central concept in modern condensed matter physics used to describe conventional metals. A state of this universality class consists of well-defined quasiparticles, which occupy a finite number of states within a Fermi volume defined in momentum space. According to the Luttinger’s theorem, such a state encloses a Fermi volume proportional to the electron density modulo the filled bands if no symmetry is broken. In the past decades strong deviations from Fermi liquid theory have been observed in the scaling behavior of thermodynamic observables in different materials, among which are for instance cuprates and iron pnictides. This thesis is about two prototypical systems that can not be described by Fermi liquid theory. The first part of this work investigates two-dimensional effective models, which are in a Mott insulating state at half-filling and have a finite conductivity when doped with holes. In such a Mott insulating state, the charge carriers are strongly localized due to the Coulomb repulsion. Hence, strong correlations are assumed to have a strong impact on the formation of the ground state, also in the regime of small hole concentration. In a first project we have used a so-called spinon dopon mean-field theory to represent the two-dimensional Fermi-Hubbard model with strong on-site repulsive interaction in effective degrees of freedom, in which the holes can be embedded into a quantum spin liquid. The corresponding SU(2) invariant ground state belongs to the class of fractionalized Fermi liquids. In a second project we investigate a quantum dimer model, an effective model based on a Hilbert space spanned by short range singlets and bound states of holes and spins. The focus here is on the calculation of the hole-part of the electron spectral function by using exact diagonalization and its comparison with two analytic methods, a diagrammatic computation based on the Bethe-Salpeter equation and a so-called two-mode approximation. The electron spectral function shows a similar analytic form in momentum space between nodal and antinodal point when compared to results from photoemission spectroscopy experiments on cuprates. Furthermore, in a subsequent work we calculate the exact ground state of the quantum dimer model along a certain parameter line. In order to analyze the behavior of the elctron spectral function when increasing the density of holes, we investigate the Fermi-Hubbard model in a current project using a dynamical mean-field approach. To solve the 4-site cluster impurity problem, we use a numerical renormalization group approach. However, numerical limitations force us to restrict the analysis to spin-polarized baths. According to the spectral data, the system is similar to the SU(2) invariant case at half-filling in a Mott insulating state and posseses a momentum-selective energy gap at finite doping. The topology of the Fermi surface shows a Lifshitz transition when increasing the hole concentration. Here, the curvature of the Fermi surface changes from electron- to hole-like. For comparison we apply the dynamical mean-field theory also to the quantum dimer model and observe that the electron spectral functions at finite doping are qualitatively similar to that of the two-dimensional Fermi-Hubbard model. The second half of the work is about Tomonaga-Luttinger liquid theory, which is used to describe the low-energy effective degrees of freedom of one-dimensional systems. Here, we first discuss a conceptional extension of the operator-based bosonization theory for one-dimensional systems. This extension is especially suited for inhomogeneous one-dimensional systems. First, we investigate a one-dimensional system with a local interation potential and compute an exact solution of the single particle propagator at T = 0. The critical exponent of the single particle propagator has an unconventional form as a function of the microscopic Tomonaga-Luttinger parameters, which is not covered by the original Luttinger paradigm postulated by F. Duncan M. Haldane. In a second project on one-dimensional systems, we study the impact of scattering processes among bosonic low-energy excitations on the thermalization process. Such scattering processes are irrelevant on large length scales, however strongly affect the dynamics. In our analytic analysis we focus on a experimental setup, where a one-dimensional Bose gas is instantaneously splitted in two identical, however strongly correlated, halves of one-dimensional electronic systems. In the following, the corresponding non-equilibrium state runs through multiple regimes in time, such as a metastable prethermalization regime. However, above a certain threshold in time such scattering processes cause an effective thermalization of the system. In order to demonstrate this, we compute the kinetic equation in the Keldysh field integral formalism from a diagrammatic expansion based on a self-consistent Born approximation.
We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $(U\ensuremath{\gg}t)$ and weak-coupling $(U\ensuremath{\ll}t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\stackrel{\ensuremath{\rightarrow}}{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of 1/41/41/4 as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity [C(T,U)][C(T,U)][C(T,U)] for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than 11%.1 In addition, we find that in 2D, the C(T,U)C(T,U)C(T,U) curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as 3He,{}^{3}\mathrm{He},3He, UBe3,{\mathrm{UBe}}_{3},UBe3​, and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.