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arXiv:2304.08623 · Physical review. B./Physical review. B

Location and thermal evolution of the pseudogap due to spin fluctuations

Mengxing Ye, Zhentao Wang, Rafael M. Fernandes, Andrey V. Chubukov

Published Sep 26, 2023• 8 citations
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Abstract

We study pseudogap behavior in a metal near a spin density wave (SDW) instability due to thermal magnetic fluctuations. We consider the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model on a square lattice at a finite doping, at intermediate coupling strength, and analyze the thermal evolution of the electron spectral function between a SDW ordered state at low temperatures and a normal Fermi liquid at high temperatures. We argue that for proper description of the pseudogap one needs to sum up infinite series of diagrams for both the fermionic self-energy and the SDW order parameter in the SDW state or the magnetic correlation length in the paramagnetic state. We use the eikonal approach to sum up an infinite series of diagrammatic contributions from thermal fluctuations. Earlier studies found that in the SDW state the spectral function ${A}_{\mathbit{k}}(\ensuremath{\omega})$ of a hot fermion at a finite $T$ is exponentially small below the energy scale $\mathrm{\ensuremath{\Delta}}(T)$, which scales with SDW order and vanishes at the ordering temperature ${T}_{N}$, and has a hump at a larger frequency ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{PG}}$, comparable to the zero-temperature SDW gap $\mathrm{\ensuremath{\Delta}}(T=0)$. We argue that the hump, which we associate with the pseudogap, survives in some $T$ range above ${T}_{N}$. We show that this range is split by regions of strong and weak pseudogap behavior. In the first region, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{PG}}$ is weakly temperature dependent, despite the fact that it comes from thermal fluctuations. Such a behavior has been seen in numerical studies of the Hubbard model. We show that to obtain it one needs to go beyond the one-loop approximation and sum up the infinite series of diagrams. In the second regime, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{PG}}$ decreases with increasing $T$ and eventually vanishes. We further argue that a magnetic pseudogap at a finite $T$ emerges only if the ground state is magnetically ordered. We present the phase diagram and apply the results to high-${T}_{c}$ cuprates.

DOI
10.1103/physrevb.108.115156

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We calculate the fermionic spectral function ${A}_{k}(\ensuremath{\omega})$ in the spiral spin density wave (SDW) state of the Hubbard model on a quasi-2D triangular lattice at small but finite temperature TTT. The spiral SDW order $\mathrm{\ensuremath{\Delta}}(T)$ develops below T=TNT={T}_{N}T=TN​ and has momentum $\mathbf{K}=(4\ensuremath{\pi}/3,0)$. We pay special attention to fermions with momenta k\mathbf{k}k, for which k\mathbf{k}k and k+K\mathbf{k}+\mathbf{K}k+K are close to Fermi surface in the absence of SDW. At the mean-field level, ${A}_{k}(\ensuremath{\omega})$ for such fermions has peaks at $\ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}\mathrm{\ensuremath{\Delta}}(T)$ at T<TNT<{T}_{N}T<TN​ and displays a conventional Fermi liquid behavior at T>TNT>{T}_{N}T>TN​. We show that this behavior changes qualitatively beyond mean field due to singular self-energy contributions from thermal fluctuations in a quasi-2D system. We use a nonperturbative eikonal approach and sum up infinite series of thermal self-energy terms. We show that ${A}_{k}(\ensuremath{\omega})$ shows peak/dip/hump features at T<TNT<{T}_{N}T<TN​, with the peak position at $\mathrm{\ensuremath{\Delta}}(T)$ and hump position at $\mathrm{\ensuremath{\Delta}}(T=0)$. Above TN{T}_{N}TN​, the hump survives up to T=Tp>TNT={T}_{p}>{T}_{N}T=Tp​>TN​, and in between TN{T}_{N}TN​ and Tp{T}_{p}Tp​ the spectral function displays the pseudogap behavior. We show that the difference between Tp{T}_{p}Tp​ and TN{T}_{N}TN​ is controlled by the ratio of in-plane and out-of-plane static spin susceptibilities, which determines the combinatoric factors in the diagrammatic series for the self-energy. For certain values of this ratio, Tp=TN{T}_{p}={T}_{N}Tp​=TN​, i.e., the pseudogap region collapses. In this last case, thermal fluctuations are logarithmically singular, yet they do not give rise to pseudogap behavior. Our computational method can be used to study pseudogap physics due to thermal fluctuations in other systems.
It is generally believed that in a 2D metal, whose ground state is antiferromagnetically ordered with Q=(π,π){\bf Q} = (π,π)Q=(π,π), thermal (static) magnetic fluctuations give rise to precursor behavior above TNT_NTN​, in which the spectral function of a hot fermion (the one for which k{\bf k}k and k+Q{\bf k} + {\bf Q}k+Q are Fermi surface points) contains two peaks, separated by roughly the same energy as in the antiferromagnetically ordered state. The two peaks persist in some range of T>TNT >T_NT>TN​ and eventually merge into a single peak at zero frequency. This behavior is obtained theoretically by departing from free fermions in a paramagnet and evaluating the dressed fermionic Green's function by summing up infinite series of diagrams with contributions from thermal magnetic fluctuations. We show, following [Y.M. Vilk and A.-M. S. Tremblay, J. Phys. I France 7{\bf 7}7 1309 (1997)] that keeping vertex renormalization diagrams in these series is crucial as other terms only broaden the spectral function of a hot fermion, but do not shift its maximum away from zero frequency. As the consequence, the magnetic pseudogap should be treated as an input for theories that neglect vertex corrections, like, e.g., Eliashberg theory for magnetically-mediated superconductivity. We also analyze the potential pseudogap behavior at T=0T=0T=0. We argue that it may exist, but only at a finite correlation length, and not as a precursor to antiferromagnetism.
We investigate the doped two-dimensional Hubbard model at finite temperature using controlled diagrammatic Monte Carlo calculations allowing for the computation of spectral properties in the infinite-size limit and, crucially, with arbitrary momentum resolution. We show that three distinct regimes are found as a function of doping and interaction strength, corresponding to a weakly correlated metal with properties close to those of the non-interacting system, a correlated metal with strong interaction effects including a reshaping of the Fermi surface, and a pseudogap regime at low doping in which quasiparticle excitations are selectively destroyed near the antinodal regions of momentum space. We study the physical mechanism leading to the pseudogap and show that it forms both at weak coupling when the magnetic correlation length is large and at strong coupling when it is shorter. In both cases, we show that spin-fluctuation theory can be modified in order to account for the behavior of the non-local component of the self-energy. We discuss the fate of the pseudogap as temperature goes to zero and show that, remarkably, this regime extrapolates precisely to the ordered stripe phase found by ground-state methods. This handshake between finite temperature and ground-state results significantly advances the elaboration of a comprehensive picture of the physics of the doped Hubbard model.
Exact diagrammatic quantum Monte Carlo (DiagMC) results for the nearest-neighbor Hubbard model motivate a closer study of the pseudogap. Using the improved two-particle self-consistent approach (TPSC+), we analyze Energy (EDC) and Momentum Distribution Curves (MDC) simultaneously. We show that Fermi-liquid terminology breaks down in the pseudogap regime, requiring a distinction between true and false Fermi surfaces and zero-energy quasiparticle (ZEQ) lines. A false Fermi surface has a momentum-space spectral maximum but a frequency-space depression at zero energy, while a false ZEQ line violates the standard quasiparticle condition ∂Σ′(k,ω)/∂ω∣ω=0<0\partial Σ'(\mathbf{k},ω)/\partial ω|_{ω=0}<0∂Σ′(k,ω)/∂ω∣ω=0​<0.The pseudogap is driven by critical thermal spin fluctuations, which occur in two dimensions because of the Mermin-Wagner theorem. Commensurate fluctuations first open an antinodal pseudogap, leaving true Fermi arcs. With decreasing temperature, the Fermi surface evolves into hole- and electron-like false Fermi surfaces. Incommensurate fluctuations generate hot spots near the diagonal, where the pseudogap persists to the quantum critical point (QCP), whereas at kAN\mathbf{k}_{AN}kAN​ it disappears before the QCP doping. There, the spectrum has two precursor antiferromagnetic (AFM) bands, both in the unoccupied (ω>0ω>0ω>0) region. We benchmark TPSC+ against DiagMC results for the Matsubara spectral proxy −Im[G(k,iπT)]/π-\mathrm{Im}[\mathcal{G}(\mathbf{k},iπT)]/π−Im[G(k,iπT)]/π. TPSC+ underestimates pseudogap suppression in the strong-interaction regime, reproducing DiagMC behavior at lower temperatures or doping. This proxy is equivalent to the spectral function with thermal broadening η=πTη=πTη=πT, which obscures features when πTπTπT is not the smallest energy scale. Using A(k,0)A(\mathbf{k},0)A(k,0), we find that hole-like Fermi surfaces emerge at any interaction strength at low temperature, even at weak coupling.
Qualitative changes in thermodynamic and single-particle properties characterize the transition between the pseudogapped electronic liquid and the Fermi liquid. Recent cold-atom experiments on a simulator of the Hubbard model with nearest-neighbor hoppings [L. H. Kendrick , .] showed that the isothermal compressibility κ ( δ ) has a maximum as a function of doping δ . Here, we use the two-particle self-consistent plus approach to explain these experiments and connect the maximum in κ ( δ ) to the transformation of the single-particle spectrum from the pseudogapped to the metallic regime. This elucidates the nature of the pseudogap. Specifically, the maximum in κ ( δ ) practically coincides with the doping at which the precursor of the lower ( π , π ) spin-density-wave (SDW) band at the antinodal point crosses the zero-frequency ω = 0 . The Knight shift χ sp ( 0 , 0 ) ( δ ) , as a function of doping, should also have a maximum. In addition, we demonstrate that two-particle self-consistent plus correctly predicts a maximum in the temperature dependence of the Knight shift χ sp ( 0 , 0 ) ( T ) , consistent with recent ultracold-atom experiments [T. Chalopin , .]. The maxima in both quantities should exist, at sufficiently low temperatures ( T ), in both the intermediate U ≈ U Mott and weak U < U Mott interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at δ max ( T ) can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at δ = δ max . We predict that, at low T , the multiple peaks in the spin susceptibility in the incommensurate case lead to more than two SDW precursor peaks in the spectral function and density of states. By allowing access to parameter regimes relevant to cuprates—including further-neighbor hopping ( t ′ , t ′′ ) and low temperatures, our work provides a high-impact tool for further studies by the broader community.
We generalize the dynamical-mean field (DMFT) approximation by including into the DMFT equations some length scale $\ensuremath{\xi}$ via a momentum dependent external self-energy ${\ensuremath{\Sigma}}_{\mathbf{k}}$. This external self-energy describes nonlocal dynamical correlations induced by the short-ranged collective spin density wave--like antiferromagnetic spin (or the charge density wave--like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with a finite correlation length. This generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with the nearest and the next nearest neighbor hopping. The effective single impurity problem in this generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ is solved by the numerical renormalization group. Both types of the strongly correlated metals, namely: (i) The doped Mott insulator and (ii) the case of the bandwidth $W\ensuremath{\lesssim}U$ (UUU---value of the local Coulomb interaction) are considered. The densities of states, the spectral functions, and the angle resolved photoemission spectra calculated within the $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ show a pseudogap formation near the Fermi level of the quasiparticle band.