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arXiv:1905.11412 · Physical review. B./Physical review. B

Hubbard model on a triangular lattice: Pseudogap due to spin density wave fluctuations

Mengxing Ye, Andrey V. Chubukov

Published Jul 30, 2019• 13 citations
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Abstract

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 $T$. The spiral SDW order $\mathrm{\ensuremath{\Delta}}(T)$ develops below $T={T}_{N}$ and has momentum $\mathbf{K}=(4\ensuremath{\pi}/3,0)$. We pay special attention to fermions with momenta $\mathbf{k}$, for which $\mathbf{k}$ and $\mathbf{k}+\mathbf{K}$ 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 $T<{T}_{N}$ and displays a conventional Fermi liquid behavior at $T>{T}_{N}$. 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 $T<{T}_{N}$, with the peak position at $\mathrm{\ensuremath{\Delta}}(T)$ and hump position at $\mathrm{\ensuremath{\Delta}}(T=0)$. Above ${T}_{N}$, the hump survives up to $T={T}_{p}>{T}_{N}$, and in between ${T}_{N}$ and ${T}_{p}$ the spectral function displays the pseudogap behavior. We show that the difference between ${T}_{p}$ and ${T}_{N}$ 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, ${T}_{p}={T}_{N}$, 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.

DOI
10.1103/physrevb.100.035135

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We calculate the fermionic spectral function Ak(\ensuremathω){A}_{k}(\ensuremath{\omega})Ak​(\ensuremathω) 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 \ensuremathΔ(T)\mathrm{\ensuremath{\Delta}}(T)\ensuremathΔ(T) develops below T=TNT={T}_{N}T=TN​ and has momentum K=(4\ensuremathπ/3,0)\mathbf{K}=(4\ensuremath{\pi}/3,0)K=(4\ensuremathπ/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, Ak(\ensuremathω){A}_{k}(\ensuremath{\omega})Ak​(\ensuremathω) for such fermions has peaks at \ensuremathω=\ifmmode±\else\textpm\fi\ensuremathΔ(T)\ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}\mathrm{\ensuremath{\Delta}}(T)\ensuremathω=\ifmmode±\else\textpm\fi\ensuremathΔ(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 Ak(\ensuremathω){A}_{k}(\ensuremath{\omega})Ak​(\ensuremathω) shows peak/dip/hump features at T<TNT<{T}_{N}T<TN​, with the peak position at \ensuremathΔ(T)\mathrm{\ensuremath{\Delta}}(T)\ensuremathΔ(T) and hump position at \ensuremathΔ(T=0)\mathrm{\ensuremath{\Delta}}(T=0)\ensuremathΔ(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.
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 TTT is exponentially small below the energy scale $\mathrm{\ensuremath{\Delta}}(T)$, which scales with SDW order and vanishes at the ordering temperature TN{T}_{N}TN​, 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 TTT range above TN{T}_{N}TN​. 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 TTT and eventually vanishes. We further argue that a magnetic pseudogap at a finite TTT emerges only if the ground state is magnetically ordered. We present the phase diagram and apply the results to high-Tc{T}_{c}Tc​ cuprates.
We have studied the pseudogap formation in the single-particle spectra of the half-filling two-dimensional Hubbard model. Using a Green's function with the one-loop self-energy correction of the spin and charge fluctuations, we have numerically calculated the self-energy, the spectral function, and the density of states in the weak-coupling regime at finite temperature. Pseudogap formations have been observed in both the density of states and the spectral function at the Fermi level. The pseudogap in the spectral function is explained by the non-Fermi-liquid-like nature of the self-energy. The anomalous behavior in the self-energy is caused by both the strong antiferromagnetic spin fluctuation and the nesting condition on the non-interacting Fermi surface. In the present approximation, we find a logarithmic singularity in the integrand of the imaginary part of the self-energy. Concerning the energy dependence of the spectral function and the self-energy, two theorems are proved. They give a necessary condition in the self-energy to produce the pseudogap at the Fermi level. The pseudogap in the spectral function is highly momentum dependent on the Fermi surface. It opens initially in the (±π,0)(\pm π,0)(±π,0), (0,±π)(0,\pm π)(0,±π) regions as the normal state pseudogap observed in the high-TcT_cTc​ superconductors and if the interaction is increased, it spreads to other Fermi surface sectors. The anisotropy of the pseudogap is produced by the low-energy enhancement of the spin excitation around Q=(π,π){\bf Q}=(π,π)Q=(π,π) and the flatness of the band dispersion around the saddle point.
In the half-filled one-orbital Hubbard model on a square lattice, we find pseudogap-like features in the form of two-peak structures associated with the momentum-resolved spectral function. These features exist within the temperature window ${T}_{N}\ensuremath{\lesssim}T\ensuremath{\lesssim}{T}^{*}$, where TN{T}_{N}TN​ is the N\'eel temperature and T∗{T}^{*}T∗ is the temperature below which there exists a well-formed dip in the density of state. Inside the window ${T}_{N}\ensuremath{\lesssim}T\ensuremath{\lesssim}{T}^{*}$, the peak-to-peak separation in the two-peak structure of the momentum-resolved spectral function grows on moving away from the point ($\ensuremath{\pi}/2,\ensuremath{\pi}/2$) along the normal state Fermi surface toward $(\ensuremath{\pi},0)$, a behavior with remarkable similarities to what is observed in the ddd-wave state and pseudogap phase of high-Tc{T}_{c}Tc​ cuprates. We unveil these features by using a parallelized cluster-based Monte Carlo method for simulating the magnetic order parameter fields on a superlattice. The method enables us to access the momentum-resolved single-particle spectral function corresponding to a lattice size of $\ensuremath{\approx}240 \ifmmode\times\else\texttimes\fi{}$ 240 with an almost negligible finite-size effect.
We have studied the pseudogap formation in the single-particle spectra of the half-filling two-dimensional Hubbard model. Using a Green's function with the one-loop self-energy correction of the spin and charge fluctuations, we have numerically calculated the self-energy, the spectral function, and the density of states in the weak-coupling regime at finite temperatures. Pseudogap formations have been observed in both the density of states and the spectral function at the Fermi level. The pseudogap in the spectral function is explained by the non-Fermi-liquid-like nature of the self-energy. The anomalous behavior in the self-energy is caused by both the strong antiferromagnetic spin fluctuation and the nesting condition on the non-interacting Fermi surface. In the present approximation, we find a logarithmic singularity in the integrand of the self-energy imaginary part. The pseudogap in the spectral function is highly momentum dependent on the Fermi surface. This anisotropy of the pseudogap is produced by the flatness of the band dispersion around the saddle point rather than the nesting condition on the Fermi level.
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.
In the half-filled one-orbital Hubbard model on a square lattice, we find pseduogap features in the form of two-peak structures associated with the momentum-resolved spectral function, which exists within the temperature window TN≲T≲T∗T_N \lesssim T \lesssim T^*TN​≲T≲T∗. T∗T^*T∗ is the temperature below which there exists a well-formed dip in the density of state. Inside the window TN≲T≲T∗T_N \lesssim T \lesssim T^*TN​≲T≲T∗, the peak-to-peak separation in the two-peak structure of the momentum-resolved spectral function rises on moving away from the point (π/2,π/2π/2, π/2π/2,π/2) along the normal state Fermi surface towards (π,0)(π, 0)(π,0), a behavior remarkably similar to what is observed in the pseudogap phase. We unveil these features by using a parallelized cluster-based Monte-Carlo method for simulating the magnetic order parameter fields on a superlattice, which enables us to access the momentum-resolved single-particle spectral function corresponding to a lattice size of ∼\sim∼ 240 ×\times× 240 with almost negligible finite-size effects.