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arXiv:1712.05396 · Physical review. B./Physical review. B

t−t′−J model in one dimension using extremely correlated Fermi-liquid theory and time-dependent density matrix renormalization group

Peizhi Mai, Steven R. White, B. Sriram Shastry

Published Jul 6, 2018• 7 citations
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Abstract

We study the one-dimensional $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ 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 quantum fluctuations inherent in one dimension. The momentum distribution function, the spectral function, and the excitation dispersion of these two methods also compare well.

DOI
10.1103/physrevb.98.035108

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We study the one-dimensional t\ensuremath-t\ensuremath′\ensuremath-Jt\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}Jt\ensuremath-t\ensuremath′\ensuremath-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 quantum fluctuations inherent in one dimension. The momentum distribution function, the spectral function, and the excitation dispersion of these two methods also compare well.
In this work, we apply the extremely correlated Fermi liquid (ECFL) theory into studying the ttt-t′t't′-JJJ model in one and two dimensions. In 1-d, we combine ECFL and time-dependent density matrix renormalization group (tDMRG) method. 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 aTomonaga-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 bystrong quantum fluctuations inherent in one dimension. In 2-d, low energy properties of the metallic state of the two-dimensional ttt-JJJ model are presented for second neighbor hopping with hole-doping (t′leq0t'\\leq0t′leq0) and electron-doping (t′>0t'>0t′>0), with various superexchange energy JJJ. The density and temperature dependent spectral properties, resistivity and Hall response are calculated. The spectral features display high thermal sensitivity at modest T for density ngeq0.8n\\geq 0.8ngeq0.8, implying a suppression of the effective Fermi-liquid temperature by two orders of magnitude relative to the bare bandwidth. Flipping the sign of t′t't′ or varying doping delta\\deltadelta changes the curvature of the resistivity versus T curves between convex and concave, consistent to experimental findings.
The one-dimensional (1D) Yang-Gaudin model--an integrable δδδ-function interacting Fermi gas, serves as a paradigm in quantum many-body physics, encompassing phenomena from spin-charge separation to the Luther-Emery liquid. However, a consistent description of the Luther-Emery liquid and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like pairing states in the 1D attractive Fermi gas remains elusive. In this work, we develop a universal Tomonaga-Luttinger liquid (TLL) theory to describe the FFLO state both for the case of weak and strong binding between fermions of opposite spins. We rigorously derive a low-energy effective Hamiltonian using bosonization, revealing the emergence of a two-component TLL: one exhibiting spin-charge coupling in the weak binding regime, and another featuring charge-charge separation in the strong binding regime. For the weak binding regime, we further derive renormalization-group equations for the sine-Gordon term in the spin sector and show that this term undergoes a relevant-irrelevant phase transition driven by the magnetic field. We further analyze the dynamical correlation functions of the FFLO pairing state based on the effective Hamiltonian we constructed. Finally, we discuss experiments using ultracold atoms to elucidate the Luther-Emery liquid behavior and the subtle phenomena of spin-charge coupling and charge-charge separation.
It is shown that the energy (ε)(\varepsilon)(ε) and momentum (k)(k)(k) dependences of the electron self-energy function $ Σ(k, \varepsilon + i0) \equiv Σ^{R}(k, \varepsilon) $ are, $ {\rm Im} Σ^{R} (k, \varepsilon) = -a\varepsilon^{2}|\varepsilon - ξ_{k}|^{- γ(k)} $ where aaa is some constant, ξk=ε(k)−μ,ε(k)ξ_{k} = \varepsilon(k)-μ, \varepsilon(k)ξk​=ε(k)−μ,ε(k) being the band energy, and the critical exponent $ γ(k) $, which depends on the curvature of the Fermi surface at $ k $, satisfies, $ 0 \leq γ(k) \leq 1 $. This leads to a new type of electron liquid, which is the Fermi liquid in the limit of $ \varepsilon, ξ_{k} \rightarrow 0 $ but for $ ξ_{k} \neq 0 $ has a split one-particle spectra as in the Tomonaga-Luttinger liquid.
We report a two-step density-matrix renormalization-group computation of the equal-time single-particle Green's function, the density-density correlations, and the low-frequency spectral weight function of a spinless fermion model in an anisotropic two-dimensional lattice at half filling. We find that at weak couplings the density-density correlations have the universal decay of a Fermi liquid; the spectral weight function displays a sharp quasiparticle peak. But in the vicinity of a quantum critical point, these correlations strongly deviate from a Fermi-liquid prediction and a pseudogap opens in the spectral weight function.
We present detailed results from a recent microscopic theory of extremely correlated Fermi liquids, applied to the ttt-JJJ model in two dimensions, developed recently by Shastry [Phys. Rev. Lett. 107, 056403 (2011); Phys. Rev. B 87, 125124 (2013)]. The second-order theory in the parameter $\ensuremath{\lambda}$, related to the density, is argued to be quantitatively valid in the overdoped regime for $0\ensuremath{\le}n\ensuremath{\lesssim}0.75$, with nnn denoting the particle density. The calculation involves the self-consistent solution of equations for an auxiliary Fermi liquid Green's function and an adaptive spectral weight. We present numerical results at low as well as high TTT, at various low to intermediate densities in the normal phase, using a minimal set of band parameters relevant to the cuprate superconductors. We display the momentum space occupation function mk{m}_{k}mk​, energy dispersion curves locating the peaks of spectral functions, the optical conductivity, relaxation rates for quasiparticles, and the electronic spectral functions on an absolute scale. The line shapes have an asymmetric shape and a broad background that is also seen in experiments, and our calculations validate approximate recent versions of the theory. The results also display the experimentally noted high-energy kink and provide an in-depth understanding of its origin and dependence on band parameters.
We present time-dependent density-matrix renormalization-group (DMRG) results for strongly interacting one-dimensional fermionic systems at finite temperature. When interactions are strong the characteristic spin energy can be greatly suppressed relative to the characteristic charge energy, allowing for the possibility of spin-incoherent Luttinger liquid physics when the temperature is high compared to the spin energy, but small compared to the charge energy. Using DMRG we compute the spectral properties of the $t\text{\ensuremath{-}}J$ model at arbitrary temperatures with respect to both spin and charge energies. We study the full crossover from the Luttinger liquid regime to the spin-incoherent regime, focusing on small J/tJ/tJ/t, where the signatures of spin-incoherent behavior are more manifest. Our method allows us to access the analytically intractable regime where temperature is on the order of the spin energy, $T\ensuremath{\sim}J$. Our results should be helpful in the interpretation of experiments that may be in the crossover regime, $T\ensuremath{\sim}J$, and apply to one-dimensional cold atomic gases where finite-temperature effects are appreciable. The technique may also be used to guide the development of analytical approximations for the crossover regime.