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arXiv:0705.3824 · Physical Review B

Field-dependent quasiparticles in the infinite-dimensional Hubbard model

Johannes Bauer, Alex C. Hewson

Published Jul 25, 2007• 29 citations
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Abstract

We present dynamical mean-field theory (DMFT) results for the local spectral densities of the one- and two-particle response functions for the infinite-dimensional Hubbard model in a magnetic field. We look at the different regimes corresponding to half-filling, near half-filling, and well away from half-filling, for intermediate and strong values of the local interaction $U$. The low energy results are analyzed in terms of quasiparticles with field-dependent parameters. These renormalized parameters are determined by two different methods, both based on numerical renormalization group (NRG) calculations, and we find good agreement. Away from half-filling, the quasiparticle weights, ${z}_{\ensuremath{\sigma}}(H)$, differ according to the spin type $\ensuremath{\sigma}=\ensuremath{\uparrow}$ or $\ensuremath{\sigma}=\ensuremath{\downarrow}$. Using the renormalized parameters, we show that DMFT-NRG results for the local longitudinal and transverse dynamic spin susceptibilities in an arbitrary field can be understood in terms of repeated scattering of these quasiparticles. We also check Luttinger's theorem for the Hubbard model and find it to be satisfied in all parameter regimes and for all values of the magnetic field.

DOI
10.1103/physrevb.76.035118

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We present dynamical mean-field theory (DMFT) results for the local spectral densities of the one- and two-particle response functions for the infinite-dimensional Hubbard model in a magnetic field. We look at the different regimes corresponding to half-filling, near half-filling, and well away from half-filling, for intermediate and strong values of the local interaction UUU. The low energy results are analyzed in terms of quasiparticles with field-dependent parameters. These renormalized parameters are determined by two different methods, both based on numerical renormalization group (NRG) calculations, and we find good agreement. Away from half-filling, the quasiparticle weights, z\ensuremathσ(H){z}_{\ensuremath{\sigma}}(H)z\ensuremathσ​(H), differ according to the spin type \ensuremathσ=\ensuremath↑\ensuremath{\sigma}=\ensuremath{\uparrow}\ensuremathσ=\ensuremath↑ or \ensuremathσ=\ensuremath↓\ensuremath{\sigma}=\ensuremath{\downarrow}\ensuremathσ=\ensuremath↓. Using the renormalized parameters, we show that DMFT-NRG results for the local longitudinal and transverse dynamic spin susceptibilities in an arbitrary field can be understood in terms of repeated scattering of these quasiparticles. We also check Luttinger's theorem for the Hubbard model and find it to be satisfied in all parameter regimes and for all values of the magnetic field.
We calculate the renormalized parameters for the quasiparticles and their interactions for the Hubbard model in the paramagnetic phase as deduced from the low-energy Fermi-liquid fixed point using the results of a numerical renormalization-group calculation (NRG) and dynamical mean-field theory (DMFT). Even in the low-density limit there is significant renormalization of the local quasiparticle interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$, in agreement with estimates based on the two-particle scattering theory of J. Kanamori [Prog. Theor. Phys. 30, 275 (1963)]. On the approach to the Mott transition we find a finite ratio for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$, where $2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$ is the renormalized bandwidth, which is independent of whether the transition is approached by increasing the on-site interaction UUU or on increasing the density to half filling. The leading ${\ensuremath{\omega}}^{2}$ term in the self-energy and the local dynamical spin and charge susceptibilities are calculated within the renormalized perturbation theory (RPT) and compared with the results calculated directly from the NRG-DMFT. We also suggest, more generally from the DMFT, how an approximate expression for the $\mathbf{q},\ensuremath{\omega}$ spin susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ can be derived from repeated quasiparticle scattering with a local renormalized scattering vertex.
We extend the renormalized quasiparticle description of the symmetric Anderson model in a magnetic field HHH, developed in earlier work, to the nonsymmetric model. The renormalized parameters are deduced from the low energy numerical renormalization group (NRG) fixed point for arbitrary field values. We find quasiparticle resonance widths, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}}_{\ensuremath{\sigma}}(H)$, which depend in general on the spin $\ensuremath{\sigma}$ as well as HHH. The low temperature static properties can be expressed completely in terms of these parameters, which can also be used as inputs for a renormalized perturbation theory. We show that taking into account repeated quasiparticle scattering gives results for the longitudinal and transverse dynamic spin susceptibilities which are in very good agreement with those obtained from direct NRG calculations.
Away from half filling and above its Neel temperature, the three-dimensional Hubbard model is a paramagnetic metal with antiferromagnetic spin fluctuations, where controlled momentum-resolved results have been lacking. Using connected-determinant diagrammatic Monte Carlo, we compute the momentum-resolved self-energy for n = 0.4-0.95, T/t = 0.05-0.33 and U/t = 4-8, and the full-zone spin susceptibility at twelve of these points, focusing on U/t = 4. At U/t = 4 and toward half filling the spin fluctuations become strong: for n = 0.7-0.9 at T/t = 0.14 they enhance the susceptibility peak up to sevenfold and raise the momentum-dependent part of the scattering rate to four to five times its second-order value. They are driven by an effective spin interaction screened to about 2U/3 and nearly uniform over the zone. Yet the peak height is set by the slightly smaller value of this interaction at the peak wave vector, and near magnetic order this small difference changes the height by a large factor. Two-particle self-consistent theory reproduces the peak because its sum rule fixes its single constant interaction near this value, but for n = 0.7-0.9 at T/t = 0.14 its self-energy overestimates the local scattering rate and underestimates the momentum-dependent part. Though strong, the fluctuations are short ranged, with a correlation length of at most about two lattice spacings, and for n = 0.7-0.9 and T/t >= 0.14 the scattering rate near the Fermi surface varies by at most about 1% at fixed band energy, without hot spots. At U/t = 4 dynamical mean-field theory (DMFT) reproduces the local scattering rate and quasiparticle weight within a few percent, and the nonlocal part it misses is approximately a function of the band energy. Strong spin fluctuations thus coexist with nearly isotropic quasiparticles, and the results provide a benchmark for extensions of DMFT and for spin-fluctuation theories.
A systematic study of the effect of magnetic field (h) on the Hubbard model has been carried out at half-filling within dynamical mean field theory. In agreement with previous studies, we find a zero temperature itinerant metamagnetic transition, reflected in the discontinuous changes in magnetization as well as in the hysteresis, from a paramagnetic (PM) metallic state to a polarized quasi-ferromagnetic (QFM) state, at intermediate and large interaction strengths (U). The jump in magnetization vanishes smoothly with decreasing interaction strength, and at a critical U, the transition becomes continuous. The region of 'coexistence' of the PM and QFM solutions in the field-U plane obtained in this study agrees quantitatively with recent numerical renormalization group calculations, thus providing an important benchmark. We highlight the changes in dynamics and quasiparticle weight across this transition. The effective mass increases sharply as the transition is approached, exhibiting a cusp-like singularity at the critical field, and decreases with field monotonically beyond the transition. We conjecture that the first order metamagnetic transition is a result of the competition between Kondo screening, that tries to quench the local moments, and Zeeman coupling, which induces polarization and hence promotes local moment formation. A comparison of our theoretical results with experiments on (3)He indicate that a theory of (3)He based on the half-filled Hubbard model places it in a regime of intermediate interaction strength.
We study the magnetic field-driven metal-to-insulator transition in half-filled Hubbard model on the Bethe lattice, using the dynamical mean-field theory by solving the quantum impurity problem with density-matrix renormalization group algorithm. The method enables us to obtain a high-resolution spectral densities in the presence of a magnetic field. It is found that the Kondo resonance at the Fermi level splits at relatively high magnetic field: the spin-up and -down components move away from the Fermi level and finally form a spin-polarized band insulator. By calculating the magnetization and spin susceptibility, we clarify that an applied magnetic field drives a transition from a paramagnetic metallic phase to a band insulating phase. In the weak interaction regime, the nature of the transition is continuous and captured by the Stoner's description, while in the strong interaction regime the transition is very likely to be metamagnetic, evidenced by the hysteresis curve. Furthermore, we determine the phase boundary by tracking the kink in the magnetic susceptibility, and the steplike change of the entanglement entropy and the entanglement gap closing. Interestingly, the phase boundaries determined from these two different ways are largely consistent with each other.
We contrast the transport properties (dc resistivity, Seebeck coefficient), optical conductivity, spectral functions, dynamical magnetic susceptibility, and the nuclear magnetic resonance 1/T11/{T}_{1}1/T1​ spin-lattice relaxation rate of the repulsive and attractive infinite-dimensional Hubbard models in the paramagnetic phase for a generic band filling. The calculations are performed in a wide temperature interval using the dynamical mean-field theory with the numerical renormalization group as the impurity solver. The attractive case exhibits significantly more complex temperature dependencies which can be explained by the behavior of the half-filled Hubbard model in external magnetic field with constant magnetization, to which the attractive Hubbard model maps through the partial particle-hole transformation. The resistivity is nonmonotonous for the strongly attractive case: it peaks significantly above the Mott-Ioffe-Regel value at a temperature Tmax{T}_{\mathrm{max}}Tmax​ where the quasiparticle band disappears. For both signs of UUU we find particle-hole asymmetry in the self-energy at low energies, but with the opposite kind of excitations having longer lifetime. This leads to a strong suppression of the slope of the Seebeck coefficient in the attractive case rather than an enhancement as in the repulsive case. The spin-lattice relaxation rate in the strongly attractive case has a nonmonotonic temperature dependence, thereby revealing the pairing fluctuations.