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Physical review. B, Condensed matter

NMR relaxation rate in the Hubbard model in high dimensions

Dai S. Hirashima

Published Feb 1, 1994• 3 citations
Publisher

Abstract

The NMR relaxation rate in the Hubbard model in high dimensions is investigated. First, by an analysis of the response functions, it is shown that, in the infinite-dimensional models, the intersite correlation does not directly affect the local response of the system. A simple approximation to consider the effects of the finite dimensions is then presented, and effects of the antiferromagnetic (AF) correlation on the NMR relaxation rate in the finite-dimensional Hubbard model is studied by treating the interaction by the self-consistent second-order perturbation theory. It is found that at low temperatures the AF correlation seriously affects the temperature dependence of the NMR relaxation rate in contrast to the case in the infinite dimensions.

DOI
10.1103/physrevb.49.3612

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The NMR relaxation rate in the Hubbard model in high dimensions is investigated. First, by an analysis of the response functions, it is shown that, in the infinite-dimensional models, the intersite correlation does not directly affect the local response of the system. A simple approximation to consider the effects of the finite dimensions is then presented, and effects of the antiferromagnetic (AF) correlation on the NMR relaxation rate in the finite-dimensional Hubbard model is studied by treating the interaction by the self-consistent second-order perturbation theory. It is found that at low temperatures the AF correlation seriously affects the temperature dependence of the NMR relaxation rate in contrast to the case in the infinite dimensions.
Dynamical response functions of the Hubbard model on an infinite dimensional lattice (D=\ensuremath{\infty}) are investigated by the self-consistent second-order perturbation theory with respect to the Coulomb interaction U. The vertex corrections are properly considered so that the conservation laws are satisfied, i.e., the f-sum rule is satisfied. The explicit expression for the f-sum rule is obtained in the case of infinite dimensions. It is found that even in the second-order perturbation theory the electronic states show many-body features at finite temperature and/or frequency regions: the system crosses over from the Fermi-liquid state at the low temperature and low-frequency region to the incoherent regime at high temperatures.
The influence of local disorder on the thermodynamics of interacting electrons is studied within the infinite-dimensional disordered Hubbard model.Using a finite-temperature quantum Monte Carlo method, the averaged local moment and staggered susceptibility are calculated and the magnetic phase diagram at half-filling is constructed.From the averaged compressibility in the paramagnetic and antiferromagnetic phase we determine the metalinsulator transitions of the system.A rich transition scenario is revealed.Quite unexpectedly the disorder is found to stabilize the magnetic order in the strong-coupling limit.Randomness can significantly influence the low-temperature behaviour of interacting systems.In particular, it may lead to new phases which have no analogue in non-random systems [l].In the case of electrons, the simplest lattice model including both interactions and random potentials is the disordered Hubbard model.It is now known that even in the .mean-field.limit d + CQ [2,3] the Hubbard interaction remains dynamical [4] and leads to a highly non-trivial single-site problem with infinitely many coupled quantum degrees of freedom [5].This problem is, in fact, equivalent to an Anderson impurity model complemented by a self-consistency condition [6] and is thus amenable to numerical investigations [7] within a finite-temperature quantum Monte Carlo approach [81.In the absence of disorder this technique was already used by several groups to investigate the magnetic phase diagram [7,9], the Mott-Hubbard transition [lo, 1 1 1 and lately also superconductivity in a two-band version [12] of the Hubbard model in d = C Q ; thereby important new insight was gained.In this letter we wish to concentrate on the combined effect of disorder and interaction.We will study the effect of diagonal disorder on the low-temperature properties of the Hubbard model in d = CQ , in particular the competition between correlation-induced
Strong antiferromagnetic correlations can radically change the behavior of an otherwise paramagnetic system, inducing anomalous effects in the single particle properties and driving the behavior of spin-spin response functions. In order to investigate such physics, we have studied the momentum distribution function of the two-dimensional Hubbard model at low doping, low temperatures and high values of the on-site Coulomb repulsion. The interpretation of the results has greatly benefited from the parallel analysis of the filling, temperature and interaction dependences of the spin-spin correlation function, the antiferromagnetic correlation length and the pole of the spin-spin propagator. On reducing doping or temperature and on increasing the interaction strength, the correlations become stronger and stronger and the Fermi surface develops hole and electron pockets and shows a dual nature (small-large).
Recent developments in density matrix renormalization group/matrix product state techniques allow today for a quantitative computation of dynamical correlations at finite temperature in low-dimensional quantum spin systems. By combining numerical results and field theoretical asymptotic methods, e.g., bosonization, it is now possible to give a full description of the dynamical quantity one is interested in for the range of temperatures experimentally accessible. This enhances of course the possibility of one-to-one comparisons with experiments without having to resort to uncontrolled approximations. Here, the authors focus on the temperature dependence of the NMR spin-lattice relaxation rate 1/T11/{T}_{1}1/T1​ in spin-12\frac{1}{2}21​ chain systems. This quantity is directly related to local, dynamical spin-spin correlation functions. The results agree in the low-temperature regime with bosonization predictions, and show interesting deviations from this behavior at intermediate temperatures.
The effect of electron correlations in the infinite-dimension (d=∞) Hubbard model at finite temperatures has been investigated with the use of a theory which was recently proposed by the present author generalizing the slave-boson functional-integral method. Sublattice magnetizations, double occupancies and band-narrowing factors are computed as functions of the interaction, U, and the temperature, T. Calculated results are exact in the limits of U→0 and U→∞, and in the limits of T=0 and T→∞ they are expected to be very accurate owing to the nature of our mean-field-type theory for infinite-dimension systems. Similarities and differences between the calculated results for d=3 and d=∞ are also discussed.
Anomalies are found in the resistivity \ensuremath{\rho} and NMR rate 1/T1{\mathit{T}}_{1}T1​ of the infinite-dimensional Hubbard model using quantum Monte Carlo calculations and the noncrossing approximation. For temperatures greater than the ``Kondo scale'' T0{\mathit{T}}_{0}T0​, we obtain 1/T1{\mathit{T}}_{1}T1​\ensuremath{\sim}a+bT and \ensuremath{\rho}\ensuremath{\sim}c+dT (a, b, c, d constants). For temperatures T\ensuremath{\ll}T0{\mathit{T}}_{0}T0​ we infer from a saturation of the effective mass of the carriers that the ground state of the system is a Fermi liquid.