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arXiv:cond-mat/0102100 · Physical review. B, Condensed matter

Ground-state magnetization for interacting fermions in a disordered potential: Kinetic energy, exchange interaction, and off-diagonal fluctuations

Philippe Jacquod, A. Douglas Stone

Published Nov 13, 2001• 35 citations
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Abstract

We study a model of n interacting fermions in a disordered potential, which is assumed to generate uniformly fluctuating interaction matrix elements. We show that the ground-state magnetization is systematically decreased by off-diagonal fluctuations of the interaction matrix elements. This effect is neglected in the Stoner picture of itinerant ferromagnetism in which the ground-state magnetization is simply determined by the balance between ferromagnetic exchange and kinetic energy, and increasing the interaction strength always favors ferromagnetism. The physical origin of the demagnetizing effect of interaction fluctuations is the larger number K of final states available for interaction-induced scattering in the lower-spin sectors of the Hilbert space. We analyze the energetic role played by these fluctuations in the limits of small and large interactions U. In the small-$U$ limit we use second-order perturbation theory and identify explicitly transitions which are allowed for minimal spin and forbidden for higher spin. These transitions then on average lower the energy of the minimal spin ground state with respect to higher spin; we analytically evaluate the size of this reduction and find it to give a contribution ${\ensuremath{\Delta}}^{s}\ensuremath{\propto}{\mathrm{nU}}^{2}/\ensuremath{\Delta}$ to the spin gap between the two lowest-spin ground states. In terms of an average effective Hamiltonian, these contributions induce a ${\mathrm{nU}}^{2}{S}^{2}/\ensuremath{\Delta}$ term which decreases the strength of the ferromagnetic exchange, thereby delaying the onset of Stoner ferromagnetism, and generate a second, larger S term $\ensuremath{\propto}{S}^{3},$ which results in a saturation of the ground-state spin before full polarization is achieved, in contrast to the Stoner scenario. For large interactions U we amplify on our earlier work [Ph. Jacquod and A. D. Stone, Phys. Rev. Lett. 84, 3938 (2000)] which showed that the broadening of the many-body density of states is proportional to $\sqrt{K}U$ and hence favors minimal spin. Numerical results are presented in both limits. After evaluating the effect of fluctuations, we discuss the competition between fluctuations plus kinetic energy and the exchange energy. We finally present numerical results for specific microscopic models and relate them to our generic model of fluctuations. We discuss the different physical situations to which such models may correspond, the importance of interaction fluctuations, and hence the relevance of our results to these situations and recall an experimental setup which we proposed in an earlier work to measure the importance of interaction fluctuations on the ground-state spin of lateral quantum dots in the Coulomb blockade regime.

DOI
10.1103/physrevb.64.214416

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We study a model of n interacting fermions in a disordered potential, which is assumed to generate uniformly fluctuating interaction matrix elements. We show that the ground-state magnetization is systematically decreased by off-diagonal fluctuations of the interaction matrix elements. This effect is neglected in the Stoner picture of itinerant ferromagnetism in which the ground-state magnetization is simply determined by the balance between ferromagnetic exchange and kinetic energy, and increasing the interaction strength always favors ferromagnetism. The physical origin of the demagnetizing effect of interaction fluctuations is the larger number K of final states available for interaction-induced scattering in the lower-spin sectors of the Hilbert space. We analyze the energetic role played by these fluctuations in the limits of small and large interactions U. In the small-UUU limit we use second-order perturbation theory and identify explicitly transitions which are allowed for minimal spin and forbidden for higher spin. These transitions then on average lower the energy of the minimal spin ground state with respect to higher spin; we analytically evaluate the size of this reduction and find it to give a contribution \ensuremathΔs\ensuremath∝nU2/\ensuremathΔ{\ensuremath{\Delta}}^{s}\ensuremath{\propto}{\mathrm{nU}}^{2}/\ensuremath{\Delta}\ensuremathΔs\ensuremath∝nU2/\ensuremathΔ to the spin gap between the two lowest-spin ground states. In terms of an average effective Hamiltonian, these contributions induce a nU2S2/\ensuremathΔ{\mathrm{nU}}^{2}{S}^{2}/\ensuremath{\Delta}nU2S2/\ensuremathΔ term which decreases the strength of the ferromagnetic exchange, thereby delaying the onset of Stoner ferromagnetism, and generate a second, larger S term \ensuremath∝S3,\ensuremath{\propto}{S}^{3},\ensuremath∝S3, which results in a saturation of the ground-state spin before full polarization is achieved, in contrast to the Stoner scenario. For large interactions U we amplify on our earlier work [Ph. Jacquod and A. D. Stone, Phys. Rev. Lett. 84, 3938 (2000)] which showed that the broadening of the many-body density of states is proportional to KU\sqrt{K}UK​U and hence favors minimal spin. Numerical results are presented in both limits. After evaluating the effect of fluctuations, we discuss the competition between fluctuations plus kinetic energy and the exchange energy. We finally present numerical results for specific microscopic models and relate them to our generic model of fluctuations. We discuss the different physical situations to which such models may correspond, the importance of interaction fluctuations, and hence the relevance of our results to these situations and recall an experimental setup which we proposed in an earlier work to measure the importance of interaction fluctuations on the ground-state spin of lateral quantum dots in the Coulomb blockade regime.
In a Hartree-Fock picture, itinerant ferromagnetism results from a competition between kinetic and exchange energy, with the magnetized state being favored at large interaction strength. In a recent paper [1] we showed that in contrast to this average effect, fluctuations of off-diagonal interaction matrix elements systematically reduce the ground-state magnetization. When the interaction dominates, the occurence of a non-zero ground-state magnetization depends on the ratio λ between average exchange energy and the fluctuation amplitude of the off-diagonal matrix elements, and a nonzero critical value λc is necessary to magnetize the ground state. We extend these results by numerical studies of λ for standard tight-binding models which indicate a regime of intermediate disorder where off-diagonal fluctuations should play an important role for ground-state magnetization. We also emphasize the presence of strong correlations between minimal eigenvalues of different magnetization blocks which further reduce the probability of a nonzero ground-state spin.
We study a generic model of interacting fermions in a finite-size disordered system. We show that the off-diagonal interaction matrix elements induce density of state fluctuations which generically favor a minimum spin ground state at large interaction amplitude, U. This effect competes with the exchange effect which favors large magnetization at large U, and it suppresses this exchange magnetization in a large parameter range. When off-diagonal fluctuations dominate, the model predicts a spin gap which is larger for odd-spin ground states as for even spin, suggesting a simple experimental signature of this off-diagonal effect in Coulomb blockade transport measurements.
What are the ground states of an interacting, low-density electron system? In the absence of disorder, it has long been expected that as the electron density is lowered, the exchange energy gained by aligning the electron spins should exceed the enhancement in the kinetic (Fermi) energy, leading to a (Bloch) ferromagnetic transition. At even lower densities, another transition to a (Wigner) solid, an ordered array of electrons, should occur. Experimental access to these regimes, however, has been limited because of the absence of a material platform that supports an electron system with very high quality (low disorder) and low density simultaneously. Here we explore the ground states of interacting electrons in an exceptionally clean, two-dimensional electron system confined to a modulation-doped AlAs quantum well. The large electron effective mass in this system allows us to reach very large values of the interaction parameter [Formula: see text], defined as the ratio of the Coulomb to Fermi energies. As we lower the electron density via gate bias, we find a sequence of phases, qualitatively consistent with the above scenario: a paramagnetic phase at large densities, a spontaneous transition to a ferromagnetic state when [Formula: see text] surpasses 35, and then a phase with strongly nonlinear current-voltage characteristics, suggestive of a pinned Wigner solid, when [Formula: see text] exceeds [Formula: see text] However, our sample makes a transition to an insulating state at [Formula: see text], preceding the onset of the spontaneous ferromagnetism, implying that besides interaction, the role of disorder must also be taken into account in understanding the different phases of a realistic dilute electron system.
The nature of the ground state of degenerate two dimensional (2D) fermions at zero magnetic field is an outstanding open problem, which has not been deciphered despite decades of research. In the absence of disorder, the ground state is believed to be determined by the interplay between the kinetic energy, E f, and the inter-particle interaction energy, Ec = e2/ica, where a = (nn)~1^ 2 is the typical inter-particle distance, n is the areal particle density, and k is the host dielectric constant. The relative importance of the two energy scales is characterized by rs = a/ao, with ao being the Bohr radius. For electrons in a single band, rs = Ec/ E f while for the (100) surface of silicon rs = Ec/2E p , due to the two-fold valley degeneracy. At very high densities (rs <£ 1) a 2D system approaches the paramagnetic limit of a non-interacting degenerate gas characterized by the Pauli susceptibility, xo • As the density is reduced, the growing ferromagnetic correlations lead to substantial enhancement of the spin susceptibility, The system is predicted to remain paramagnetic up to rs « 20, where numerical calculations [1] find a quantum phase transition to a ferromagnetic liquid phase [2]. At lower density, rs & 37 [3], the Coulomb correlations are predicted to lead to a second phase transition to an anti-ferromagnetic quantum Wigner crystal. The energy balance between the ferro and paramagnetic states is very subtle, therefore the density window where ferromagnetism can take place is small [1].
We show that the crossover from the weak interaction limit towards the strong interaction limit may be accompanied by a delocalization effect in one dimensional disordered quantum models. The spin degrees of freedom are frozen and the spatial wave functions remain symmetric or antisymmetric when the strength U of a short range interaction is varied. The study concerns the excited states for two interacting particles and the ground state for a finite density of carriers. First, for two particles in a chain of length L, we establish a duality transformation mapping the behavior at weak U onto the behavior at strong U. For intermediate U, the mixing of the one body states and the interaction induced delocalization effect are maximum. Furthermore, if L ≈︂ L1 (the one particle localization length), the system becomes weakly chaotic with critical spectral statistics. This weak chaos is related to the multifractality of the interaction matrix. For two particles starting close to each other, localization is reached in two steps. Before the time t1 necessary to propagate over L1 , U de-favors the propagation. On the contrary, U favors a very slow delocalization after t1 , characterized by a log(t) spreading of the center of mass. Similarly, the curvatures of the energy levels with respect to an enclosed magnetic flux decrease as a function of U for L < L1 and increase for L > L1 . The changes of the curvatures can be described by a conductance-like single scaling parameter. Second, using the density renormalization group algorithm, we have studied the ground state energy of a finite density of spinless fermions and its change under twisted boundary conditions. For a large disorder, a charge reorganization is induced by the interaction: When the system becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the ground state sensitivity can be enhanced by orders of magnitude. In contrast, no enhancement occurs at weaker disorder, when there are many particles on a scale L1.
Orbital magnetization is known empirically to play an important role in several magnetic phenomena, such as permanent magnetism and ferromagnetic superconductivity. Within the recently developed ``modern theory of orbital magnetization,'' theoretical insight has been gained into the nature of this often neglected contribution to magnetism but is based on an underlying mean-field approximation. From this theory, a few treatments have emerged which also take into account correlations beyond the mean-field approximation. Here, we apply the scheme developed in a previous work [F. Aryasetiawan et al., Phys. Rev. B 93, 161104(R) (2016)] to the spin-12\frac{1}{2}21​ Haldane-Hubbard model to investigate the effect of charge fluctuations on the orbital magnetization within the GWGWGW approximation. Qualitatively, we are led to distinguish between two quite different situations: (i) When the lattice potential is larger than the nearest-neighbor hopping, the correlations are found to boost the orbital magnetization. (ii) If the nearest-neighbor hopping is instead larger than the lattice potential, the correlations reduce the magnetization. The boost and reduction are identified to stem from interband and intraband correlations, respectively, and the relative importance of the two varies with the strength of the lattice potential. We finally study graphene with parameters obtained from first principles.