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.
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 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 KU 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.
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.
The decay of (disorder-averaged) static spin correlation functions at T = 0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling J ∆ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At ∆ = 0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N = 100 sites. At ∆ ̸ = 0 Lanczos diagonalizations are carried out for chains with up to N = 22 sites. The longitudinal correlation function 〈Sz 0Sz r 〉 is found to exhibit a power-law decay with an exponent that varies with ∆ and, for nonzero ∆, also with the width of the λi-distribution. The results for the transverse correlation function 〈S x 0 Sx r 〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on. 1 Typeset using REVTEXThe combination of randomness and quantum fluctuations is well-known to be a fertile ground for interesting physical phenomena including Anderson localization. In onedimensional (1D) tight-binding systems the rule is that disorder always leads to localization. However, if the randomness is purely off-diagonal, the localization length diverges at the band center, 1 which is bound to affect the decay law of correlation functions. One particular off-diagonally disordered fermion model, the 1D half-filled tight-binding model with random hopping, is equivalent to the special XX case ( ∆ = 0) of the 1D s = 1/2 XXZ model with random transverse exchange coupling, described by the Hamiltonian H = J ∑
Models of many-body localization (MBL) can be represented as tight-binding models in the many-body Hilbert space (Fock space). We explore the role of correlations between matrix elements of the effective Fock-space Hamiltonians in the scaling of MBL critical disorder Wc(n) with the size n of the system. For this purpose, we consider five models, which all have the same distributions of diagonal (energy) and off-diagonal (``hopping'') Fock-space matrix elements but different Fock-space correlations. These include quantum-dot (QD) and one-dimensional (1D) MBL models, their modifications (uQD and u1D models) with removed correlations of off-diagonal matrix elements, as well a quantum random energy model (QREM) with no correlations at all. Our numerical results are in full consistency with analytical arguments predicting ${n}^{3/4}{(lnn)}^{\ensuremath{-}1/4}\ensuremath{\lesssim}{W}_{c}\ensuremath{\lesssim}nlnn$ for the scaling of Wc(n) in the QD model (we find ${W}_{c}\ensuremath{\sim}n$ numerically), ${W}_{c}(n)\ensuremath{\sim}\text{const}$ for the 1D model, ${W}_{c}\ensuremath{\sim}nlnn$ for the uQD and u1D models without off-diagonal correlations, and ${W}_{c}\ensuremath{\sim}{n}^{1/2}lnn$ for QREM. The key difference between the QD and 1D models is in the structure of correlations of many-body energies. Removing off-diagonal Fock-space correlations makes both these models ``maximally chaotic''. Our findings demonstrate that the scaling of Wc(n) for MBL transitions is governed by a combined effect of Fock-space correlations of diagonal and off-diagonal matrix elements.
In this work we investigate themes related to many-body systems in which multiple ground states are accessible, a condition known as frustration. Frustration can arise in a number of contexts, and we consider the consequences of this situation with some examples from condensed-matter physics. In some magnetic materials interactions between spins are such that no single spin configuration provides a unique ground state. In the class of frustrated magnets where the number of ground states is extensive, thermal fluctuations are strong even at temperatures significantly below the interaction strength. At such temperatures spins are highly correlated, and small perturbations may have profound consequences. In this thesis we provide an example of this. By considering classical n-component spins with nearest-neigbour exchange on a frustrated octahedral lattice we show that – in the limit where exchange interactions are large – the system is in a disordered, correlated phase where correlations have the form of a dipole field. This is termed a Coulomb phase. From this phase we induce an ordering transition, lifting the degeneracy with weak, additional short-range interactions. By studying the transition in the solvable limit of n → ∞, we discover that the transition has identical thermodynamics to that of a magnetic system interacting through long-range, dipolar forces. Finally, we provide a more apposite characterisation of the transition, where the high-temperature side of the transition is described through the fluctuations of solenoidal fields, and the ordering corresponds to a condensation of these fields. In a separate part of the thesis, we investigate the influence of disorder on frustrated lattices. We study a two-dimensional tight-binding model with nearest-neighbour hopping and on-site disorder. Restricting the allowed states to being those from the low-lying manifold of ground states, the disorder feeds through to act as effective disorder in the hopping terms, which decay algebraically with distance. The quasi-long range nature of this effective hopping leads to a situation in which the resultant single-particle eigenstates are critical, and we probe their behaviour numerically with a transfer matrix calculation.
Motivated by recent quantum Monte Carlo (QMC) simulations of the quantum Kagome ice model by Juan Carrasquilla et al., [Nat. Commun., 6, 7421 (2015)], we study the ground-state properties of this model on the triangular lattice. In the presence of a magnetic field h, the Hamiltonian possesses competing interactions between a Z2-invariant easy-axis ferromagnetic interaction ${J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$ and a frustrated Ising term Jz. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the Z2-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large h, an easy-axis canted ferromagnet (CFM) with broken Z2 symmetry for small h and dominant ${J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$, a ferrosolid phase with broken translational and Z2 symmetries for small h and dominant Jz, and two lobes with $m=\ensuremath{\langle}{S}_{z}\ensuremath{\rangle}=\ifmmode\pm\else\textpm\fi{}1/6$ for small h and dominant Jz. We show that quantum fluctuations are suppressed in this model, hence the large-S expansion gives an accurate picture of the ground-state properties. When quantum fluctuations are introduced, we show that the ferrosolid state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for ${J}_{z}\ensuremath{\rightarrow}\ensuremath{\infty}$. With nonzero magnetic field the classical lobes acquire a finite magnetic susceptibility with no Sz order. We present the trends of the ground-state energy and the magnetizations. We also present a detail analysis of the CFM.