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arXiv:0906.4870 · Physical Review B

Ground-state properties of the disordered Hubbard model in two dimensions

M. Pezzoli, Federico Becca

Published Feb 5, 2010• 26 citations
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Abstract

We study the interplay between electron correlation and disorder in the two-dimensional Hubbard model at half filling by means of a variational wave function that can interpolate between Anderson and Mott insulators. We give a detailed description of our improved variational state and explain how the physics of the Anderson-Mott transition can be inferred from equal-time correlations functions, which can be easily computed within the variational Monte Carlo scheme. The ground-state phase diagram is worked out in both the paramagnetic and the magnetic sector. Whereas in the former a direct second-order Anderson-Mott transition is obtained, when magnetism is allowed variationally, we find evidence for the formation of local magnetic moments that order before the Mott transition. Although the localization length increases before the Mott transition, we have no evidence for the stabilization of a true metallic phase. The effect of a frustrating next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ is also studied in some detail. In particular, we show that ${t}^{\ensuremath{'}}$ has two primary effects. The first one is the narrowing of the stability region of the magnetic Anderson insulator, also leading to a first-order magnetic transition. The second and most important effect of a frustrating hopping term is the development of a ``glassy'' phase at strong couplings, where many paramagnetic states, with disordered local moments, may be stabilized.

DOI
10.1103/physrevb.81.075106

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In this thesis we consider a variational wave function approach as a possible route to describe the competition between disorder and strong electron-electron interaction in two dimensions. In particular we aim to obtain a transparent and physically intuitive understanding of the competition between these two localizing forces within the simplest model where they both are active, namely the disordered Hubbard model at half filling and in a square lattice. Our approach is based on an approximate form of the ground-state wave function, which we believe contains the physically relevant ingredients for a correct description of both the Mott and the Anderson insulators, where electrons are localized by the Coulomb repulsion and by disorder, respectively. For strongly interacting fermionic systems, a standard variational wave function is constructed by a correlation term acting on a Slater determinant, the latter being an uncorrelated metallic state. Previous variational calculations showed that a long-range density-density correlation factor, so called Jastrow factor, is needed to correctly describe the Mott insulator [9]. This term, which is collective by definition, correlates spatially charge uctuations, thus preventing their free motion that would otherwise imply metallic conductance. For this reason, our variational wave function does include such a term. Anderson localization is instead mostly a matter of single-particle wave functions, hence it pertains to the uncorrelated Slater determinant which the Jastrow factor acts onto. We consider both the case in which we enforce paramagnetism in the wave function and the case in which we allow for magnetic ordering. Summarizing briefly our results, we find that, when the variational wave function is forced to be paramagnetic, the Anderson insulator to Mott insulator transition is continuous. This transition can be captured by studying several quantities. In particular, a novel one that we have identified and that is easily accessible variationally is the disconnected density-density fluctuation at long wavelength, defined by lim where ^nq is the Fourier transform of the charge density at momentum q, (...) denotes quantum average at fixed disorder and the overbar represents the average over disorder configurations. We find that Ndisc q!0 is everywhere finite in the Anderson insulator and vanishes critically at the Mott transition, staying zero in the Mott insulator. When magnetism is allowed and the hopping only connects nearest neighbor sites, upon increasing interaction the paramagnetic Anderson insulator first turns antiferromagnetic and finally the magnetic and compressible Anderson insulator gives way to an incompressible antiferromagnetic Mott insulator. The optimized uncorrelated Slater determinant is always found to be the eigenstate of a disordered non-interacting effective Hamiltonian, which suggests that the model is never metallic. Finally, when magnetism is frustrated by a next to nearest neighbor hopping, the overall sequence of phases does not change. However, the paramagnetic to magnetic transition within the Anderson insulator basin of stability turns first order. Indeed, within the magnetically ordered phase, we find many almost degenerate paramagnetic states with well defined local moments. This is suggestive of an emerging glassy behavior when the competition between disorder and strong correlation is maximum.
We study the ground-state phase diagram of the Anderson-Hubbard model with correlated hopping at half-filling in one dimension. The Hamiltonian has a local Coulomb repulsion UUU and a disorder potential with local energies randomly distributed in the interval $(\ensuremath{-}W,+W)$ with equal probability, acting on the singly occupied sites. The hopping process which modifies the number of doubly occupied sites is forbidden. The hopping between nearest-neighbor singly occupied and empty sites or between singly occupied and doubly occupied sites has the same amplitude ttt. We identify three different phases as functions of the disorder amplitude WWW and Coulomb interaction strength U>0U>0U>0. When U<4tU<4tU<4t the system shows a metallic phase: (i) only when no disorder is present W=0W=0W=0 or an Anderson-localized phase, (ii) when disorder is introduced $W\ensuremath{\ne}0$. When U>4tU>4tU>4t the Anderson-localized phase survives as long as disorder effects dominate on the interaction effects, otherwise a Mott-insulator phase (iii) arises. Phases (i) and (ii) are characterized by a finite density of doublons and a vanishing charge gap among the ground state and the excited states. Phase (iii) is characterized by the vanishing density of doublons and a finite gap for the charge excitations.
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
We study ground-state properties of the two-dimensional Hubbard model at half filling by improving variational Monte Carlo method and by implementing quantum-number projection and multi-variable optimization. The improved variational wave function enables a highly accurate description of the Mott transition and strong fluctuations in metals. We clarify how anomalous metals appear near the first-order Mott transition. The double occupancy stays nearly constant as a function of the on-site Coulomb interaction in the metallic phase near the Mott transition in agreement with the previous unbiased results. This unconventional metal at half filling is stabilized by a formation of ``electron-like pockets'' coexisting with an arc structure, which leads to a prominent differentiation of electrons in momentum space. An abrupt collapse of the ``pocket'' and ``arc'' drives the first-order Mott transition.
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We investigate the ground-state phase diagram of the half-filled one-dimensional Hubbard model with next-nearest-neighbor hopping using the density-matrix renormalization-group technique as well as an unrestricted Hartree-Fock approximation. We find commensurate and incommensurate disordered magnetic insulating phases and a spin-gapped metallic phase in addition to the one-dimensional Heisenberg phase. At large on-site Coulomb repulsion U, we make contact with the phase diagram of the frustrated Heisenberg chain, which has spin-gapped phases for sufficiently large frustration. For weak U, sufficiently large next-nearest-neighbor hopping t2{t}_{2}t2​ leads to a band structure with four Fermi points rather than two, producing a spin-gapped metallic phase. As U is increased in this regime, the system undergoes a Mott-Hubbard transition to a frustrated antiferromagnetic insulator.