The insulating state of matter: a geometrical theory

The ground state and transport properties of the Lieb lattice flat band in the presence of an attractive Hubbard interaction are considered. It is shown that the superfluid weight can be large even for an isolated and strictly flat band. Moreover the superfluid weight is proportional to the interaction strength and to the quantum metric, a band structure quantity derived solely from the flat-band Bloch functions. These predictions are amenable to verification with ultracold gases and may explain the anomalous behaviour of the superfluid weight of high-Tc superconductors.

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“…The same quantity M R appears in the theory of the polarization [1,60] and current [61] fluctuations in band insulators.…”

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confidence: 71%

“…In the absence of disorder and interactions the ground state of a flat band is insulating at any filling [1]. However, interactions and disorder lead to a reconstruction of the ground state whose properties are often hard to predict.…”

mentioning

confidence: 99%

Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band

et al. 2016

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“…This fluctuation dissipation relation is valid for noninteracting electrons with any boundary conditions 42 and for generic interacting systems with periodic boundaries 24,26 , the former scenario being pertinent to our study. By considering the generalized Einstein relation at low temperature 43,44 , that is…”

Section: Insulation Vs Localizationmentioning

confidence: 95%

“…where α, β correspond to spatial coordinates; ρ D (r, r ′ ) = 2P (r, r ′ ) is the one-particle density matrix for a Slater determinant, which in turn is given by 26 …”

Section: Insulation Vs Localizationmentioning

confidence: 99%

Conduction in quasiperiodic and quasirandom lattices: Fibonacci, Riemann, and Anderson models

2016

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We study the ground state conduction properties of noninteracting electrons in aperiodic but non-random one-dimensional models with chiral symmetry, and make comparisons against Anderson models with non-deterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all singleparticle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the half-filled case; however, at various specific fillings ρ, including the values ρ = 1/g n , where g is the golden ratio and n is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at half-filling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-centre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted.

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“…as well. It is known that-in the large-system limit-it converges to a finite value in insulators, while it diverges in metals [17]. Simulations and heuristic arguments altogether suggest that the metallic divergence is of the order L in any dimension: d = 1, 2 or 3 [18][19][20].…”

Section: Arxiv:170208885v1 [Cond-matmes-hall] 28 Feb 2017mentioning

confidence: 99%

Locality of the anomalous Hall conductivity

Marrazzo

Resta

2017

Phys. Rev. B

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The geometrical intrinsic contribution to the anomalous Hall conductivity (AHC) of a metal is commonly expressed as a reciprocal-space integral: as such, it only addresses unbounded and macroscopically homogeneous samples. Here we show that the geometrical AHC has an equivalent expression as a local property. We define a "geometrical marker" which actually probes the AHC in inhomogeneous systems (e.g. heterojunctions), as well as in bounded samples. The marker may even include extrinsic contributions of geometrical nature.PACS numbers: 72.15.Eb,72.20.My The Hall conductivity is "anomalous" whenever it is nonzero in absence of an applied magnetic field. The phenomenon requires the absence of time-reversal symmetry: it was discovered by Hall himself in 1881 in ferromagnetic metals. The possibility of observing anomalous Hall conductivity (AHC) in insulators was pointed out in 1988 by Haldane, who proposed a model Hamiltonian where the AHC is nonzero and quantized [1]; the AHC value is determined by the topology of the electronic ground state. In metals extrinsic mechanisms are essential to make the longitudinal dc conductivity finite. In absence of timereversal symmetry extrinsic mechanisms contribute to the AHC as well: these go under the name of side-jump and skew-scattering [2]. Since the early 2000s [3,4] it became clear that an intrinsic effect, only dependent on the ground wavefunction of the pristine crystal, provides an important additional contribution to the AHC. The latter contribution is geometrical in nature; its expression is the nonquantized version of the corresponding formula for insulators. In this work we only address the geometrical/topological AHC in metals/insulators, which is customarily expressed as the Fermi-volume integral of the Berry curvature, Eq. (4) below. The standard approach requires an unbounded crystalline sample where the orbitals have the Bloch form; this was extended in Ref. [5] to "dirty" metals in a supercell framework, where the geometric contribution includes some extrinsic effects.Recent work has demonstrated the locality of AHC, although in the insulating case only [6]. One does not require lattice periodicity and reciprocal-space paraphernalia: the AHC can be defined and computed for bounded samples and/or for macroscopically inhomogeneous systems (e.g. heterojunctions). The extension to the metallic case is not obvious, since one of the reasons for the locality of the (quantized) AHC is the r-space exponential decay of the one-body density matrix in insulators ("nearsightedness" [7]). In metals instead such decay is only power-law, which hints to a possibly different behavior. Furthermore the Hall current in insulators may only flow in the edge region, while in metals the current flows through the bulk of the sample as well. Our major result is that even in metals the AHC is a local property: for a bounded sample it can be expressed in terms of the one-body density matrix, evaluated in the sample bulk. We show this by means of simulations on model two-dimensional b...

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The insulating state of matter: a geometrical theory

Cited by 229 publications

References 74 publications

Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band

Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band

Conduction in quasiperiodic and quasirandom lattices: Fibonacci, Riemann, and Anderson models

Locality of the anomalous Hall conductivity

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