dc conductivity as a geometric phase

The zero frequency conductivity ($D_c$), the criterion to distinguish between conductors and insulators is expressed in terms of a geometric phase. $D_c$ is also expressed using the formalism of the modern theory of polarization. The tenet of Kohn [{\it Phys. Rev.} {\bf 133} A171 (1964)], namely, that insulation is due to localization in the many-body space, is refined as follows. Wavefunctions which are eigenfunctions of the total current operator give rise to a finite $D_c$ and are therefore metallic. They are also delocalized. Several examples which corroborate the results are presented, as well as a numerical implementation. The formalism is also applied to the Hall conductance, and the quantization condition for zero Hall conductance is derived to be $\frac{e\Phi_B}{N h c} = \frac{Q}{M}$, with $Q$ and $M$ integers.Comment: minor changes compared to previous version, and reference adde

Drude weight, Meissner weight, rotational inertia of bosonic superfluids: how are they distinguished?

The Drude weight, the quantity which distinguishes metals from insulators, is proportional to the second derivative of the ground state energy with respect to a flux at zero flux. The same expression also appears in the definition of the Meissner weight, the quantity which indicates superconductivity, as well as in the definition of non-classical rotational inertia of bosonic superfluids. It is shown that the difference between these quantities depends on the interpretation of the average momentum term, which can be understood as the expectation value of the total momentum (Drude weight), the sum of the expectation values of single momenta (rotational inertia of a superfluid), or the sum over expectation values of momentum pairs (Meissner weight). This distinction appears naturally when the current from which the particular transport quantity is derived is cast in terms of shift operators

Cumulants associated with geometric phases

The Berry phase can be obtained by taking the continuous limit of a cyclic product -\mbox{Im} \ln \prod_{I=0}^{M-1} \langle \Psi_0({\boldsymbol \xi}_I)|\Psi_0({\boldsymbol \xi}_{I+1})\rangle, resulting in the circuit integral i \oint \mbox{d}{\boldsymbol \xi} \cdot \langle \Psi_0({\boldsymbol \xi})|\nabla_{\boldsymbol \xi}|\Psi_0({\boldsymbol \xi}\rangle. Considering a parametrized curve ${\boldsymbol \xi}(\chi)$ we show that the product $\prod_{I=0}^{M-1} \langle \Psi_0(\chi_I)|\Psi_0( \chi_{I+1})\rangle$ can be equated to a cumulant expansion. The first contributing term of this expansion is the Berry phase itself, the other terms are the associated spread, skew, kurtosis, etc. The cumulants are shown to be gauge invariant. It is also shown that these quantities can be expressed in terms of an operator.Comment: text + 1 figur

Drude and Superfluid Weights in Extended Systems: the Role of Discontinuities and δ\delta-peaks in the One and Two-Body Momentum Densities

The question of conductivity is revisited. Using the total momentum shift operator to construct the perturbed many-body Hamiltonian and ground state wave function the second derivative of the ground state energy with respect to the perturbing field is expressed in terms of the one and two-body momentum densities. The distinction between the adiabatic and envelope function derivatives, hence that between the Drude and superfluid weights can be introduced in a straightforward manner. It is shown that a discontinuity in the momentum density leads to a contribution to the Drude weight, but not the superfluid weight, however a $\delta$-function contribution in the two-body momentum density (such as in the BCS wave-funtion) contributes to both quantities. The connection between the discontinuity in the momentum density and localization is also demonstrated.Comment: some changes in the text, to appear in the Journal of the Physical Society of Japan, reference will be added as soon as it is availabl

Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems

In this work we focus on two questions. One, we complement the machinary to calculate geometric phases along adiabatic cycles as follows. The geometric phase is a line integral along an adiabatic cycle, and if the cycle encircles a degeneracy point, the phase becomes non-trivial. If the cycle crosses the degeneracy point the phase diverges. We construct quantities which are well-defined when the path crosses the degeneracy point. We do this by constructing a generalized Bargmann invariant, and noting that it can be interpreted as a cumulant generating function, with the geometric phase being the first cumulant. We show that particular ratios of cumulants remain finite for cycles crossing a set of isolated degeneracy points. The cumulant ratios take the form of the Binder cumulants known from the theory of finite size scaling in statistical mechanics (we name them geometric Binder cumulants). Two, we show that the machinery developed can be applied to perform finite size scaling in the context of the modern theory of polarization. The geometric Binder cumulants are size independent at gap closure points or regions with closed gap (Luttinger liquid). We demonstrate this by model calculations for a one-dimensional topological model, several two-dimensional models, and a one-dimensional correlated model. In the case of two dimensions we analyze to different situations, one in which the Fermi surface is one-dimensional (a line), and two cases in which it is zero dimensional (Dirac points). For the geometric Binder cumulants the gap closure points can be found by one dimensional scaling even in two dimensions. As a technical point we stress that only certain finite difference approximations for the cumulants are applicable, since not all approximation schemes are capable of extracting the size scaling information in the case of a closed gap system

Fluctuations, uncertainty relations, and the geometry of quantum state manifolds

The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher order geometric tensors (affine connection, Riemann curvature tensor) can be obtained in terms of gauge invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the $SU(1,1)$ group. In our examples non-trivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the $SU(1,1)$ group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states non-trivial geometry results