Papers by Gabriele la Nave
Physical Review B
The topological non-triviality of insulating phases of matter are by now well understood through ... more The topological non-triviality of insulating phases of matter are by now well understood through topological K-theory where the indices of the Dirac operators are assembled into topological classes. We consider in the context of the Kitaev chain a notion of a generalized Dirac operator where the associated Clifford algebra is centrally extended. We demonstrate that the central extension is achieved via taking rational operator powers of Pauli matrices that appear in the corresponding BdG Hamiltonian. Doing so introduces a pseudo-metallic component to the topological phase diagram within which the winding number is valued in Q. We find that this phase hosts a mode that remains extended in the presence of weak disorder, motivating a topological interpretation of a non-integral winding number. We remark that this is in correspondence with Ref. [1] demonstrating that projective Dirac operators defined in the absence of spin C structure have rational indices.
Bulletin of the American Physical Society, Mar 15, 2021
While the Mott transition from a Fermi liquid is correctly believed to obtain without the breakin... more While the Mott transition from a Fermi liquid is correctly believed to obtain without the breaking of any continuous symmetry, we show that in fact a discrete emergent ℤ_2 symmetry of the Fermi surface is broken. The extra ℤ_2 symmetry of the Fermi liquid appears to be little known although it was pointed out by Anderson and Haldane and we use it here to classify all possible Fermi liquids topologically by invoking K-homology. It is this ℤ_2 symmetry breaking that signals the onset of particle-hole asymmetry, a widely observed phenomenon in strongly correlated systems. In addition from this principle, we are able to classify which interactions suffice to generate the ℤ_2-symmetry-broken phase. As this is a symmetry breaking in momentum space, the local-in-momentum space interaction of the Hatsugai-Kohmoto (HK) model suffices as well as the Hubbard interaction as it contains the HK interaction. Both lie in the same universality class as can be seen from exact diagonalization. We then...
In this paper we extend the notion of Futaki invariant to big and nef classes in such a way that ... more In this paper we extend the notion of Futaki invariant to big and nef classes in such a way that it defines a continuous function on the cone up to the boundary. We apply this concept to prove that reduced normal crossing singularities are sufficient to check K-semistability. A similar improvement on Donaldson's lower bound for Calabi energy is given.
We prove a regularity result for Monge-Ampère equations degenerate along smooth divisor on Kaehle... more We prove a regularity result for Monge-Ampère equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of β-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical singularities along divisors and give a geometric sufficient condition on the divisor for its boundedness.
From the partition function for two classes of classically non-local actions containing the fract... more From the partition function for two classes of classically non-local actions containing the fractional Laplacian, we show that as long as there exists a suitable (non-local) Hilbert-space transform the underlying action can be mapped onto a purely local theory. In all such cases the partition function is equivalent to that of a local theory and an area law for the entanglement entropy obtains. When such a reduction fails, the entanglement entropy deviates strongly from an area law and can in some cases scale as the volume. As these two criteria are coincident, we conjecture that they are equivalent and provide the ultimate test for locality of Gaussian theories rather than a simple inspection of the explicit operator content.
We show explicitly that the full structure of IIB string theory is needed to remove the non-local... more We show explicitly that the full structure of IIB string theory is needed to remove the non-localities that arise in boundary conformal theories that border hyperbolic spaces on AdS_5. Specifically, using the Caffarelli/Silvestricaffarelli, Graham/Zworskigraham, and Chang/Gonzalezchang:2010 extension theorems, we prove that the boundary operator conjugate to bulk p-forms with negative mass in geodesically complete metrics is inherently a non-local operator, specifically the fractional conformal Laplacian. The non-locality, which arises even in compact spaces, applies to any degree p-form such as a gauge field. We show that the boundary theory contains fractional derivatives of the longitudinal components of the gauge field if the gauge field in the bulk along the holographic direction acquires a mass via the Higgs mechanism. The non-locality is shown to vanish once the metric becomes incomplete, for example, either 1) asymptotically by adding N transversely stacked Dd-branes or 2) e...
While it is a standard result in field theory that the scaling dimension of conserved currents an... more While it is a standard result in field theory that the scaling dimension of conserved currents and their associated gauge fields are determined strictly by dimensional analysis and hence cannot change under any amount of renormalization, it is also the case that the standard conservation laws for currents, dJ=0, remain unchanged in form if any differential operator that commutes with the total exterior derivative, [d,Ŷ]=0, multiplies the current. Such an operator, effectively changing the dimension of the current, increases the allowable gauge transformations in electromagnetism and is at the heart of Nöther's second theorem. We review here our recent work on one particular instance of this theorem, namely fractional electromagnetism and highlight the holographic dilaton models that exhibit such behavior and the physical consequences this theory has for charge quantization. Namely, the standard electromagnetic gauge and the fractional counterpart cannot both yield integer values...
We present here a theory of fractional electro-magnetism which is capable of describing phenomeno... more We present here a theory of fractional electro-magnetism which is capable of describing phenomenon as disparate as the non-locality of the Pippard kernel in superconductivity and anomalous dimensions for conserved currents in holographic dilatonic models. The starting point for our analysis is the observation that the standard current conservation equations remain unchanged if any differential operator that commutes with the total exterior derivative multiplies the current. Such an operator, effectively changing the dimension of the current, increases the allowable gauge transformations in electromagnetism and is at the heart of N\"other's second theorem. Here we develop a consistent theory of electromagnetism that exploits this hidden redundancy in which the standard gauge symmetry in electromagnetism is modified by the rotationally invariant operator, the fractional Laplacian. We show that the resultant theories all allow for anomalous (non-traditional) scaling dimensions...
We discuss some classification results for Ricci solitons, that is, self similar solutions of the... more We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by
Springer Proceedings in Mathematics & Statistics, 2020
While it is a standard result in field theory that the scaling dimension of conserved currents an... more While it is a standard result in field theory that the scaling dimension of conserved currents and their associated gauge fields are determined strictly by dimensional analysis and hence cannot change under any amount of renormalization, it is also the case that the standard conservation laws for currents, d J = 0, remain unchanged in form if any differential operator that commutes with the total exterior derivative, [d,Ŷ ] = 0, multiplies the current. Such an operator, effectively changing the dimension of the current, increases the allowable gauge transformations in electromagnetism and is at the heart of Nöther's second theorem. We review here our recent work on one particular instance of this theorem, namely fractional electromagnetism and highlight the holographic dilaton models that exhibit such behavior and the physical consequences this theory has for charge quantization. Namely, the standard electromagnetic gauge and the fractional counterpart cannot both yield integer values of Planck's constant when they are integrated around a closed loop, thereby leading to a breakdown of charge quantization.
arXiv: High Energy Physics - Theory, 2020
Using the recently developed fractional Virasoro algebra [1], we construct the equivalent operato... more Using the recently developed fractional Virasoro algebra [1], we construct the equivalent operator product expansions for nonlocal quantum field theories in which the nonlocality is provided by the fractional Laplacian as has been shown to be relevant in the long-range Ising model. We find that the OPE's of a general nonlocal CFT are of the form $T_k(z)\Phi(w) \sim \frac{ h_\gamma \Phi}{(z-w)^{1+\gamma}}+\frac{\partial_w^\gamma \Phi}{z-w},$ and $T_k(z)T_k(w) \sim \frac{ c_kZ_\gamma}{(z-w)^{3\gamma+1}}+\frac{(1+\gamma ) T_k(w)}{(z-w)^{1+\gamma}}+\frac{\partial^\gamma_w T_k}{z-w}$ which naturally results in a central charge, $c_k$, that is state-dependent and hence not a constant. In fact, our work indicates that only those theories which are truly nonlocal have state-dependent central charges. All others can be mapped onto an equivalent Gaussian one using a suitable field redefinition.
arXiv: Algebraic Geometry, 2002
In this note we show how to find the stable model of a one-parameter family of elliptic surfaces ... more In this note we show how to find the stable model of a one-parameter family of elliptic surfaces with sections. More specifically, we perform the log Minimal Model Program in an explicit manner by means of toric geometry, in each such one parameter family. This way we obtain an explicit combinatorial description of the surfaces that may occur at the boundary of moduli (as well as a new proof of the completeness of the moduli space)
arXiv: Differential Geometry, 2013
We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bound... more We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups. Our main technique is modeled on Donaldson's version of H\"ormander technique to produce (almost) holomorphic sections which we use to construct destabilizing sections.
arXiv: High Energy Physics - Theory, 2019
Holographic entanglement entropy and the first law of thermodynamics are believed to decode the g... more Holographic entanglement entropy and the first law of thermodynamics are believed to decode the gravity theory in the bulk. In particular, assuming the Ryu-Takayanagi (RT)\cite{ryu-takayanagi} formula holds for ball-shaped regions on the boundary around CFT vacuum states implies\cite{Nonlinear-Faulkner} a bulk gravity theory equivalent to Einstein gravity through second-order perturbations. In this paper, we show that the same assumptions can also give rise to second-order Lovelock gravity. Specifically, we generalize the procedure in \cite{Nonlinear-Faulkner} to show that the arguments there also hold for Lovelock gravity by proving through second-order perturbation theory, the entropy calculated using the Wald formula\cite{Wald_noether} in Lovelock also obeys an area law (at least up to second order). Since the equations for second-order perturbations of Lovelock gravity are different in general from the second-order perturbation of the Einstein-Hilbert action, our work shows that...
Bulletin of the American Physical Society, 2019
We discuss some classification results for Ricci solitons, that is, self similar solutions of the... more We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.
arXiv: High Energy Physics - Theory, 2017
Using holographic renormalization coupled with the Caffarelli/Silvestre\cite{caffarelli} extensio... more Using holographic renormalization coupled with the Caffarelli/Silvestre\cite{caffarelli} extension theorem, we calculate the precise form of the boundary operator dual to a bulk scalar field rather than just its average value. We show that even in the presence of interactions in the bulk, the boundary operator dual to a bulk scalar field is an anti-local operator, namely the fractional Laplacian. The propagator associated with such operators is of the general power-law (fixed by the dimension of the scalar field) type indicative of the absence of particle-like excitations at the Wilson-Fisher fixed point or the phenomenological unparticle construction. Holographic renormalization also allows us to show how radial quantization can be extended to such non-local conformal operators.
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Papers by Gabriele la Nave