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 $mathbb Z_2$ symmetry of the Fermi surface is broken. The extra $mathbb Z_2$ symme
try 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 $mathbb Z_2$ 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 $mathbb Z_2$-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 use the Bott topological invariant to establish the stability of a Luttinger surface. Our proof demonstrates that the strongly coupled fixed point only corresponds to those Luttinger surfaces with co-dimension $p+1$ with $p$ odd. Because they both lie in the same universality class, we conclude that the Hubard and HK models are controlled by this fixed point.
Using the recently developed fractional Virasoro algebra cite{la_nave_fractional_2019}, we construct a class of nonlocal CFTs with OPEs 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 $T_k(z)T_k(
w) sim frac{ c_kZ_gamma}{(z-w)^{3gamma+1}}+frac{(1+gamma ) T_k(w)}{(z-w)^{1+gamma}}+frac{partial^gamma_w T_k}{z-w}$ which naturally results in a central charge, $c_k$, that is state-dependent, with $k$ indexing a particular grading. Our work indicates that only those theories which are nonlocal have state-dependent central charges, regardless of the pseudo-differential operator content of their action. All others, including certain fractional Laplacian theories, can be mapped onto an equivalent local one using a suitable covering/field redefinition. In addition, we discuss various perturbative implications of deformations of fractional CFTs that realize a fractional Virasoro algebra through the lense of a degree/state-dependent refinement of the 2 dimensional C-theorem.
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 ar
ound CFT vacuum states impliescite{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 formulacite{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 the holographic area law cannot determine a unique bulk theory even for second-order perturbations assuming only RT on ball-shaped regions. It is anticipated that RT on all subregions is expected to encode the full non-linear Einstein equations on asymptotically AdS spacetimes.
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 lo
cal 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 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 mo
dels. 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 Nothers 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 of the gauge field and the associated current. Using well known extension theorems and the membrane paradigm, we show that either the boundary (UV) or horizon (IR) theory of holographic dilatonic models are both described by such fractional electromagnetism. We also show that the non-local Pippard kernel introduced to solve the problem of the Meissner effect in elemental superconductors can also be formulated as a special case of fractional electromagnetism. We show that the standard charge quantization rules fail when the gauge field acquires an anomalous dimension. The breakdown of charge quantization is discussed extensively in terms of the experimentally measurable modified Aharonov-Bohm effect in the strange metal phase of the cuprate superconductors.
