Exact holographic mapping and emergent space-time geometry

In this paper, we propose an {\it exact holographic mapping} which is a unitary mapping from the Hilbert space of a lattice system in flat space (boundary) to that of another lattice system in one higher dimension (bulk). By defining the distance in the bulk system from two-point correlation functions, we obtain an emergent bulk space-time geometry that is determined by the boundary state and the mapping. As a specific example, we study the exact holographic mapping for $(1+1)$-dimensional lattice Dirac fermions and explore the emergent bulk geometry corresponding to different boundary states including massless and massive states at zero temperature, and the massless system at finite temperature. We also study two entangled one-dimensional chains and show that the corresponding bulk geometry consists of two asymptotic regions connected by a worm-hole. The quantum quench of the coupled chains is mapped to dynamics of the worm-hole. In the end we discuss the general procedure of applying this approach to interacting systems, and other open questions.Comment: 15 pages, 7 figure

Topological Quantum Computation Based on Chiral Majorana Fermions

Chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain two-dimensonal topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, Majorana zero mode in the bulk of the corresponding topological matter, is known to be applicable in topological quantum computations. Here we show that the propagation of chiral Majorana fermions lead to the same unitary transformation as that in the braiding of Majorana zero modes, and propose a new platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can utilize quantum coherent chiral Majorana fermions to implement the Hadamard gate and the phase gate, and the junction conductance yields a natural readout for the qubit state.Comment: Accepted for publication at PNA

Space-time random tensor networks and holographic duality

In this paper we propose a space-time random tensor network approach for understanding holographic duality. Using tensor networks with random link projections, we define boundary theories with interesting holographic properties, such as the Renyi entropies satisfying the covariant Hubeny-Rangamani-Takayanagi formula, and operator correspondence with local reconstruction properties. We also investigate the unitarity of boundary theory in spacetime geometries with Lorenzian signature. Compared with the spatial random tensor networks, the space-time generalization does not require a particular time slicing, and provides a more covariant family of microscopic models that may help us to understand holographic duality.Comment: 31 pages, 9 figure

Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK

In many-body chaotic systems, the size of an operator generically grows in Heisenberg evolution, which can be measured by certain out-of-time-ordered four-point functions. However, these only provide a coarse probe of the full underlying operator growth structure. In this article we develop a methodology to derive the full growth structure of fermionic systems, that also naturally introduces the effect of finite temperature. We then apply our methodology to the SYK model, which features all-to-all $q$-body interactions. We derive the full operator growth structure in the large $q$ limit at all temperatures. We see that its temperature dependence has a remarkably simple form consistent with the slowing down of scrambling as temperature is decreased. Furthermore, our finite-temperature scrambling results can be modeled by a modified epidemic model, where the thermal state serves as a vaccinated population, thereby slowing the overall rate of infection.Comment: 31 pages, 10 figure

Tunable circular dichroism due to the chiral anomaly in Weyl semimetals

Weyl semimetals are a three dimensional gapless topological phase in which bands intersect at arbitrary points -- the Weyl nodes -- in the Brillouin zone. These points carry a topological quantum number known as the \emph{chirality} and always appear in pairs of opposite chiralities. The notion of chirality leads to anomalous non-conservation of chiral charge, known as the \emph{chiral anomaly}, according to which charge can be pumped between Weyl nodes of opposite chiralities by an electromagnetic field with non-zero $\boldsymbol{E}\cdot\boldsymbol{B}$. Here, we propose probing the chiral anomaly by measuring the optical activity of Weyl semimetals via circular dichroism. In particular, we observe that applying such an electromagnetic field on this state gives it a non-zero gyrotropic coefficient or a Hall-like conductivity, which may be detectable by routine circular dichroism experiments. This method also serves as a diagnostic tool to discriminate between Weyl and Dirac semimetals; the latter will give a null result. More generally, any experiment that probes a bulk correlation function that has the same symmetries as the gyrotropic coefficient can detect the chiral anomaly as well as differentiate between Dirac and Weyl semimetals.Comment: Replaced Kubo calculation of dielectric tensor by a more intuitive semiclassical calculation. Fixed error in assumptions about various time scales, which changed the prediction from a Faraday effect to circular dichrois

Characterizing eigenstate thermalization via measures in the Fock space of operators

The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in various regimes, by mapping general interacting quantum systems on regular lattices onto a single particle living on a high-dimensional graph. By numerically analyzing deviations from ETH behavior in the non-integrable Ising model, we propose a quantity that we call the $n$-$weight$ to democratically characterize the average deviations for all operators residing on a given number of sites, irrespective of their spatial structure. It appears to have a simple scaling form, that we conjecture to hold true for all non-integrable systems. A closely related quantity, that we term the $n$-$distinguishability$, tells us how well two states can be distinguished if only $n$-site operators are measured. Along the way, we discover that complicated operators on average are worse than simple ones at distinguishing between neighboring eigenstates, contrary to the naive intuition created by the usual statements of the ETH that few-body (many-body) operators acquire the same (different) expectation values in nearby eigenstates at finite energy density. Finally, we sketch heuristic arguments that the ETH originates from the limited ability of simple operators to distinguish between quantum states of a system, especially when the states are subject to constraints such as roughly fixed energy with respect to a local Hamiltonian.Comment: 9 pages, 5 figures; Typos fixed, references adde

Eternal traversable wormhole

We construct a nearly-$AdS_2$ solution describing an eternal traversable wormhole. The solution contains negative null energy generated by quantum fields under the influence of an external coupling between the two boundaries. In parallel, we discuss two SYK systems coupled by a relevant interaction. The physics of the two cases is very similar. They both share a "gravitational" subsector which is identical. The solution within this subsector sets the stage for dynamics which is almost conformal invariant. We study this system in detail, both in gravity and in the SYK model. The coupled SYK models have an interesting phase diagram at finite temperature, displaying the usual Hawking-Page transition between the thermal AdS phase at low temperature and the black hole phase at high temperature. Interestingly, these two phases are continuously connected in the microcannonical ensemble.Comment: 50 +21 pages. 23 figures. V2: Slightly expanded discussion of the thermal phases. References added. V3 Latex issue fixe

Fractional Statistics and the Butterfly Effect

Fractional statistics and quantum chaos are both phenomena associated with the non-local storage of quantum information. In this article, we point out a connection between the butterfly effect in (1+1)-dimensional rational conformal field theories and fractional statistics in (2+1)-dimensional topologically ordered states. This connection comes from the characterization of the butterfly effect by the out-of-time-order-correlator proposed recently. We show that the late-time behavior of such correlators is determined by universal properties of the rational conformal field theory such as the modular S-matrix and conformal spins. Using the bulk-boundary correspondence between rational conformal field theories and (2+1)-dimensional topologically ordered states, we show that the late time behavior of out-of-time-order-correlators is intrinsically connected with fractional statistics in the topological order. We also propose a quantitative measure of chaos in a rational conformal field theory, which turns out to be determined by the topological entanglement entropy of the corresponding topological order.Comment: Published version, 1+25 pages, 10 figure

An integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind

In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind.Comment: 10 page

Quantum Oscillations in Weyl and Dirac Semimetal Ultra-Thin Films

We show that a thin film of Weyl or Dirac semimetal with a strong in-plane magnetic field becomes a novel two-dimensional Fermi liquid with interesting properties. The Fermi surface in this system is strongly anisotropic, which originates from a combination of chiral bulk channels and the Fermi arcs. The area enclosed by the Fermi surface depends strongly on the in-plane magnetic field component parallel to the Weyl/Dirac node splitting, which leads to unusual behavior in quantum oscillations when the magnetic field is tilted out of the plane. We estimate the oscillation frequencies and the regimes where such effects could be seen in Cd$_3$As$_2$, Na$_3$Bi, and TaAs.Comment: 4.5 pages, 4 figures and 2 pages of appendix with 2 figure