We find half-wormhole solutions in Jackiw-Teitelboim gravity by allowing the geometry to end on a spacetime D-brane with specific boundary conditions. This theory also contains a Euclidean wormhole which leads to a factorization problem. We propose t
hat half-wormholes provide a gravitational picture for how factorization is restored and show that the Euclidean wormhole emerges from averaging over the boundary conditions. The wormhole is known to be dual to a Sachdev-Ye-Kitaev (SYK) model with random complex couplings. We find that the free energy of the half-wormhole is strikingly similar to that of a single realization of this SYK model. These results suggest that the gravitational path integral computes an average over spacetime D-brane boundary conditions.
Isolated islands in two-dimensional strongly-disordered and strongly-coupled superconductors become optically active inducing sub-gap collective excitations in the ac conductivity. Here, we investigate the fate of these excitations as a function of t
he disorder strength in the experimentally relevant case of weak electron-phonon coupling. An explicit calculation of the ac conductivity, that includes vertex corrections to restore gauge symmetry, reveals the existence of collective sub-gap excitations, related to phase fluctuations and therefore identified as the Goldstone modes, for intermediate to strong disorder. As disorder increases, the shape of the sub-gap excitation transits from peaked close to the spectral gap to a broader distribution reaching much smaller frequencies. Phase-coherence still holds in part of this disorder regime. The requirement to observe sub-gap excitations is not the existence of isolated islands acting as nano-antennas but rather the combination of a sufficiently inhomogeneous order parameter with a phase fluctuation correlation length smaller than the system size. Our results indicate that, by tuning disorder, the Goldstone mode may be observed experimentally in metallic superconductors based for instance on Al, Sn, Pb or Nb.
We study a two-site Sachdev-Ye-Kitaev (SYK) model with complex couplings, and identify a low temperature transition to a gapped phase characterized by a constant in temperature free energy. This transition is observed without introducing a coupling b
etween the two sites, and only appears after ensemble average over the complex couplings. We propose a gravity interpretation of these results by constructing an explicit solution of Jackiw-Teitelboim (JT) gravity with matter: a two-dimensional Euclidean wormhole whose geometry is the double trumpet. This solution is sustained by imaginary sources for a marginal operator, without the need of a coupling between the two boundaries. As the temperature is decreased, there is a transition from a disconnected phase with two black holes to the connected wormhole phase, in qualitative agreement with the SYK observation. The expectation value of the marginal operator is an order parameter for this transition. This illustrates in a concrete setup how a Euclidean wormhole can arise from an average over field theory couplings.
In a recent comment to the paper Chaotic Integrable transition in the SYK model, it was claimed that, in a certain region of parameters, the Lyapunov exponent of the N Majoranas Sachdev-Ye-Kitaev model with a quadratic perturbation, is always positiv
e. This implies that the model is quantum chaotic. In this reply, we show that the employed perturbative formalism breaks down precisely in the range of parameters investigated in the comment due to a lack of separation of time scales. Moreover, based on recent analytical results, we show that for any large and fixed N, the model has indeed a chaotic-integrable transition that invalidate the results of the comment.
Eigenstate multifractality is a distinctive feature of non-interacting disordered metals close to a metal-insulator transition, whose properties are expected to extend to superconductivity. While multifractality in three dimensions (3D) only develops
near the critical point for specific strong-disorder strengths, multifractality in 2D systems is expected to be observable even for weak disorder. Here we provide evidence for multifractal features in the superconducting state of an intrinsic weakly disordered single-layer NbSe$_2$ by means of low-temperature scanning tunneling microscopy/spectroscopy. The superconducting gap, characterized by its width, depth and coherence peaks amplitude, shows a characteristic spatial modulation coincident with the periodicity of the quasiparticle interference pattern. Spatial inhomogeneity of the superconducting gap width, proportional to the local order parameter in the weak-disorder regime, follows a log-normal statistical distribution as well as a power-law decay of the two-point correlation function, in agreement with our theoretical model. Furthermore, the experimental singularity spectrum f($alpha$) shows anomalous scaling behavior typical from 2D weakly disordered systems.
We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $E approx 0$, discrete symmetries alter qualitatively the spect
ral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $E approx 0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E approx 0$, the spectral density grows exponential with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E approx 0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalues separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universality characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/t^2$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.
We investigate the holographic dual of a probe scalar in an asymptotically Anti-de-Sitter (AdS) disordered background which is an exact solution of Einsteins equations in three bulk dimensions. Unlike other approaches to model disorder in holography,
we are able to explore quantum wave-like interference effects between an oscillating or random source and the geometry. In the weak-disorder limit, we compute analytically and numerically the one-point correlation function of the dual field theory for different choices of sources and backgrounds. The most interesting feature is the suppression of the one-point function in the presence of an oscillating source and weak random background. We have also computed analytically and numerically the two-point function in the weak disorder limit. We have found that, in general, the perturbative contribution induces an additional power-law decay whose exponent depends on the distribution of disorder. For certain choices of the gravity background, this contribution becomes dominant for large separations which indicates breaking of perturbation theory and the possible existence of a phase transition induced by disorder.
We show analytically that the spectral density of the $q$-body Sachdeev-Ye-Kitaev (SYK) model agrees with that of Q-Hermite polynomials with Q a non-trivial function of $q ge 2$ and the number of Majorana fermions $N gg 1$. Numerical results, obtaine
d by exact diagonalization, are in excellent agreement with the analytical spectral density even for relatively small $N sim 8$. For $N gg 1$ and not close to the edge of the spectrum, we find the macroscopic spectral density simplifies to $rho(E) sim exp[2arcsin^2(E/E_0)/log eta]$, where $eta$ is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions. This spectral density reproduces the known result for the free energy in the large $q$ and $N$ limit. In the infrared region, where the SYK model is believed to have a gravity-dual, the spectral density is given by $rho(E) sim sinh[2pi sqrt 2 sqrt{(1-E/E_0)/(-log eta)}]$. It therefore has a square-root edge, as in random matrix ensembles, followed by an exponential growth, a distinctive feature of black holes and also of low energy nuclear excitations. Results for level-statistics in this region confirm the agreement with random matrix theory. Physically this is a signature that, for sufficiently long times, the SYK model and its gravity dual evolve to a fully ergodic state whose dynamics only depends on the global symmetry of the system. Our results strongly suggest that random matrix correlations are a universal feature of quantum black holes and that the SYK model, combined with holography, may be relevant to model certain aspects of the nuclear dynamics.
We study spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, a variant of the $k$-body embedded random ensembles studied for several decades in the context of nuclear physics and quantum chaos. We show analytically that the fourth a
nd sixth order energy cumulants vanish in the limit of large number of particles $N to infty$ which is consistent with a Gaussian spectral density. However, for finite $N$, the tail of the average spectral density is well approximated by a semi-circle law. The specific heat coefficient, determined numerically from the low temperature behavior of the partition function, is consistent with the value obtained by previous analytical calculations. For energy scales of the order of the mean level spacing we show that level statistics are well described by random matrix theory. Due to the underlying Clifford algebra of the model, the universality class of the spectral correlations depends on $N$. For larger energy separations we identify an energy scale that grows with $N$, reminiscent of the Thouless energy in mesoscopic physics, where deviations from random matrix theory are observed. Our results are a further confirmation that the Sachdev-Ye-Kitaev model is quantum chaotic for all time scales. According to recent claims in the literature, this is an expected feature in field theories with a gravity-dual.
We study the dynamics in a one dimensional hard-core Bose gas with power-law hopping after an abrupt reduction of the hopping range using the time-dependent density-matrix renormalization group (t-DMRG) and bosonization techniques. In particular, we
focus on the destruction of the Bose-Einstein condensate (BEC), which is present in the initial state in the thermodynamic limit. We argue that this type of quench is akin to a sudden reduction in the effective dimensionality $d$ of the system (from $d > 1$ to $d = 1$). We identify two regimes in the evolution of the BEC fraction. For short times the decay of the BEC fraction is Gaussian while for intermediate to long times, it is well described by a stretched exponential with an exponent that depends on the initial effective dimensionality of the system. These results are potentially relevant for cold trapped-ion experiments which can simulate an equivalent of hard-core bosons, i.e. spins, with tunable long-range interactions.
