Search

We study the meson spectrum of the N = 4 supersymmetric Yang-Mills theory with N = 2 fundamental hypermultiplets for a finite electric field by using the D3/D7 model. The spectrum for scalar and vector mesons is computed by analyzing the (quasi-)normal modes for the fluctuations of the D7-brane embedding and gauge fields. In the presence of an electric field, two different phases in the background are realized: the meson and dissociation phases. In this paper, we analyze the meson spectrum of scalar and vector mesons for all ranges of the electric field and explore the effect of the electric field on the meson spectrum, that is, the Stark effect. In the meson spectrum, we observe the avoided crossing between different levels due to the coupling of fluctuations via the electric field. [ABSTRACT FROM AUTHOR]

In the two-component Ginzburg-Landau theory of superfluidity, a pair of fractional vortices form a composite type of topological defect, usually referred to as a baby skyrmion. In this paper, we initiate the construction of such a baby skyrmion in the holographic model of two-component superfluids. As a result, two types of baby skyrmion configurations are found, where the monopole-type of one is constructed directly by solving the static equations of motion while the dipole-type of one is obtained by resorting to the time evolution method. In addition, we find that the existence of these two types of baby skyrmion depends on the inter-component coupling, reminiscent of the situation in the baby skyrmion model. [ABSTRACT FROM AUTHOR]

In this paper, we study the fine structure of entanglement in holographic two-dimensional boundary conformal field theories (BCFT) in terms of the spatially resolved quasilocal extension of entanglement entropy — entanglement contour. We find that the boundary induces discontinuities in the contour revealing hidden localization-delocalization patterns of the entanglement degrees of freedom. Moreover, we observe the formation of “islands” where the entanglement contour vanishes identically implying that these regions do not contribute to the entanglement at all. We argue that these phenomena are the manifestation of the entanglement islands recently discussed in the literature. We apply the entanglement contour proposal to the recently discussed BCFT black hole models reproducing the Page curve — moving mirror model and the pair of BCFT in the thermofield double state. From the viewpoint of entanglement contour, the Page curve also carries the imprint of strong delocalization caused by dynamical entanglement islands. [ABSTRACT FROM AUTHOR]

In this paper, we study the phase structure of two Sachdev-Ye-Kitaev models (L-system and R-system) coupled by a simple interaction, with imperfectly correlated disorder. When the disorder of the two systems is perfectly correlated, J i 1 ⋯ i q L = J i 1 ⋯ i q R , this model is known to exhibit a phase transition at a finite temperature between the two-black hole phase at high temperature and the traversable wormhole phase at low temperature. We find that, as the correlation J i 1 ⋯ i q L = J i 1 ⋯ i q R is decreased, the critical temperature becomes lower. At the same time, the transmission between the L-system and R-system in the low-temperature phase becomes more suppressed, while the chaos exponent of the whole system becomes larger. Interestingly we also observe that when the correlation is smaller than some q-dependent critical value the phase transition completely disappears in the entire parameter space. At zero temperature, the energy gap becomes larger as we decrease the correlation. We also use a generalized thermofield double state as a variational state. Interestingly, this state coincides with the ground state in the large q limit. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mixed-state entanglement in a model of p-wave superconductivity phase transition using holographic methods. We calculate several entanglement measures, including holographic entanglement entropy (HEE), mutual information (MI), and entanglement wedge cross-section (EWCS). Our results show that these measures display critical behavior at the phase transition points, with the EWCS exhibiting opposite temperature behavior compared to the HEE. Furthermore, we explore the behavior of thermodynamics and holographic quantum information at the zeroth-order phase transition point and find that it is opposite to that observed in the first-order phase transition. Additionally, we find that the critical exponents of all entanglement measures are twice those of the condensate. Our findings also suggest that the EWCS is a more sensitive indicator of the critical behavior of phase transitions than the HEE. Lastly, we uncover a universal inequality in the growth rates of EWCS and MI near critical points in thermal phase transitions, such as p-wave and s-wave superconductivity, suggesting that MI captures more information than EWCS when a phase transition first occurs. [ABSTRACT FROM AUTHOR]

In this paper we study the chaos exponent, the exponential growth rate of the out-of-time-ordered four point functions, in a two coupled SYK models which exhibits a first order phase transition between the high temperature black hole phase and the low temperature gapped phase interpreted as a traversable wormhole. We see that as the temperature decreases the chaos exponent exhibits a discontinuous fall-off from the value of order the universal bound 2π/β at the critical temperature of the phase transition, which is consistent with the expected relation between black holes and strong chaos. Interestingly, the chaos exponent is small but non-zero even in the wormhole phase. This is surprising but consistent with the observation on the decay rate of the two point function [1], and we found the chaos exponent and the decay rate indeed obey the same temperature dependence in this regime. We also studied the chaos exponent of a closely related model with single SYK term, and found that the chaos exponent of this model is always greater than that of the two coupled model in the entire parameter space. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the internal structure of a charged hairy black hole solution in the non-linear massive gravity. We first consider the impact of various configurations of massive gravity on the condensate operator and then probe the black hole interior dynamics. Like a standard holographic superconductor system, just below the critical temperature, the interior evolves through several distinct epochs, including a collapse of the Einstein-Rosen bridge, Josephson oscillations of the scalar field, and finally a Kasner (or Kasner inversion) cosmology. However, for the large massive gravity parameter, we see distinguishing features for the interior dynamics. In this regime, at a given temperature, the Einstein-Rosen bridge collapse and subsequent Josephson oscillations epochs completely disappear from the interior dynamics and the final Kasner cosmology epoch starts exactly after the would-be inner horizon and the system does not experience the Kasner inversion epoch. [ABSTRACT FROM AUTHOR]

Based on previous work that topologically nontrivial gapless modes in relativistic hydrodynamics could be found by weakly breaking the energy momentum conservation, in this paper, we study the holographic system which produces the same hydrodynamic modes. In the hydrodynamic system, one possibility to obtain the energy momentum non-conservation is to couple the system to external gravitational fields, i.e. to observe the system in a special non-inertial frame. Similar to what happens in the hydrodynamic system, a non-inertial frame version of holography indeed produces the same topologically nontrivial gapless hydrodynamic modes. We also generalize the study of topological modes in relativistic hydrodynamics to the case with one extra U(1) current and find that more complicated topological phase diagrams could exist when we consider more possibilities of the mass terms. We also discuss the possible underlying mechanism for this topological change in the spectrum when being observed in a non-inertial reference frame. [ABSTRACT FROM AUTHOR]

In this paper, we propose a concrete teleportation protocol in the SYK model based on a particle traversing a wormhole. The required operations for the communication, and insertion and extraction of the qubit, are all simple operators in terms of the basic qubits. We determine the effectiveness of this protocol, and find a version achieves almost perfect fidelity. Many features of semiclassical traversable wormholes are manifested in this setup. [ABSTRACT FROM AUTHOR]

An important open question in black hole thermodynamics is about the existence of a "mass gap" between an extremal black hole and the lightest near-extremal state within a sector of fixed charge. In this paper, we reliably compute the partition function of Reissner-Nordström near-extremal black holes at temperature scales comparable to the conjectured gap. We find that the density of states at fixed charge does not exhibit a gap; rather, at the expected gap energy scale, we see a continuum of states. We compute the partition function in the canonical and grand canonical ensembles, keeping track of all the fields appearing through a dimensional reduction on S2 in the near-horizon region. Our calculation shows that the relevant degrees of freedom at low temperatures are those of 2d Jackiw-Teitelboim gravity coupled to the electromagnetic U(1) gauge field and to an SO(3) gauge field generated by the dimensional reduction. [ABSTRACT FROM AUTHOR]

In this paper we study the Wiedemann-Franz laws for transport in 2+1 dimensions, and the action of Sl(2, ℤ) on this transport, for theories with an AdS/CMT dual. We find that Sl(2, ℤ) restricts the RG-like flow of conductivities and that the Wiedemann-Franz law is L ¯ = κ ¯ / Tσ = cg 4 2 π / 3 , from the weakly coupled gravity dual. In a self-dual theory this value is also the value of L = κ/(Tσ) in the weakly coupled field theory description. Using the formalism of a 0+1 dimensional effective action for both generalized SY Kq models and the AdS4 gravity dual, we calculate the transport coefficients and show how they can be matched at large q. We construct a generalization of this effective action that is invariant under Sl(2, ℤ) and can describe vortex conduction and integer quantum Hall effect. [ABSTRACT FROM AUTHOR]

We study phase transitions and critical phenomena in nonequilibrium steady states controlled by an electric field. We employ the D3/D7 model in the presence of a charge density and electric field at finite temperatures. The system undergoes the first-order and the second-order phase transitions under the variation of the electric field in the presence of dissipation. We numerically find that the critical exponents which we define for the nonequilibrium phase transition in this model take the mean-field values. [ABSTRACT FROM AUTHOR]

We construct a holographic model describing the striped superconductor (SSC), which is characterized by the presence of pair density waves (PDW). We explicitly demonstrate that the SSC phase is implemented as the intertwined phase of charge density waves (CDW) order and uniform superconducting (SC) order. The interplay of PDW order, CDW order as well as the uniform SC order in SSC phase is studied. It is found that the PDW order is prominent when both CDW order and uniform SC order are balanced. The critical temperature of CDW becomes higher in the presence of the uniform SC order, but its charge density amplitude is suppressed. On the other hand, the SC order is not sensitive to the presence of CDW order. We also demonstrate that among all the possible solutions, the black hole in SSC phase has the lowest free energy and thus is thermodynamically favored. [ABSTRACT FROM AUTHOR]

A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results: In a companion paper [1], these results are put in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel way to perturbatively calculate Rényi relative divergences Dγ(ρ‖σ) = trργσ1 − γ and related quantities without using replica trick as well as analytic continuation. We explicitly determine the form of the perturbative term at any order by an integral along the modular flow of the unperturbed state. By applying the prescription to a class of reduced density matrices in conformal field theory, we find that the second order term of certain linear combination of the divergences has a holographic expression in terms of bulk symplectic form, which is a one parameter generalization of the statement "Fisher information = Bulk canonical energy". [ABSTRACT FROM AUTHOR]

The theory of superconductivity can be divided into two groups depending on whether it has multi-kink solutions. For example, the BCS theory and the Gross-Neveu model have metastable multi-kink solutions whereas the conventional Ginzburg-Landau theory without higher-derivative interactions does not have any multi-kink solutions. In this paper, we systematically examine the solutions of the holographic superconductor model to find out which group the model falls into. We show that the holographic superconductor model has metastable multi-kink solutions. In this sense, we find that the holographic superconductor model falls into the category of the BCS theory and the Gross- Neveu model. We also find that the holographic superconductor model has kink crystalline condensates which are well-fitted by the Jacobi elliptic functions. [ABSTRACT FROM AUTHOR]

Recently an effective membrane theory valid in a “hydrodynamic limit” was proposed to describe entanglement dynamics of chaotic systems based on results in random quantum circuits and holographic gauge theories. In this paper, we show that this theory is robust under a large set of generalizations. In generic quench protocols we find that the membrane couples geometrically to hydrodynamics, joining quenches are captured by branes in the effective theory, and the entanglement of time evolved local operators can be computed by probing a time fold geometry with the membrane. We also demonstrate that the structure of the effective theory does not change under finite coupling corrections holographically dual to higher derivative gravity and that subleading orders in the hydrodynamic expansion can be incorporated by including higher derivative terms in the effective theory. [ABSTRACT FROM AUTHOR]

We study critical phenomena at a tricritical point associated with a chiral phase transition which emerges in the D3/D7 model in the presence of a finite baryon number density and an external magnetic field. We numerically determine critical exponents related to the thermodynamic quantities and correlation functions. We find that the values of the critical exponents agree with the mean-field values. The scaling relations between the critical exponents are satisfied, implying that the scaling hypothesis for the free energy and the correlation functions hold. Our results indicate that the critical phenomena at the tricritical point in the D3/D7 model are well described by the conventional Landau theory. [ABSTRACT FROM AUTHOR]

We study the entanglement entropy associated with a holographic RG flow from AdS7 to AdS4 × ℍ3, where ℍ3 is a 3-dimensional hyperbolic manifold with curvature κ. The dual six-dimensional RG flow is disconnected from Lorentz-invariant flows. In this context we address various notions of central charges and identify a monotonic candidate c-function that captures IR aspects of the flow. The UV behavior of the holographic entanglement entropy and, in particular its universal term, display an interesting dependence on the curvature, κ. We then contrast our holographic results with existing field theory computations in six dimensions and find a series of new corrections in curvature to the universal term in the entanglement entropy. [ABSTRACT FROM AUTHOR]

We investigate the link between quantum position-verification (QPV) and holography established in [1] using holographic quantum error correcting codes as toy models. By inserting the "temporal" scaling of the AdS metric by hand via the bulk Hamiltonian interaction strength, we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based verification is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another. [ABSTRACT FROM AUTHOR]

Pole-skipping refers to the special phenomenon that the pole and the zero of a retarded two-point Green’s function coincide at certain points in momentum space. We study the pole-skipping phenomenon in holographic Green’s functions of boundary operators that are dual to massive p-form fields and the dRGT massive gravitational fields in the AdS black hole background. Pole-skipping points for these systems are computed using the near horizon method. The relation between the pole-skipping points of massive fields and their massless counterparts is revealed. In particular, as the field mass m is varied from zero to non-zero, the pole-skipping phenomenon undergoes an abrupt change with doubled pole-skipping points found in the massive case. This arises from the breaking of gauge invariance due to the mass term and the consequent appearance of more degrees of freedom. We recover the gauge invariance using the Stueckelberg formalism by introducing auxiliary dynamical fields. The extra pole-skipping points are identified to be associated with the Stueckelberg fields. We also observe that, as the mass varies, some pole-skipping points of the wave number q may move from a non-physical region with complex q to a physical region with real q. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop the holographic mean field theory for strongly interacting fermion systems. We investigate various types of the symmetry-breakings and their effect on the spectral function. We found analytic expressions of fermion Green’s functions in the probe-limit for all types of tensor order parameter fields. We classified the spectral shapes and singularity types from the analytic Green’s function. We calculated the fermions spectral function in the full backreacted background and then compared it with the analytic results to show the reliability of analytic results in the probe limit. The fact that all the main features of the spectral features in the current condensed matter physics including gaps of s-,p- waves, nodal rings and nodal shells, the flat band of dimension 1,2,3, can be obtained in the absence of the lattice as consequences of the order and symmetry breaking pattern, is a pleaseant surprise.

We develop computational tools necessary to extend the application of Krylov complexity beyond the simple Hamiltonian systems considered thus far in the literature. As a first step toward this broader goal, we show how the Lanczos algorithm that iteratively generates the Krylov basis can be augmented to treat coherent states associated with the Jacobi group, the semi-direct product of the 3-dimensional real Heisenberg-Weyl group H1, and the symplectic group, Sp(2, ℝ) ≃ SU(1, 1). Such coherent states are physically realized as squeezed states in, for example, quantum optics [1]. With the Krylov basis for both the SU(1, 1) and Heisenberg-Weyl groups being well understood, their semi-direct product is also partially analytically tractable. We exploit this to benchmark a scheme to numerically compute the Lanczos coefficients which, in principle, generalizes to the more general Jacobi group Hn ⋊ Sp(2n, ℝ). [ABSTRACT FROM AUTHOR]

In this work we introduce a method for calculating holographic duals of BCFTs with more than two boundaries. We apply it to calculating the dynamics of entanglement entropy in a 1+1d CFT that is instantaneously split into multiple segments and calculate the entanglement entropy as a function of time for the case of two splits, showing that our approach reproduces earlier results for the double split case. Our manuscript lays the groundwork for future calculations of the entanglement entropy for more than two splits and systems at nonzero temperature. [ABSTRACT FROM AUTHOR]

We introduce a holographic model that exhibits a coexistence state of the Weyl semimetal and the topological nodal line state, providing us with a valuable tool to investigate the system’s behavior in the strong coupling regime. Nine types of bulk solutions exhibiting different IR behaviors have been identified, corresponding to nine different types of boundary states. These nine states include four distinct phases, namely the Weyl-nodal phase, the gap-nodal phase, the Weyl gap phase and the gap-gap phase, four phase boundaries, which are the Weyl-Dirac phase, the gap-Dirac phase, the Dirac-gap phase and the Dirac-nodal phase, and finally a double critical point. A phase diagram is plotted that exhibits qualitative similarity to the one obtained in the weak coupling limit. The anomalous Hall conductivity, which serves as an order parameter, and the free energy are calculated, with the latter showing the continuity of the topological phase transitions within the system. Our study highlights the similarities and differences in such a topological system between the weak and strong coupling regimes, paving the way for further experimental observations. [ABSTRACT FROM AUTHOR]

We investigate spatially inhomogeneous solutions in a top-down holographic model: the D3/D7 model which provides a holographic description of the chiral phase transition for a finite external magnetic field, chemical potential, and temperature. We numerically find a domain wall (or kink) solution in the three dimensional space, which incorporates between the chiral symmetry broken phase at the spatial infinity, under the homogeneous sources. Along with the inhomogeneity of the chiral condensate, the charge density is also spatially modulated. The modulated charge density and finite magnetic field lead to the chiral edge current close to the domain wall. We explore the dependences of those profiles on the chemical potential and temperature near the first and second order phase transition points. Our results indicate that the inhomogeneous solutions we found are in good agreement with those obtained by the Ginzburg-Landau theory in the vicinity of the transition points. [ABSTRACT FROM AUTHOR]

Abstract Flat band electronic systems exhibit a rich landscape of correlation-driven phases, both at the charge neutrality and finite electronic density, featuring exotic electromagnetic and thermodynamic responses. Motivated by these developments, in this paper, we explicitly include the effects of the chemical potential in a holographic model featuring approximately flat bands. In particular, we explore the phase diagram of this holographic flat band system as a function of the chemical potential. We find that at low temperatures and densities, the system features a nematic phase, transitioning into the Lifshitz phase as the chemical potential or temperature increases. To further characterize the ensuing phases, we investigate the optical conductivity and find that this observable shows strong anisotropies in the nematic phase.

Abstract We study the meson spectrum of the N $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory with N $$ \mathcal{N} $$ = 2 fundamental hypermultiplets for a finite electric field by using the D3/D7 model. The spectrum for scalar and vector mesons is computed by analyzing the (quasi-)normal modes for the fluctuations of the D7-brane embedding and gauge fields. In the presence of an electric field, two different phases in the background are realized: the meson and dissociation phases. In this paper, we analyze the meson spectrum of scalar and vector mesons for all ranges of the electric field and explore the effect of the electric field on the meson spectrum, that is, the Stark effect. In the meson spectrum, we observe the avoided crossing between different levels due to the coupling of fluctuations via the electric field.

In this paper we re-investigate the model of the anti-de Sitter gravity coupled to Maxwell and charged scalar fields, which has been studied as the gravitational dual to a superconductor for a long time since the famous work [Phys. Rev. Lett.101 (2008) 031601]. By numerical method, we present a novel family of solutions of holographical superconductor with excited states, and find there exists a lower critical temperature in the corresponding excited state. Moreover, we study the condensate and conductivity in the excited states. It is very interesting that the conductivity σ of each excited state has an additional pole in Im[σ] and a delta function in Re[σ] arising at the low temperature inside the gap, which is just the evidence of the existence of excited states. [ABSTRACT FROM AUTHOR]

Using covariant expansions, recent work showed that pole skipping happens in general holographic theories with bosonic fields at frequencies i(lb − s)2πT, where lb is the highest integer spin in the theory and s takes all positive integer values. We revisit this formalism in theories with gauge symmetry and upgrade the pole-skipping condition so that it works without having to remove the gauge redundancy. We also extend the formalism by incorporating fermions with general spins and interactions and show that their presence generally leads to a separate tower of pole-skipping points at frequencies i(lf − s)2πT, lf being the highest half-integer spin in the theory and s again taking all positive integer values. We also demonstrate the practical value of this formalism using a selection of examples with spins 0, 1 2 , 1, 3 2 , 2. [ABSTRACT FROM AUTHOR]

We study the entanglement transition in monitored Brownian SYK chains in the large-N limit. Without measurement the steady state n-th Rényi entropy is obtained by summing over a class of solutions, and is found to saturate to the Page value in the n → 1 limit. In the presence of measurements, the analytical continuation n → 1 is performed using the cyclic symmetric solution. The result shows that as the monitoring rate increases, a continuous von Neumann entanglement entropy transition from volume-law to area-law occurs at the point of replica symmetry unbreaking. [ABSTRACT FROM AUTHOR]

We revisit the case of a real scalar field in global AdS4 subject to a periodic driving. We address the issue of adiabatic preparation and deformation of a time-periodic solution dual to a Floquet condensate. Then we carefully study the case of driving close to the normal mode resonant frequencies. We examine different slow protocols that adiabatically change the amplitude and/or the frequency of the driving. Traversing a normal mode frequency has very different results depending upon the sense of the frequency modulation. Generally, in the growing sense, the geometry reaches a periodically-modulated state, whereas in the opposite one, it collapses into a black hole. We study the suppression points. These are periodic solutions that are dual to a scalar field with vanishing v.e.v., 〈ϕ〉 = 0, instead of vanishing source. We also investigate quasi-periodic solutions that are prepared by driving with a combination of two normal resonant frequencies. We observe that, while the driving is on, the non-linear cascading towards higher frequencies is strongly suppressed. However, once the driving is switched off, the cascading takes over again, and in some cases, it eventually brings the solution to a collapse. Finally, we study the driving by a non-coherent thermal ensemble of resonant drivings that model stochastic noise. Our numerical results suggest the existence of stable regular solutions at sufficiently low temperature and a transition to collapse above some threshold. [ABSTRACT FROM AUTHOR]

We use holography to compute the exact form of retarded Green's functions for a scalar operator with conformal dimension ∆ in a thermal CFT and in its related counterpart with chemical potential in R1 × H3. In our analysis, we recast the wave equation of a scalar field in the normal form of Heun's equation in the dual gravity theories described by the AdS hyperbolic blackhole and its charged version. Heun's equation is identified to the semiclassical limit of the BPZ equation for a five-point correlator with one degenerate field insertion in the Liouville theory on the Riemann sphere. The crossing symmetry of conformal block in the Liouville theory eventually gives rise to a set of connection formulas among the solutions of Heun's equation evaluated at different regular singularities. We use the connection formula to reproduce the leading order behaviors of the scalar field near the horizon as well as near the boundary and achieve the exact form of the retarded thermal Green's function. We show a recipe to obtain the exact retarded Green's function for a thermal CFT dual to AdS blackbrane from a high-temperature limit accompanied by a complex mapping on AdS hyperbolic blackhole. Moreover, we show the retarded Green's function for the boundary CFT of Rindler AdS spacetime admits a free integer parameter. [ABSTRACT FROM AUTHOR]

We employ the holographic quench technique to drive Einstein-Maxwell-scalar (EMs) black holes out of equilibrium and study the real-time dynamics therein. From the fully nonlinear dynamical simulations, a dynamically unstable Reissner-Nordström anti-de Sitter (RN-AdS) black hole can be scalarized spontaneously after an arbitrarily small quench. On the other hand, a dynamically stable scalarized black hole can be descalarized after a quench of sufficient strength. Interestingly, on the way to descalarization, the scalarized black hole behaves like a holographic superfluid, undergoing a dynamical transition from oscillatory to non-oscillatory decay. Such behaviors are related to the spectrums of quasi-normal modes of scalarized black holes, where the dominant mode migrates toward the imaginary axis with increasing quench strength. In addition, due to the ℤ2-symmetry preserved by the model, the ground state is degenerate. We find that there exists a threshold for the quench strength that induces a dynamical transition of the gravitational system from one degenerate ground state to the other. Near the threshold, the gravitational system is attracted to an excited state, that is, a RN-AdS black hole with dynamical instability. [ABSTRACT FROM AUTHOR]

We find, in the AdS/CFT, a source on the boundary which generates one wave packet drawing a null geodesic inside the bulk. Once such a wave packet dives into the bulk, it comes back to the boundary after a specific time, at which the expectation value of the corresponding boundary operator finally stands up. Since this behavior strongly reflects the existence of the holographic spacetime, our technique will be helpful in identifying holographic materials. [ABSTRACT FROM AUTHOR]

Under the Heisenberg evolution in chaotic quantum systems, initially simple operators evolve into complicated ones and ultimately cover the whole operator space. We study the growth of the operator "size" in this process, which is related to the out-of-time-order correlator (OTOC). We derive the full time evolution of the size distribution in large N quantum mechanics of Majorana fermions. As examples, we apply the formalism to the Brownian SYK model (infinite temperature) and the large q SYK model (finite temperature). [ABSTRACT FROM AUTHOR]