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Gu, Yingfei, Kitaev, Alexei, and Zhang, Pengfei
 Subjects

High Energy Physics  Theory and Condensed Matter  Strongly Correlated Electrons
 Abstract

Outoftimeorder correlators (OTOCs) are a standard measure of quantum chaos. Of the four operators involved, one pair may be regarded as a source and the other as a probe. A usual approach, applicable to large$N$ systems such as the SYK model, is to replace the actual source with some meanfield perturbation and solve for the probe correlation function on the double Keldysh contour. We show how to obtain the OTOC by combining two such solutions for perturbations propagating forward and backward in time. These dynamical perturbations, or scrambling modes, are considered on the thermofield double background and decomposed into a coherent and an incoherent part. For the large$q$ SYK, we obtain the OTOC in a closed form. We also prove a previously conjectured relation between the Lyapunov exponent and highfrequency behavior of the spectral function.
Comment: 35 pages, 9 figures. v2: a typo fixed, a reference added. v3: two typos fixed, a reference added
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Wen, Xueda, Gu, Yingfei, Vishwanath, Ashvin, and Fan, Ruihua
 Subjects

Condensed Matter  Statistical Mechanics, Condensed Matter  Strongly Correlated Electrons, High Energy Physics  Theory, and Mathematical Physics
 Abstract

In this sequel (to [Phys. Rev. Res. 3, 023044(2021)], arXiv:2006.10072), we study randomly driven $(1+1)$ dimensional conformal field theories (CFTs), a family of quantum manybody systems with soluble nonequilibrium quantum dynamics. The sequence of driving Hamiltonians is drawn from an independent and identically distributed random ensemble. At each driving step, the deformed Hamiltonian only involves the energymomentum density spatially modulated at a single wavelength and therefore induces a M\"obius transformation on the complex coordinates. The nonequilibrium dynamics is then determined by the corresponding sequence of M\"obius transformations, from which the Lyapunov exponent $\lambda_L$ is defined. We use Furstenberg's theorem to classify the dynamical phases and show that except for a few \emph{exceptional points} that do not satisfy Furstenberg's criteria, the random drivings always lead to a heating phase with the total energy growing exponentially in the number of driving steps $n$ and the subsystem entanglement entropy growing linearly in $n$ with a slope proportional to central charge $c$ and the Lyapunov exponent $\lambda_L$. On the contrary, the subsystem entanglement entropy at an exceptional point could grow as $\sqrt{n}$ while the total energy remains to grow exponentially. In addition, we show that the distributions of the operator evolution and the energy density peaks are also useful characterizations to distinguish the heating phase from the exceptional points: the heating phase has both distributions to be continuous, while the exceptional points could support finite convex combinations of Dirac measures depending on their specific type. In the end, we compare the field theory results with the lattice model calculations for both the entanglement and energy evolution and find remarkably good agreement.
Comment: 65 pages, 1 table, many figures; This is part 2 on randomly driven CFTs
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Zhang, Pengfei, Gu, Yingfei, and Kitaev, Alexei
 J. High Energ. Phys. 2021, 94 (2021)
 Subjects

High Energy Physics  Theory and Condensed Matter  Strongly Correlated Electrons
 Abstract

We argue that "stringy" effects in a putative gravitydual picture for SYKlike models are related to the branching time, a kinetic coefficient defined in terms of the retarded kernel. A bound on the branching time is established assuming that the leading diagrams are ladders with thin rungs. Thus, such models are unlikely candidates for subAdS holography. In the weak coupling limit, we derive a relation between the branching time, the Lyapunov exponent, and the quasiparticle lifetime using two different approximations.
Comment: 31 pages, 3 figures
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Fan, Ruihua, Gu, Yingfei, Vishwanath, Ashvin, and Wen, Xueda
 SciPost Phys. 10, 049 (2021)
 Subjects

High Energy Physics  Theory, Condensed Matter  Statistical Mechanics, Condensed Matter  Strongly Correlated Electrons, and Mathematical Physics
 Abstract

In this work, we study nonequilibrium dynamics in Floquet conformal field theories (CFTs) in 1+1D, in which the driving Hamiltonian involves the energymomentum density spatially modulated by an arbitrary smooth function. This generalizes earlier work which was restricted to the sinesquare deformed type of Floquet Hamiltonians, operating within a $\mathfrak{sl}_2$ subalgebra. Here we show remarkably that the problem remains soluble in this generalized case which involves the full Virasoro algebra, based on a geometrical approach. It is found that the phase diagram is determined by the stroboscopic trajectories of operator evolution. The presence/absence of spatial fixed points in the operator evolution indicates that the driven CFT is in a heating/nonheating phase, in which the entanglement entropy grows/oscillates in time. Additionally, the heating regime is further subdivided into a multitude of phases, with different entanglement patterns and spatial distribution of energymomentum density, which are characterized by the number of spatial fixed points. Phase transitions between these different heating phases can be achieved simply by changing the duration of application of the driving Hamiltonian. We demonstrate the general features with concrete CFT examples and compare the results to lattice calculations and find remarkable agreement.
Comment: 36 pages, 11 figures
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Zhang, Pengfei and Gu, Yingfei
 SciPost Phys. 9, 079 (2020)
 Subjects

Condensed Matter  Quantum Gases and Condensed Matter  Strongly Correlated Electrons
 Abstract

We study the quantum dynamics of BoseEinstein condensates when the scattering length is modulated periodically or quasiperiodically in time within the Bogoliubov framework. For the periodically driven case, we consider two protocols where the modulation is a squarewave or a sinewave. In both protocols for each fixed momentum, there are heating and nonheating phases, and a phase boundary between them. The two phases are distinguished by whether the number of excited particles grows exponentially or not. For the quasiperiodically driven case, we again consider two protocols: the squarewave quasiperiodicity, where the excitations are generated for almost all parameters as an analog of the Fibonaccitype quasicrystal; and the sinewave quasiperiodicity, where there is a finite measure parameter regime for the nonheating phase. We also plot the analogs of the Hofstadter butterfly for both protocols.
Comment: 23 pages, 7 figures
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Chen, Xiao, Gu, Yingfei, and Lucas, Andrew
 SciPost Phys. 9, 071 (2020)
 Subjects

Quantum Physics, Condensed Matter  Strongly Correlated Electrons, and High Energy Physics  Theory
 Abstract

We study quantum manybody systems with a global U(1) conservation law, focusing on a theory of $N$ interacting fermions with charge conservation, or $N$ interacting spins with one conserved component of total spin. We define an effective operator size at finite chemical potential through suitably regularized outoftimeordered correlation functions. The growth rate of this densitydependent operator size vanishes algebraically with charge density; hence we obtain new bounds on Lyapunov exponents and butterfly velocities in charged systems at a given density, which are parametrically stronger than any LiebRobinson bound. We argue that the density dependence of our bound on the Lyapunov exponent is saturated in the charged SachdevYeKitaev model. We also study random automaton quantum circuits and Brownian SachdevYeKitaev models, each of which exhibit a different density dependence for the Lyapunov exponent, and explain the discrepancy. We propose that our results are a cartoon for understanding Planckianlimited energyconserving dynamics at finite temperature.
Comment: 17+3 pages; 3+1 figures; v2, v3: minor changes
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Wen, Xueda, Fan, Ruihua, Vishwanath, Ashvin, and Gu, Yingfei
 Phys. Rev. Research 3, 023044 (2021)
 Subjects

Condensed Matter  Statistical Mechanics, Condensed Matter  Mesoscale and Nanoscale Physics, Condensed Matter  Strongly Correlated Electrons, High Energy Physics  Theory, and Mathematical Physics
 Abstract

In this paper and its sequel, we study nonequilibrium dynamics in driven 1+1D conformal field theories (CFTs) with periodic, quasiperiodic, and random driving. We study a soluble family of drives in which the Hamiltonian only involves the energymomentum density spatially modulated at a single wavelength. The resulting time evolution is then captured by a M\"obius coordinate transformation. In this Part I, we establish the general framework and focus on the first two classes. In periodically driven CFTs, we generalize earlier work and study the generic features of entanglement/energy evolution in different phases, i.e. the heating, nonheating phases and the phase transition between them. In quasiperiodically driven CFTs, we mainly focus on the case of driving with a Fibonacci sequence. We find that (i) the nonheating phases form a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents (which characterize the growth rate of the entanglement entropy and energy) exhibit selfsimilarity, and can be arbitrarily small; (iii) the heating phase exhibits periodicity in the location of spatial structures at the Fibonacci times; (iv) one can find exactly the nonheating fixed point, where the entanglement entropy/energy oscillate at the Fibonacci numbers, but grow logarithmically/polynomially at the nonFibonacci numbers; (v) for certain choices of driving Hamiltonians, the nonheating phases of the Fibonacci driving CFT can be mapped to the energy spectrum of electrons propagating in a Fibonacci quasicrystal. In addition, another quasiperiodically driven CFT with an AubryAndr\'e like sequence is also studied. We compare the CFT results to lattice calculations and find remarkable agreement.
Comment: 82 pages, many figures; reference added
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Guo, Haoyu, Gu, Yingfei, and Sachdev, Subir
 Annals of Physics 418, 168202 (2020)
 Subjects

Condensed Matter  Strongly Correlated Electrons and High Energy Physics  Theory
 Abstract

The most puzzling aspect of the 'strange metal' behavior of correlated electron compounds is that the linear in temperature resistivity often extends down to low temperatures, lower than natural microscopic energy scales. We consider recently proposed deconfined critical points (or phases) in models of electrons in large dimension lattices with random nearestneighbor exchange interactions. The criticality is in the class of SachdevYeKitaev models, and exhibits a time reparameterization soft mode representing gravity in dual holographic theories. We compute the low temperature resistivity in a large $M$ limit of models with SU($M$) spin symmetry, and find that the dominant temperature dependence arises from this soft mode. The resistivity is linear in temperature down to zero temperature at the critical point, with a coefficient universally proportional to the product of the residual resistivity and the coefficient of the linear in temperature specific heat. We argue that the time reparameterization soft mode offers a promising and generic mechanism for resolving the strange metal puzzle.
Comment: 35 pages, 4 figures
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Gu, Yingfei, Kitaev, Alexei, Sachdev, Subir, and Tarnopolsky, Grigory
 JHEP 02 (2020) 157
 Subjects

High Energy Physics  Theory and Condensed Matter  Strongly Correlated Electrons
 Abstract

We describe numerous properties of the SachdevYeKitaev model for complex fermions with $N\gg 1$ flavors and a global U(1) charge. We provide a general definition of the charge in the $(G,\Sigma)$ formalism, and compute its universal relation to the infrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the manybody density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.
Comment: v2: 75 pages, 14 figures, new appendix A.6 and references added; v1: 73 pages, 13 figures, 1 spooky propagator
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Gu, Yingfei and Qi, XiaoLiang
 Subjects

Condensed Matter  Statistical Mechanics, Condensed Matter  Strongly Correlated Electrons, Computer Science  Computational Complexity, and Mathematics  Combinatorics
 Abstract

Recently, Hao Huang proved the Sensitivity Conjecture, an important result about complexity measures of Boolean functions. We will discuss how this simple and elegant proof turns out to be closely related to physics concepts of the JordanWigner transformation and Majorana fermions. This note is not intended to contain original results. Instead, it is a translation of the math literature in a language that is more familiar to physicists, which helps our understanding and hopefully may inspire future works along this direction.
Comment: 13 pages, 7 figures
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11. Emergent Spatial Structure and Entanglement Localization in Floquet Conformal Field Theory [2019]

Fan, Ruihua, Gu, Yingfei, Vishwanath, Ashvin, and Wen, Xueda
 Phys. Rev. X 10, 031036 (2020)
 Subjects

Condensed Matter  Strongly Correlated Electrons, Condensed Matter  Quantum Gases, and High Energy Physics  Theory
 Abstract

We study the energy and entanglement dynamics of $(1+1)$D conformal field theories (CFTs) under a Floquet drive with the sinesquare deformed (SSD) Hamiltonian. Previous work has shown this model supports both a nonheating and a heating phase. Here we analytically establish several robust and `superuniversal' features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and antichiral peak, which leads to an exponential growth in the total energy. The peak locations are set by fixed points of the M\"obius transformation. Second, all of the quantum entanglement is shared between these two peaks. In each driving period, a number of Bell pairs are generated, with one member pumped to the chiral peak, and the other member pumped to the antichiral peak. These Bell pairs are localized and accumulate at these two peaks, and can serve as a source of quantum entanglement. Third, in both the heating and nonheating phases we find that the total energy is related to the half system entanglement entropy by a simple relation $E(t)\propto c \exp \left( \frac{6}{c}S(t) \right)$ with $c$ being the central charge. In addition, we show that the nonheating phase, in which the energy and entanglement oscillate in time, is unstable to small fluctuations of the driving frequency in contrast to the heating phase. Finally, we point out an analogy to the periodically driven harmonic oscillator which allows us to understand global features of the phases, and introduce a quasiparticle picture to explain the spatial structure, which can be generalized to setups beyond the SSD construction.
Comment: 41 pages, 19 figures
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Guo, Haoyu, Gu, Yingfei, and Sachdev, Subir
 Phys. Rev. B 100, 045140 (2019)
 Subjects

Condensed Matter  Strongly Correlated Electrons, Condensed Matter  Disordered Systems and Neural Networks, Condensed Matter  Statistical Mechanics, and High Energy Physics  Theory
 Abstract

We compute the transport and chaos properties of lattices of quantum SachdevYeKitaev islands coupled by single fermion hopping, and with the islands coupled to a large number of local, low energy phonons. We find two distinct regimes of linearintemperature ($T$) resistivity, and describe the crossover between them. When the electronphonon coupling is weak, we obtain the `incoherent metal' regime, where there is nearmaximal chaos with front propagation at a butterfly velocity $v_B$, and the associated diffusivity $D_{\rm chaos} = v_B^2/(2 \pi T)$ closely tracks the energy diffusivity. On the other hand, when the electronphonon coupling is strong, and the linear resistivity is largely due to nearelastic scattering of electrons off nearly free phonons, we find that the chaos is far from maximal and spreads diffusively. We also describe the crossovers to low $T$ regimes where the electronic quasiparticles are well defined.
Comment: 25 pages, 26 figures; v2: Added a new Sec.II and a new figure; Added clarifications; Corrected Typos
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Gu, Yingfei and Kitaev, Alexei
 J. High Energ. Phys. (2019) 2019: 75
 Subjects

High Energy Physics  Theory and Condensed Matter  Strongly Correlated Electrons
 Abstract

We derive an identity relating the growth exponent of earlytime OTOCs, the preexponential factor, and a third number called "branching time". The latter is defined within the dynamical meanfield framework, namely, in terms of the retarded kernel. This identity can be used to calculate stringy effects in the SYK and similar models, we also explicitly define "strings" in this context. As another application, we consider an SYK chain. If the coupling strength $\beta J$ is above a certain threshold and nonlinear (in the magnitude of OTOCs) effects are ignored, the exponent in the butterfly wavefront is exactly $2\pi/\beta$.
Comment: 20 pages, 7 figures
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Chaturvedi, Pankaj, Gu, Yingfei, Song, Wei, and Yu, Boyang
 Subjects

High Energy Physics  Theory
 Abstract

We discuss the connections between the complex SYK model at the conformal limit and warped conformal field theories. Both theories have an $SL(2,R) \times U(1)$ global symmetry. We present comparisons on symmetries, correlation functions, the effective action and the entropy formula. We also use modular covariance to reinterpret results in the complex SYK model.
Comment: 27 pages
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Zhang, Pengfei, Gu, Yingfei, and Kitaev, Alexei
 Journal of High Energy Physics. 2021(3)
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16. Fast scrambling on sparse graphs [2018]

Bentsen, Gregory, Gu, Yingfei, and Lucas, Andrew
 Proceedings of the National Academy of Sciences 116, 6689 (2019)
 Subjects

Condensed Matter  Strongly Correlated Electrons, Condensed Matter  Statistical Mechanics, High Energy Physics  Theory, and Quantum Physics
 Abstract

Given a quantum manybody system with fewbody interactions, how rapidly can quantum information be hidden during time evolution? The fast scrambling conjecture is that the time to thoroughly mix information among N degrees of freedom grows at least logarithmically in N. We derive this inequality for generic quantum systems at infinite temperature, bounding the scrambling time by a finite decay time of local quantum correlations at late times. Using LiebRobinson bounds, generalized SachdevYeKitaev models, and random unitary circuits, we propose that a logarithmic scrambling time can be achieved in most quantum systems with sparse connectivity. These models also elucidate how quantum chaos is not universally related to scrambling: we construct random fewbody circuits with infinite Lyapunov exponent but logarithmic scrambling time. We discuss analogies between quantum models on graphs and quantum black holes, and suggest methods to experimentally study scrambling with as many as 100 sparselyconnected quantum degrees of freedom.
Comment: 11+24 pages; 1+3 figures. v2: minor revisions, added references. v3: many revisions to improve presentation
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Fu, Wenbo, Gu, Yingfei, Sachdev, Subir, and Tarnopolsky, Grigory
 Phys. Rev. B 98, 075150 (2018)
 Subjects

Condensed Matter  Strongly Correlated Electrons and High Energy Physics  Theory
 Abstract

We describe the phases of a solvable $t$$J$ model of electrons with infiniterange, and random, hopping and exchange interactions, similar to those in the SachdevYeKitaev models. The electron fractionalizes, as in an `orthogonal metal', into a fermion $f$ which carries both the electron spin and charge, and a boson $\phi$. Both $f$ and $\phi$ carry emergent $\mathbb{Z}_2$ gauge charges. The model has a phase in which the $\phi$ bosons are gapped, and the $f$ fermions are gapless and critical, and so the electron spectral function is gapped. This phase can be considered as a toy model for the underdoped cuprates. The model also has an extended, critical, `quasiHiggs' phase where both $\phi$ and $f$ are gapless, and the electron operator $\sim f \phi$ has a Fermi liquidlike $1/\tau$ propagator in imaginary time, $\tau$. So while the electron spectral function has a Fermi liquid form, other properties are controlled by $\mathbb{Z}_2$ fractionalization and the anomalous exponents of the $f$ and $\phi$ excitations. This `quasiHiggs' phase is proposed as a toy model of the overdoped cuprates. We also describe the critical state separating these two phases.
Comment: 30 pages, 9 figures
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You, YiZhuang and Gu, Yingfei
 Phys. Rev. B 98, 014309 (2018)
 Subjects

Quantum Physics, Condensed Matter  Statistical Mechanics, and High Energy Physics  Theory
 Abstract

We introduce the concept of entanglement features of unitary gates, as a collection of exponentiated entanglement entropies over all bipartitions of input and output channels. We obtained the general formula for timedependent $n$thRenyi entanglement features for unitary gates generated by random Hamiltonian. In particular, we propose an Ising formulation for the 2ndRenyi entanglement features of random Hamiltonian dynamics, which admits a holographic tensor network interpretation. As a general description of entanglement properties, we show that the entanglement features can be applied to several dynamical measures of thermalization, including the outoftimeorder correlation and the entanglement growth after a quantum quench. We also analyze the YoshidaKitaev probabilistic protocol for random Hamiltonian dynamics.
Comment: 15 pages, 10 figures
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Guo, Haoyu, Gu, Yingfei, and Sachdev, Subir
 In
Annals of Physics July 2020 418
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Huang, Yichen and Gu, Yingfei
 Physical Review D 100 (4), 041901 (Rapid Communication), 2019
 Subjects

High Energy Physics  Theory, Condensed Matter  Statistical Mechanics, Condensed Matter  Strongly Correlated Electrons, and Quantum Physics
 Abstract

We study the entanglement entropy of eigenstates (including the ground state) of the SachdevYeKitaev model. We argue for a volume law, whose coefficient can be calculated analytically from the density of states. The coefficient depends on not only the energy density of the eigenstate but also the subsystem size. Very recent numerical results of Liu, Chen, and Balents confirm our analytical results.
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