A bound on chaos

Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas

doi:10.1007/jhep08(2016)106

Cited by 2,202 publications

(3,310 citation statements)

References 55 publications

Supporting

114

Mentioning

3,186

Contrasting

Unclassified

Order By: Relevance

“…At relatively low energies this growth matches the one expected in a theory of gravity [9,10,11], which saturates the chaos bound [12].…”

Section: Introductionsupporting

confidence: 82%

“…Also, the term comes with a sign that is forbidden by the argument of [12]. So, by itself, F h =2 would not be an allowable four point function.…”

Section: Analytic Continuation To the Chaos Regionmentioning

confidence: 99%

“…where we have split the thermal density matrix into four factors y as in [12]. In a conformal theory, this can be obtained from the Euclidean correlator on the line, by mapping to the finite temperature circle using (3.48) and then continuing to real time.…”

Section: Analytic Continuation To the Chaos Regionmentioning

confidence: 99%

“…One can imagine getting rid of the fundamental fermions by viewing the couplings j ijkl as dynamical with very slow dynamics so that they are effectively constant [19]. At the order we are working, this gives the same equations 12 . Once we make the couplings j ijkl dynamical, we can gauge the O(N ) symmetry.…”

Section: The Fermionsmentioning

confidence: 99%

See 3 more Smart Citations

Remarks on the Sachdev-Ye-Kitaev model

2016

Self Cite

View full text Add to dashboard Cite

We study a quantum mechanical model proposed by Sachdev, Ye and Kitaev. The model consists of N Majorana fermions with random interactions of a few fermions at a time. It it tractable in the large N limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies where it develops an emergent conformal symmetry. We study two and four point functions of the fundamental fermions. This provides the spectrum of physical excitations for the bilocal field.The emergent conformal symmetry is a reparametrization symmetry, which is spontaneously broken to SL(2, R), leading to zero modes. These zero modes are lifted by a small residual explicit breaking, which produces an enhanced contribution to the four point function. This contribution displays a maximal Lyapunov exponent in the chaos region (out of time ordered correlator). We expect these features to be universal properties of large N quantum mechanics systems with emergent reparametrization symmetry.This article is largely based on talks given by Kitaev [1], which motivated us to work out the details of the ideas described there.

show abstract

“…At relatively low energies this growth matches the one expected in a theory of gravity [9,10,11], which saturates the chaos bound [12].…”

Section: Introductionsupporting

confidence: 82%

“…Also, the term comes with a sign that is forbidden by the argument of [12]. So, by itself, F h =2 would not be an allowable four point function.…”

Section: Analytic Continuation To the Chaos Regionmentioning

confidence: 99%

Section: Analytic Continuation To the Chaos Regionmentioning

confidence: 99%

Section: The Fermionsmentioning

confidence: 99%

See 2 more Smart Citations

Remarks on the Sachdev-Ye-Kitaev model

2016

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is a simple model with solvable aspects which exhibits interesting connections to quantum chaos [9][10][11][12][13], black hole physics and quantum gravity in 1+1 dimensions [14][15][16][17][18][19][20]. Understanding these relations better in this model, and potential extensions of it, is an active and exciting research direction that promises to improve our knowledge of holography.…”

Section: Introduction and Conclusionmentioning

confidence: 99%

Higher dimensional generalizations of the SYK model

Berkooz

Narayan

Rozali

et al. 2017

J. High Energ. Phys.

142

127

View full text Add to dashboard Cite

We discuss a 1+1 dimensional generalization of the Sachdev-Ye-Kitaev model. The model contains N Majorana fermions at each lattice site with a nearest-neighbour hopping term. The SYK random interaction is restricted to low momentum fermions of definite chirality within each lattice site. This gives rise to an ordinary 1+1 field theory above some energy scale and a low energy SYK-like behavior. We exhibit a class of low-pass filters which give rise to a rich variety of hyperscaling behaviour in the IR. We also discuss another set of generalizations which describes probing an SYK system with an external fermion, together with the new scaling behavior they exhibit in the IR.

show abstract

Out‐of‐time‐ordered correlators in many‐body localized systems

2016

View full text Add to dashboard Cite

In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities typically spread only within a finite distance. Is it possible to detect the logarithmic light cone generated by a local perturbation from the response of a local operator at a later time? We numerically calculate various correlators in the random-field Heisenberg chain. While the equilibrium retarded correlator A(t = 0)B (t > 0) is not sensitive to the unbounded information propagation, the out-of-time-ordered corre- In the presence of disorder, localization can occur not only in single-particle systems [1], but also in interacting many-body systems [2][3][4][5][6][7][8][9][10][11][12]. The former is known as Anderson localization (AL), and the latter is called manybody localization (MBL). Neither AL nor MBL systems transfer energy, charge, or other local conserved quantities: Changes in energy or charge at position x = 0 from equilibrium can spread and lead to changes in the corresponding quantity only within a finite distance |x| < L 0 , where L 0 is the localization length.A characteristic feature that distinguishes MBL from AL lies in the dynamics of entanglement after a global quench. Initialized in a random product state at time t = 0, the half-chain entanglement entropy remains bounded in AL systems [13], but grows logarithmically with time in MBL systems [14][15][16][17][18][19][20][21]. In sharp contrast to the transport phenomena, the unbounded growth of entanglement in MBL systems suggests that information propagates throughout the system, although very slowly.The propagation of information can be formulated by adapting the Lieb-Robinson (LR) bound [22][23][24] to the present context. In particular, it is manifested as the noncommutativity of a local operator A at x = 0 and t = 0 with another local operator B at position x and evolved for some time t . In MBL systems, the operator norm of the commutator [A(0, 0), B (x, t )] is non-negligible inside a light cone whose radius is given by |x| ∼ log |t |, and decays exponentially with distance outside the light cone, i.e. [25],after averaging over disorder. Here, B (x, t ) = e i H t B (x, 0)e −i H t is the time-evolved operator; · is the operator norm (the largest singular value); C , v LR , ξ are positive constants.Is it possible to detect the logarithmic light cone (LLC) with equilibrium correlators of A(0, 0) and B (x, t )? Arguably the most straightforward approach is to measure the commutator in the LR bound (1) on equilibrium states, i.e., thermal states or eigenstates, using the Kubo formula in linear response theory: (2) where U = e −i Aτ with τ 1 is a local unitary perturbation if A is Hermitian. The first and second terms on the lefthand side of (2) are the expectation values of B in the presence and absence of the perturbation U , respectively. The difference is the effect of U observed by measuring B . Note that U and B can be, but do not have to be, chosen as the op...

show abstract

A bound on chaos

Cited by 2,202 publications

References 55 publications

Remarks on the Sachdev-Ye-Kitaev model

Remarks on the Sachdev-Ye-Kitaev model

Higher dimensional generalizations of the SYK model

Out‐of‐time‐ordered correlators in many‐body localized systems

Contact Info

Product

Resources

About