arXiv subjects

Thomas Hartman

Publications and source records attributed to Thomas Hartman.

At least 55 records · Page 3Linked to original sources

arXiv2017

Black Hole Collapse in the 1/c Expansion

We present a first-principles CFT calculation corresponding to the spherical collapse of a shell of matter in three dimensional quantum gravity. In field theory terms, we describe the equilibration process, from early times to thermalization, of a CFT following a sudden injection of energy at time t=0. By formulating a continuum version of Zamolodchikov's monodromy method to calculate conformal blocks at large central charge c, we give a framework to compute a general class of probe observables in the collapse state, incorporating the full backreaction of matter fields on the dual geometry. This is illustrated by calculating a scalar field two-point function at time-like separation and the time-dependent entanglement entropy of an interval, both showing thermalization at late times. The results are in perfect agreement with previous gravity calculations in the AdS$_3$-Vaidya geometry. Information loss appears in the CFT as an explicit violation of unitarity in the 1/c expansion, restored by nonperturbative corrections.

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arXiv2016

Averaged Null Energy Condition from Causality

Unitary, Lorentz-invariant quantum field theories in flat spacetime obey microcausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, $\int du T_{uu}$, must be positive. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to $n$-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form $\int du X_{uuu\cdots u} \geq 0$. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.

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arXiv2016

Causality Constraints in Conformal Field Theory

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interactions of low-lying operators. As an application, we use the bootstrap to rederive the well known sign constraint on the $(\partialϕ)^4$ coupling in effective field theory, from a dual CFT. We also find constraints on theories with higher spin conserved currents. Our analysis is restricted to scalar correlators, but we argue that similar methods should also impose nontrivial constraints on the interactions of spinning operators.

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arXiv2016

A New Spin on Causality Constraints

Causality in a shockwave state is related to the analytic properties of a four-point correlation function. Extending recent results for scalar probes, we show that this constrains the couplings of the stress tensor to light spinning operators in conformal field theory, and interpret these constraints in terms of the interaction with null energy. For spin-1 and spin-2 conserved currents in four dimensions, the resulting inequalities are a subset of the Hofman-Maldacena conditions for positive energy deposition. It is well known that energy conditions in holographic theories are related to causality on the gravity side; our results make a connection on the CFT side, and extend it to non-holographic theories.

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arXiv2015

Speed Limits for Entanglement

We show that in any relativistic system, entanglement entropy obeys a speed limit set by the entanglement in thermal equilibrium. The bound is derived from inequalities on relative entropy with respect to a thermal reference state. Thus the thermal state constrains far-from-equilibrium entanglement dynamics whether or not the system actually equilibrates, in a manner reminiscent of fluctuation theorems in classical statistical mechanics. A similar shape-dependent bound constrains the full nonlinear time evolution, supporting a simple physical picture for entanglement propagation that has previously been motivated by holographic calculations in conformal field theory. We discuss general quantum field theories in any spacetime dimension, but also derive some results of independent interest for thermal relative entropy in 1+1d CFT.

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arXiv2015

Entanglement Scrambling in 2d Conformal Field Theory

We investigate how entanglement spreads in time-dependent states of a 1+1 dimensional conformal field theory (CFT). The results depend qualitatively on the value of the central charge. In rational CFTs, which have central charge below a critical value, entanglement entropy behaves as if correlations were carried by free quasiparticles. This leads to long-term memory effects, such as spikes in the mutual information of widely separated regions at late times. When the central charge is above the critical value, the quasiparticle picture fails. Assuming no extended symmetry algebra, any theory with $c>1$ has diminished memory effects compared to the rational models. In holographic CFTs, with $c \gg 1$, these memory effects are eliminated altogether at strong coupling, but reappear after the scrambling time $t \gtrsim β\log c$ at weak coupling.

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arXiv2015

Topological aspects of generalized gravitational entropy

The holographic prescription for computing entanglement entropy requires that the bulk extremal surface, whose area encodes the amount of entanglement, satisfies a homology constraint. Usually this is stated as the requirement of a (spacelike) interpolating surface that connects the region of interest and the extremal surface. We investigate to what extent this constraint is upheld by the generalized gravitational entropy argument, which relies on constructing replica symmetric q-fold covering spaces of the bulk, branched at the extremal surface. We prove (at the level of topology) that the putative extremal surface satisfies the homology constraint if and only if the corresponding branched cover can be constructed for every positive integer q. We give simple examples to show that homology can be violated if the cover exists for some values of q but not others, along with some other issues.

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arXiv2014

Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches

We consider the entanglement entropy in 2d conformal field theory in a class of excited states produced by the insertion of a heavy local operator. These include both high-energy eigenstates of the Hamiltonian and time-dependent local quenches. We compute the universal contribution from the stress tensor to the single interval Renyi entropies and entanglement entropy, and conjecture that this dominates the answer in theories with a large central charge and a sparse spectrum of low-dimension operators. The resulting entanglement entropies agree precisely with holographic calculations in three-dimensional gravity. High-energy eigenstates are dual to microstates of the BTZ black hole, so the corresponding holographic calculation is a geodesic length in the black hole geometry; agreement between these two answers demonstrates that entanglement entropy thermalizes in individual microstates of holographic CFTs. For local quenches, the dual geometry is a highly boosted black hole or conical defect. On the CFT side, the rise in entanglement entropy after a quench is directly related to the monodromy of a Virasoro conformal block.

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arXiv2014

Universal Spectrum of 2d Conformal Field Theory in the Large c Limit

Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit $T \to \infty$, and a universal spectrum in the Cardy regime, $Δ\to \infty$. We show that a much stronger form of universality holds in theories with a large central charge $c$ and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all $Δ\geq c/6$. The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal `enigma' saddlepoints in the thermodynamics of 3d gravity.

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arXiv2013

Time Evolution of Entanglement Entropy from Black Hole Interiors

We compute the time-dependent entanglement entropy of a CFT which starts in relatively simple initial states. The initial states are the thermofield double for thermal states, dual to eternal black holes, and a particular pure state, dual to a black hole formed by gravitational collapse. The entanglement entropy grows linearly in time. This linear growth is directly related to the growth of the black hole interior measured along "nice" spatial slices. These nice slices probe the spacelike direction in the interior, at a fixed special value of the interior time. In the case of a two-dimensional CFT, we match the bulk and boundary computations of the entanglement entropy. We briefly discuss the long time behavior of various correlators, computed via classical geodesics or surfaces, and point out that their exponential decay comes about for similar reasons. We also present the time evolution of the wavefunction in the tensor network description.

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arXiv2013

Entanglement Entropy at Large Central Charge

Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.

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arXiv2012

Warped Conformal Field Theory

We study field theories in two spacetime dimensions invariant under a chiral scaling symmetry that acts only on right-movers. The local symmetries include one copy of the Virasoro algebra and a U(1) current algebra. This differs from the 2d conformal group, but in some respects is equally powerful in constraining the theory. In particular, the symmetries on a torus lead to modular covariance of the partition function, which is used to derive a universal formula for the asymptotic density of states. For an application we turn to the holographic description of black holes in quantum gravity, motivated by the fact that the symmetries in the near horizon geometry of any extremal black hole are identical to those of a 2d field theory with chiral scaling. We consider two examples: black holes in warped AdS_3 in topologically massive gravity, and in string theory. In both cases, the density of states in the 2d field theory reproduces the Bekenstein-Hawking entropy of black holes in the gravity theory.

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arXiv2012

Higher Spin Black Holes from CFT

Higher spin gravity in three dimensions has explicit black holes solutions, carrying higher spin charge. We compute the free energy of a charged black hole from the holographic dual, a 2d CFT with extended conformal symmetry, and find exact agreement with the bulk thermodynamics. In the CFT, higher spin corrections to the free energy can be calculated at high temperature from correlation functions of W-algebra currents.

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arXiv2011

The Gravity Dual of the Ising Model

We evaluate the partition function of three dimensional theories of gravity in the quantum regime, where the AdS radius is Planck scale and the central charge is of order one. The contribution from the AdS vacuum sector can - with certain assumptions - be computed and equals the vacuum character of a minimal model CFT. The torus partition function is given by a sum over geometries which is finite and computable. For generic values of Newton's constant G and the AdS radius L the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known CFT. For example, the partition function of pure Einstein gravity with G=3L equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the 3-state and tricritical Potts models are dual to pure higher spin theories of gravity based on SL(3) and E_6, respectively.

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arXiv2011

Higher Spin Realization of the dS/CFT Correspondence

We conjecture that Vasiliev's theory of higher spin gravity in four-dimensional de Sitter space (dS) is holographically dual to a three-dimensional conformal field theory (CFT) living on the spacelike boundary of dS at future timelike infinity. The CFT is the Euclidean Sp(N) vector model with anticommuting scalars. The free CFT flows under a double-trace deformation to an interacting CFT in the IR. We argue that both CFTs are dual to Vasiliev dS gravity but with different future boundary conditions on the bulk scalar field. Our analysis rests heavily on analytic continuations of bulk and boundary correlators in the proposed duality relating the O(N) model with Vasiliev gravity in AdS.

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arXiv2011

The Gravitational Exclusion Principle and Null States in Anti-de Sitter Space

The holographic principle implies a vast reduction in the number of degrees of freedom of quantum gravity. This idea can be made precise in AdS_3, where the the stringy or gravitational exclusion principle asserts that certain perturbative excitations are not present in the exact quantum spectrum. We show that this effect is visible directly in the bulk gravity theory: the norm of the offending linearized state is zero or negative. When the norm is negative, the theory is signaling its own breakdown as an effective field theory; this provides a perturbative bulk explanation for the stringy exclusion principle. When the norm vanishes the bulk state is null rather than physical. This implies that certain non-trivial diffeomorphisms must be regarded as gauge symmetries rather than spectrum-generating elements of the asymptotic symmetry group. This leads to subtle effects in the computation of one-loop determinants for Einstein gravity, higher spin theories and topologically massive gravity in AdS_3. In particular, heat kernel methods do not capture the correct spectrum of a theory with null states.

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arXiv2011

Partition Functions of Holographic Minimal Models

The partition function of the W_N minimal model CFT is computed in the large N 't Hooft limit and compared to the spectrum of the proposed holographic dual, a 3d higher spin gravity theory coupled to massive scalar fields. At finite N, the CFT contains additional light states that are not visible in the perturbative gravity theory. We carefully define the large N limit, and give evidence that, at N = infinity, the additional states become null and decouple from all correlation functions. The surviving states are shown to match precisely (for all values of the 't Hooft coupling) with the spectrum of the higher spin gravity theory. The agreement between bulk and boundary is partially explained by symmetry considerations involving the conjectured equivalence between the W_N algebra in the large N limit and the higher spin algebra of the Vasiliev theory.

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arXiv2011

Symmetries of Holographic Minimal Models

It was recently proposed that a large N limit of a family of minimal model CFTs is dual to a certain higher spin gravity theory in AdS_3, where the 't Hooft coupling constant of the CFT is related to a deformation parameter of the higher spin algebra. We identify the asymptotic symmetry algebra of the higher spin theory for generic 't Hooft parameter, and show that it coincides with a family of W-algebras previously discovered in the context of the KP hierarchy. We furthermore demonstrate that this family of W-algebras controls the representation theory of the minimal model CFTs in the 't Hooft limit. This provides a non-trivial consistency check of the proposal and explains part of the underlying mechanism.

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