Thomas Faulkner
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Thomas Faulkner has a total of 44 co-authors. Here are the top co-authors by number of shared publications:
- Simon Lin
New York University Abu Dhabi
11 shared publications - Chris Akers
University of Colorado
9 shared publications - Pratik Rath
University of California, Berkeley
9 shared publications - Eduardo Fradkin
Department of Physics and Materials Research Laboratory, Grainger College of Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801
4 shared publications - Huajia Wang
University of Chinese Academy of Sciences
4 shared publications - Venkatesa Chandrasekaran
School of Natural Sciences Institute for Advanced Study 1 Einstein Drive 08540 Princeton NJ USA
4 shared publications
Latest Publications
- Open Access
Gravitational Algebras with Two Areas
We study gravitational algebras on spacetimes with two extremal surfaces. In the example of a long wormhole with two asymptotic AdS boundaries and two compact extremal surfaces, we discuss the assignment of gravitational algebras to various regions bounded by the extremal surfaces and/or asymptotic boundaries. Using the split property, we construct type II algebras from the crossed product in the left exterior, right exterior, the middle ``python's lunch'' region, and their complement regions. We also study the case where only the area sum operator or area difference operator is included as part of the gravitational algebra. This can be achieved by picking the appropriate microcanonical ensemble, and these gravitational algebras can either be type II or type III depending on the region. In the case where we include only the area difference mode, the crossed product gives rise to a weight that restricts to a trace on the middle region. Differences of relative entropies with respect to this weight give differences in generalized entropies. This provides an algebraic understanding of the order parameter that controls the phase transitions between entanglement wedges. We emphasize the role of operator-valued weights used in our construction.
Open AccessGravitational Algebras with Two Areas
We study gravitational algebras on spacetimes with two extremal surfaces. In the example of a long wormhole with two asymptotic AdS boundaries and two compact extremal surfaces, we discuss the assignment of gravitational algebras to various regions bounded by the extremal surfaces and/or asymptotic boundaries. Using the split property, we construct type II algebras from the crossed product in the left exterior, right exterior, the middle ``python's lunch'' region, and their complement regions. We also study the case where only the area sum operator or area difference operator is included as part of the gravitational algebra. This can be achieved by picking the appropriate microcanonical ensemble, and these gravitational algebras can either be type II or type III depending on the region. In the case where we include only the area difference mode, the crossed product gives rise to a weight that restricts to a trace on the middle region. Differences of relative entropies with respect to this weight give differences in generalized entropies. This provides an algebraic understanding of the order parameter that controls the phase transitions between entanglement wedges. We emphasize the role of operator-valued weights used in our construction.
- Open Access
- 21 January 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (1) , 1-35
Ramp from replica trick
A bstract: We compute the spectral form factor of the modular Hamiltonian K = −ln ρ A associated to the reduced density matrix of a Haar random state. A ramp is demonstrated and we find an analytic expression for its slope. Our method involves an application of the replica trick, where we first calculate the correlator $$ \left\langle \textrm{tr}{\rho}_A^n tr{\rho}_A^m\right\rangle $$ at large bond dimension and then analytically continue the indices n, m from integers to arbitrary complex numbers. We use steepest descent methods at large modular times to extract the ramp. The large bond dimension limit of the replicated partition function is dominated by a sum over annular non-crossing permutations . We explored the similarity between our results and calculations of the spectral form factor in low dimensional gravitational theories where the ramp is determined by the double trumpet geometry. We find there is an underlying resemblance in the two calculations, when we interpret the annular non-crossing permutations as representing a discretized version of the double trumpet. Similar results are found for an equilibrated pure state in place of the Haar random state.- 21 January 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (1) , 1-35
Open AccessRamp from replica trick
A bstract: We compute the spectral form factor of the modular Hamiltonian K = −ln ρ A associated to the reduced density matrix of a Haar random state. A ramp is demonstrated and we find an analytic expression for its slope. Our method involves an application of the replica trick, where we first calculate the correlator $$ \left\langle \textrm{tr}{\rho}_A^n tr{\rho}_A^m\right\rangle $$ at large bond dimension and then analytically continue the indices n, m from integers to arbitrary complex numbers. We use steepest descent methods at large modular times to extract the ramp. The large bond dimension limit of the replicated partition function is dominated by a sum over annular non-crossing permutations . We explored the similarity between our results and calculations of the spectral form factor in low dimensional gravitational theories where the ramp is determined by the double trumpet geometry. We find there is an underlying resemblance in the two calculations, when we interpret the annular non-crossing permutations as representing a discretized version of the double trumpet. Similar results are found for an equilibrated pure state in place of the Haar random state. - Open Access
- 27 December 2024
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2024 (12) , 1-70
Reflected entropy in random tensor networks. Part III. Triway cuts
A bstract: For general random tensor network states at large bond dimension, we prove that the integer Rényi reflected entropies (away from phase transitions) are determined by minimal triway cuts through the network. This generalizes the minimal cut description of bipartite entanglement for these states. A natural extrapolation away from integer Rényi parameters, suggested by the triway cut problem, implies the holographic conjecture S R = 2EW, where S R is the reflected entropy and EW is the entanglement wedge cross-section. Minimal triway cuts can be formulated as integer programs which cannot be relaxed to find a dual maximal flow/bit-thread description. This sheds light on the gap between the existence of tripartite entanglement in holographic states and the bipartite entanglement structure motivated by bit-threads. In particular, we prove that the Markov gap that measures tripartite entanglement is lower bounded by the integrality gap of the integer program that computes the triway cut.
- 27 December 2024
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2024 (12) , 1-70
Open AccessReflected entropy in random tensor networks. Part III. Triway cuts
A bstract: For general random tensor network states at large bond dimension, we prove that the integer Rényi reflected entropies (away from phase transitions) are determined by minimal triway cuts through the network. This generalizes the minimal cut description of bipartite entanglement for these states. A natural extrapolation away from integer Rényi parameters, suggested by the triway cut problem, implies the holographic conjecture S R = 2EW, where S R is the reflected entropy and EW is the entanglement wedge cross-section. Minimal triway cuts can be formulated as integer programs which cannot be relaxed to find a dual maximal flow/bit-thread description. This sheds light on the gap between the existence of tripartite entanglement in holographic states and the bipartite entanglement structure motivated by bit-threads. In particular, we prove that the Markov gap that measures tripartite entanglement is lower bounded by the integrality gap of the integer program that computes the triway cut.
- Open Access
- 15 November 2024
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2024 (11) , 1-78
Gravitational algebras and the generalized second law
We derive the generalized second law (GSL) for arbitrary cuts of Killing horizons from the perspective of crossed-product gravitational algebras, making use of a recent proposal by one of us for the construction of local gravitational algebras. This construction relies on the existence of a state whose modular flow is geometric on the horizon. In both free and interacting quantum field theories, such states are guaranteed to exist by the properties of half-sided translations on the horizon. Using geometric identities derived from the canonical analysis of general relativity on null surfaces, we show that the crossed product entropy agrees with the generalized entropy of the horizon cut in a semiclassical limit, and further reproduce Wall’s result relating the GSL to monotonicity of relative entropy of the quantum field algebras. We also give a novel generalization of the GSL for interacting theories in asymptotically flat spacetimes involving the concept of an algebra at infinity for a half-sided translation, which accounts for triviality of the algebra of fields smeared only on the horizon. Going beyond the semiclassical limit, we compute subleading corrections to the crossed product entropy, but are unable to determine if the GSL continues to hold after accounting for these. We speculate that an improved GSL could follow from a hidden subalgebra structure of the crossed products, assuming the existence of an operator-valued weight between horizon cut algebras.
- 15 November 2024
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2024 (11) , 1-78
Open AccessGravitational algebras and the generalized second law
We derive the generalized second law (GSL) for arbitrary cuts of Killing horizons from the perspective of crossed-product gravitational algebras, making use of a recent proposal by one of us for the construction of local gravitational algebras. This construction relies on the existence of a state whose modular flow is geometric on the horizon. In both free and interacting quantum field theories, such states are guaranteed to exist by the properties of half-sided translations on the horizon. Using geometric identities derived from the canonical analysis of general relativity on null surfaces, we show that the crossed product entropy agrees with the generalized entropy of the horizon cut in a semiclassical limit, and further reproduce Wall’s result relating the GSL to monotonicity of relative entropy of the quantum field algebras. We also give a novel generalization of the GSL for interacting theories in asymptotically flat spacetimes involving the concept of an algebra at infinity for a half-sided translation, which accounts for triviality of the algebra of fields smeared only on the horizon. Going beyond the semiclassical limit, we compute subleading corrections to the crossed product entropy, but are unable to determine if the GSL continues to hold after accounting for these. We speculate that an improved GSL could follow from a hidden subalgebra structure of the crossed products, assuming the existence of an operator-valued weight between horizon cut algebras.
- Open Access
- 6 May 2024
- Published by American Physical Society (APS) in Physical Review D
- Vol. 109 (10) , L101902
Entanglement of purification in random tensor networks
The entanglement of purification is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: , where is the Renyi reflected entropy. Using this, we compute for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section , proving a previous conjecture motivated from AdS/CFT.- 6 May 2024
- Published by American Physical Society (APS) in Physical Review D
- Vol. 109 (10) , L101902
Open AccessEntanglement of purification in random tensor networks
The entanglement of purification is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: , where is the Renyi reflected entropy. Using this, we compute for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section , proving a previous conjecture motivated from AdS/CFT. - Open Access
The Page Curve for Reflected Entropy
We study the reflected entropy $S_R$ in the West Coast Model, a toy model of black hole evaporation consisting of JT gravity coupled to end-of-the-world branes. We demonstrate the validity of the holographic duality relating it to the entanglement wedge cross section away from phase transitions. Further, we analyze the important non-perturbative effects that smooth out the discontinuity in the $S_R$ phase transition. By performing the gravitational path integral, we obtain the reflected entanglement spectrum analytically. The spectrum takes a simple form consisting of superselection sectors, which we interpret as a direct sum of geometries, a disconnected one and a connected one involving a closed universe. We find that area fluctuations of $O(\sqrt{G_N})$ spread out the $S_R$ phase transition in the canonical ensemble, analogous to the entanglement entropy phase transition. We also consider a Renyi generalization of the reflected entropy and show that the location of the phase transition varies as a function of the Renyi parameter.Open AccessThe Page Curve for Reflected Entropy
We study the reflected entropy $S_R$ in the West Coast Model, a toy model of black hole evaporation consisting of JT gravity coupled to end-of-the-world branes. We demonstrate the validity of the holographic duality relating it to the entanglement wedge cross section away from phase transitions. Further, we analyze the important non-perturbative effects that smooth out the discontinuity in the $S_R$ phase transition. By performing the gravitational path integral, we obtain the reflected entanglement spectrum analytically. The spectrum takes a simple form consisting of superselection sectors, which we interpret as a direct sum of geometries, a disconnected one and a connected one involving a closed universe. We find that area fluctuations of $O(\sqrt{G_N})$ spread out the $S_R$ phase transition in the canonical ensemble, analogous to the entanglement entropy phase transition. We also consider a Renyi generalization of the reflected entropy and show that the location of the phase transition varies as a function of the Renyi parameter. - Open Access
Entanglement of Purification in Random Tensor Networks
The entanglement of purification $E_P(A\colon B)$ is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: $E_P(A\colon B)\geq \frac{1}{2}S_R^{(2)}(A\colon B)$, where $S_R^{(n)}(A\colon B)$ is the Renyi reflected entropy. Using this, we compute $E_P(A\colon B)$ for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section $EW(A\colon B)$, proving a previous conjecture motivated from AdS/CFT.Open AccessEntanglement of Purification in Random Tensor Networks
The entanglement of purification $E_P(A\colon B)$ is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: $E_P(A\colon B)\geq \frac{1}{2}S_R^{(2)}(A\colon B)$, where $S_R^{(n)}(A\colon B)$ is the Renyi reflected entropy. Using this, we compute $E_P(A\colon B)$ for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section $EW(A\colon B)$, proving a previous conjecture motivated from AdS/CFT. - Open Access
- 22 February 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (2) , 1-28
The reflected entanglement spectrum for free fermions
We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1 + 1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the ground state wave function.
- 22 February 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (2) , 1-28
Open AccessThe reflected entanglement spectrum for free fermions
We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1 + 1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the ground state wave function.
- Open Access
- 13 January 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (1) , 1-74
Reflected entropy in random tensor networks. Part II. A topological index from canonical purification
In ref. [1], we analyzed the reflected entropy (SR) in random tensor networks motivated by its proposed duality to the entanglement wedge cross section (EW) in holographic theories, $$ {S}_R=2\frac{EW}{4G} $$ . In this paper, we discover further details of this duality by analyzing a simple network consisting of a chain of two random tensors. This setup models a multiboundary wormhole. We show that the reflected entanglement spectrum is controlled by representation theory of the Temperley-Lieb algebra. In the semiclassical limit motivated by holography, the spectrum takes the form of a sum over superselection sectors associated to different irreducible representations of the Temperley-Lieb algebra and labelled by a topological index k ∈ ℤ>0. Each sector contributes to the reflected entropy an amount $$ 2k\frac{EW}{4G} $$ weighted by its probability. We provide a gravitational interpretation in terms of fixed-area, higher-genus multiboundary wormholes with genus 2k – 1 initial value slices. These wormholes appear in the gravitational description of the canonical purification. We confirm the reflected entropy holographic duality away from phase transitions. We also find important non-perturbative contributions from the novel geometries with k ≥ 2 near phase transitions, resolving the discontinuous transition in SR. Along with analytic arguments, we provide numerical evidence for our results. We finally speculate that signatures of a non-trivial von Neumann algebra, connected to the Temperley-Lieb algebra, will emerge from a modular flowed version of reflected entropy.- 13 January 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (1) , 1-74
Open AccessReflected entropy in random tensor networks. Part II. A topological index from canonical purification
In ref. [1], we analyzed the reflected entropy (SR) in random tensor networks motivated by its proposed duality to the entanglement wedge cross section (EW) in holographic theories, $$ {S}_R=2\frac{EW}{4G} $$ . In this paper, we discover further details of this duality by analyzing a simple network consisting of a chain of two random tensors. This setup models a multiboundary wormhole. We show that the reflected entanglement spectrum is controlled by representation theory of the Temperley-Lieb algebra. In the semiclassical limit motivated by holography, the spectrum takes the form of a sum over superselection sectors associated to different irreducible representations of the Temperley-Lieb algebra and labelled by a topological index k ∈ ℤ>0. Each sector contributes to the reflected entropy an amount $$ 2k\frac{EW}{4G} $$ weighted by its probability. We provide a gravitational interpretation in terms of fixed-area, higher-genus multiboundary wormholes with genus 2k – 1 initial value slices. These wormholes appear in the gravitational description of the canonical purification. We confirm the reflected entropy holographic duality away from phase transitions. We also find important non-perturbative contributions from the novel geometries with k ≥ 2 near phase transitions, resolving the discontinuous transition in SR. Along with analytic arguments, we provide numerical evidence for our results. We finally speculate that signatures of a non-trivial von Neumann algebra, connected to the Temperley-Lieb algebra, will emerge from a modular flowed version of reflected entropy. - Open Access
The Reflected Entanglement Spectrum for Free Fermions
We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1+1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the ground state wave function.
Open AccessThe Reflected Entanglement Spectrum for Free Fermions
We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1+1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the ground state wave function.