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Banerjee A, Kibe T, Mittal N, Mukhopadhyay A, Roy P. Erasure Tolerant Quantum Memory and the Quantum Null Energy Condition in Holographic Systems. PHYSICAL REVIEW LETTERS 2022; 129:191601. [PMID: 36399741 DOI: 10.1103/physrevlett.129.191601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2022] [Revised: 06/11/2022] [Accepted: 10/19/2022] [Indexed: 06/16/2023]
Abstract
Investigating principles for storage of quantum information at finite temperature with minimal need for active error correction is an active area of research. We bear upon this question in two-dimensional holographic conformal field theories via the quantum null energy condition that we have shown earlier to implement the restrictions imposed by quantum thermodynamics on such many-body systems. We study an explicit encoding of a logical qubit into two similar chirally propagating excitations of finite von Neumann entropy on a finite temperature background whose erasure can be implemented by an appropriate inhomogeneous and instantaneous energy-momentum inflow from an infinite energy memoryless bath due to which the system transits to a thermal state. Holographically, these fast erasure processes can be depicted by generalized AdS-Vaidya geometries described previously in which no assumption of specific form of bulk matter is needed. We show that the quantum null energy condition gives analytic results for the minimal finite temperature needed for the deletion which is larger than the initial background temperature in consistency with Landauer's principle. In particular, we find a simple expression for the minimum final temperature needed for the erasure of a large number of encoding qubits. We also find that if the encoding qubits are localized over an interval shorter than a specific localization length, then the fast erasure process is impossible, and furthermore this localization length is the largest for an optimal amount of encoding qubits determined by the central charge. We estimate the optimal encoding qubits for realistic protection against fast erasure. We discuss possible generalizations of our study for novel constructions of fault-tolerant quantum gates operating at finite temperature.
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Affiliation(s)
- Avik Banerjee
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Tanay Kibe
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Nehal Mittal
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Ayan Mukhopadhyay
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Pratik Roy
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
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Kibe T, Mukhopadhyay A, Roy P. Quantum Thermodynamics of Holographic Quenches and Bounds on the Growth of Entanglement from the Quantum Null Energy Condition. PHYSICAL REVIEW LETTERS 2022; 128:191602. [PMID: 35622045 DOI: 10.1103/physrevlett.128.191602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2021] [Revised: 03/20/2022] [Accepted: 04/21/2022] [Indexed: 06/15/2023]
Abstract
The quantum null energy condition (QNEC) is a lower bound on the energy-momentum tensor in terms of the variation of the entanglement entropy of a subregion along a null direction. To gain insights into quantum thermodynamics of many-body systems, we study if the QNEC restricts irreversible entropy production in quenches driven by energy-momentum inflow from an infinite memoryless bath in two-dimensional holographic theories. We find that an increase in both entropy and temperature, as implied by the Clausius inequality of classical thermodynamics, is necessary but not sufficient to not violate QNEC in quenches leading to transitions between thermal states with momentums that are dual to Banados-Teitelboim-Zanelli geometries. For an arbitrary initial state, we can determine the lower and upper bounds on the increase of entropy (temperature) for a fixed increase in temperature (entropy). Our results provide explicit instances of quantum lower and upper bounds on irreversible entropy production whose existence has been established in literature. We also find monotonic behavior of the nonsaturation of the QNEC with time after a quench, and analytically determine their asymptotic values. Our study shows that the entanglement entropy of an interval of length l always thermalizes in time l/2 with an exponent 3/2. Furthermore, we determine the coefficient of initial quadratic growth of entanglement analytically for any l, and show that the slope of the asymptotic ballistic growth of entanglement for a semi-infinite interval is twice the difference of the entropy densities of the final and initial states. We determine explicit upper and lower bounds on these rates of growth of entanglement.
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Affiliation(s)
- Tanay Kibe
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Ayan Mukhopadhyay
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Pratik Roy
- Center for Quantum Information Theory of Matter and Spacetime, and Center for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
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3
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Du X, Schlichting S. Equilibration of the Quark-Gluon Plasma at Finite Net-Baryon Density in QCD Kinetic Theory. PHYSICAL REVIEW LETTERS 2021; 127:122301. [PMID: 34597092 DOI: 10.1103/physrevlett.127.122301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2021] [Revised: 05/03/2021] [Accepted: 06/25/2021] [Indexed: 06/13/2023]
Abstract
We explore the out-of-equilibrium dynamics of the quark-gluon plasma at zero and finite net-baryon density based on an effective kinetic theory of quantum chromodynamics (QCD). By investigating the isotropization of the longitudinal pressure, we determine the relevant time and temperature scales for the onset of viscous hydrodynamics and quantify the dependence on the chemical composition of the quark-gluon plasma. By extrapolating our results to realistic coupling strength, we discuss phenomenological consequences regarding the role of the preequilibrium phase at different collision energies.
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Affiliation(s)
- Xiaojian Du
- Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany
| | - Sören Schlichting
- Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany
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5
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Mahapatra S, Priyadarshinee S, Narasimha Reddy G, Shukla B. Exact topological charged hairy black holes in AdS space in
d
dimensions. Int J Clin Exp Med 2020. [DOI: 10.1103/physrevd.102.024042] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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6
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Liu H, Sonner J. Holographic systems far from equilibrium: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2020; 83:016001. [PMID: 31634882 DOI: 10.1088/1361-6633/ab4f91] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
In this paper we give an overview of some recent progress in using holography to study various far-from-equilibrium condensed matter systems. Non-equilibrium problems are notoriously difficult to deal with, not to mention at strong coupling and when including quantum effects. Remarkably, using holographic duality one can describe and follow the real-time evolution of far-from-equilibrium systems, including those which are spatially inhomogeneous and anisotropic, by solving partial differential gravity equations. We sample developments in two broad classes of question which have recently been of much interest to the condensed matter community: non-equilibrium steady states, and quantum systems undergoing a global quench. Our discussion focuses on the main physical insights obtained from the gravity approaches, rather than comprehensive treatment of each topic or detailed descriptions of gravity calculations. The paper also includes an overview of current numerical techniques, as well as the holographic Schwinger-Keldysh approach to real-time correlation functions.
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Affiliation(s)
- Hong Liu
- Center for theoretical physics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States of America
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7
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Hampapura HR, Rolph A, Stoica B. Scrambling in two-dimensional conformal field theories with light and smeared operators. Int J Clin Exp Med 2019. [DOI: 10.1103/physrevd.99.106010] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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9
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Zeng XX, Chen DY, Li LF. Holographic thermalization and gravitational collapse in a spacetime dominated by quintessence dark energy. Int J Clin Exp Med 2015. [DOI: 10.1103/physrevd.91.046005] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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10
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Craps B, Lindgren E, Taliotis A, Vanhoof J, Zhang H. Holographic gravitational infall in the hard wall model. Int J Clin Exp Med 2014. [DOI: 10.1103/physrevd.90.086004] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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11
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Balasubramanian V, Buchel A, Green SR, Lehner L, Liebling SL. Holographic thermalization, stability of anti-de sitter space, and the Fermi-Pasta-Ulam paradox. PHYSICAL REVIEW LETTERS 2014; 113:071601. [PMID: 25170699 DOI: 10.1103/physrevlett.113.071601] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2014] [Indexed: 06/03/2023]
Abstract
For a real massless scalar field in general relativity with a negative cosmological constant, we uncover a large class of spherically symmetric initial conditions that are close to anti-de Sitter space (AdS) but whose numerical evolution does not result in black hole formation. According to the AdS/conformal field theory (CFT) dictionary, these bulk solutions are dual to states of a strongly interacting boundary CFT that fail to thermalize at late times. Furthermore, as these states are not stationary, they define dynamical CFT configurations that do not equilibrate. We develop a two-time-scale perturbative formalism that captures both direct and inverse cascades of energy and agrees with our fully nonlinear evolutions in the appropriate regime. We also show that this formalism admits a large class of quasiperiodic solutions. Finally, we demonstrate a striking parallel between the dynamics of AdS and the classic Fermi-Pasta-Ulam-Tsingou problem.
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Affiliation(s)
- Venkat Balasubramanian
- Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
| | - Alex Buchel
- Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Stephen R Green
- Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada
| | - Luis Lehner
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Steven L Liebling
- Department of Physics, Long Island University, Brookville, New York 11548, USA
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12
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Cardy J. Thermalization and revivals after a quantum quench in conformal field theory. PHYSICAL REVIEW LETTERS 2014; 112:220401. [PMID: 24949745 DOI: 10.1103/physrevlett.112.220401] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2014] [Indexed: 06/03/2023]
Abstract
We consider a quantum quench in a finite system of length L described by a 1+1-dimensional conformal field theory (CFT), of central charge c, from a state with finite energy density corresponding to an inverse temperature β≪L. For times t such that ℓ/2<t<(L-ℓ)/2 the reduced density matrix of a subsystem of length ℓ is exponentially close to a thermal density matrix. We compute exactly the overlap F of the state at time t with the initial state and show that in general it is exponentially suppressed at large L/β. However, for minimal models with c<1 (more generally, rational CFTs), at times which are integer multiples of L/2 (for periodic boundary conditions, L for open boundary conditions) there are (in general, partial) revivals at which F is O(1), leading to an eventual complete revival with F=1. There is also interesting structure at all rational values of t/L, related to properties of the CFT under modular transformations. At early times t≪(Lβ)^{1/2} there is a universal decay F∼exp(-(πc/3)Lt^{2}/β(β^{2}+4t^{2})). The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at t=nL/2 by an amount O(n^{1/2}).
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Affiliation(s)
- John Cardy
- Rudolf Peierls Centre for Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United Kingdom; All Souls College, Oxford OX1 4AL, United Kingdom; and Kavli Institute for Theoretical Physics, Santa Barbara, California 93106-4030, USA
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13
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Das SR, Galante DA, Myers RC. Universal scaling in fast quantum quenches in conformal field theories. PHYSICAL REVIEW LETTERS 2014; 112:171601. [PMID: 24836234 DOI: 10.1103/physrevlett.112.171601] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2014] [Indexed: 06/03/2023]
Abstract
We study the time evolution of a conformal field theory deformed by a relevant operator under a smooth but fast quantum quench which brings it to the conformal point. We argue that when the quench time scale δt is small compared to the scale set by the relevant coupling, the expectation value of the quenched operator scales universally as δλ/δt(2Δ-d), where δλ is the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions. We present explicit results for free scalar and fermionic field theories, supported by an analytic understanding of the leading contribution for fast quenches. Our results suggest that this scaling result, first found in holography, is in fact quite general. Our considerations also show that this limit of fast smooth quenches is quite different from an instantaneous quench from one time-independent Hamiltonian to another, where the state at the time of the quench serves as an initial condition for subsequent evolution with the final Hamiltonian.
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Affiliation(s)
- Sumit R Das
- Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA
| | - Damián A Galante
- Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Robert C Myers
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
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14
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Liu H, Suh SJ. Entanglement tsunami: universal scaling in holographic thermalization. PHYSICAL REVIEW LETTERS 2014; 112:011601. [PMID: 24483884 DOI: 10.1103/physrevlett.112.011601] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2013] [Indexed: 06/03/2023]
Abstract
We consider the time evolution of entanglement entropy after a global quench in a strongly coupled holographic system, whose subsequent equilibration is described in the gravity dual by the gravitational collapse of a thin shell of matter resulting in a black hole. In the limit of large regions of entanglement, the evolution of entanglement entropy is controlled by the geometry around and inside the event horizon of the black hole, resulting in regimes of pre-local-equilibration quadratic growth (in time), post-local-equilibration linear growth, a late-time regime in which the evolution does not carry memory of the size and shape of the entangled region, and a saturation regime with critical behavior resembling those in continuous phase transitions. Collectively, these regimes suggest a picture of entanglement growth in which an "entanglement tsunami" carries entanglement inward from the boundary. We also make a conjecture on the maximal rate of entanglement growth in relativistic systems.
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Affiliation(s)
- Hong Liu
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - S Josephine Suh
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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15
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Balasubramanian V, Bernamonti A, de Boer J, Craps B, Franti L, Galli F, Keski-Vakkuri E, Müller B, Schäfer A. Inhomogeneous thermalization in strongly coupled field theories. PHYSICAL REVIEW LETTERS 2013; 111:231602. [PMID: 24476254 DOI: 10.1103/physrevlett.111.231602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2013] [Indexed: 06/03/2023]
Abstract
To describe theoretically the creation and evolution of the quark-gluon plasma, one typically employs three ingredients: a model for the initial state, nonhydrodynamic early time evolution, and hydrodynamics. In this Letter we study the nonhydrodynamic early time evolution using the AdS/CFT correspondence in the presence of inhomogeneities. We find that the AdS description of the early time evolution is well matched by free streaming. Near the end of the early time interval where our analytic computations are reliable, the stress tensor agrees with the second order hydrodynamic stress tensor computed from the local energy density and fluid velocity. Our techniques may also be useful for the study of far-from-equilibrium strongly coupled systems in other areas of physics.
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Affiliation(s)
- V Balasubramanian
- David Rittenhouse Laboratory, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA and Laboratoire de Physique Théorique, École Normale Supérieure, 75005 Paris, France
| | - A Bernamonti
- Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
| | - J de Boer
- Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, The Netherlands
| | - B Craps
- Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay Institutes, B-1050 Brussels, Belgium
| | - L Franti
- Helsinki Institute of Physics & Department of Physics, FIN-00014 University of Helsinki, Finland
| | - F Galli
- Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay Institutes, B-1050 Brussels, Belgium
| | - E Keski-Vakkuri
- Helsinki Institute of Physics & Department of Physics, FIN-00014 University of Helsinki, Finland and Department of Physics and Astronomy, Uppsala University, SE-75108 Uppsala, Sweden
| | - B Müller
- Department of Physics, Duke University, Durham, North Carolina 27708, USA and Brookhaven National Laboratory, Upton, New York 11973, USA
| | - A Schäfer
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
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Buchel A, Myers RC, van Niekerk A. Universality of abrupt holographic quenches. PHYSICAL REVIEW LETTERS 2013; 111:201602. [PMID: 24289674 DOI: 10.1103/physrevlett.111.201602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2013] [Indexed: 06/02/2023]
Abstract
We make an analytic investigation of rapid quenches of relevant operators in d-dimensional holographic conformal field theories, which admit a dual gravity description. We uncover a universal scaling behavior in the response of the system, which depends only on the conformal dimension of the quenched operator in the vicinity of the ultraviolet fixed point of the theory. Unless the amplitude of the quench is scaled appropriately, the work done on a system during the quench diverges in the limit of abrupt quenches for operators with dimension (d/2)≤Δ<d.
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Affiliation(s)
- Alex Buchel
- Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
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17
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Yee HU. Flows and polarization of early photons with magnetic field at strong coupling. Int J Clin Exp Med 2013. [DOI: 10.1103/physrevd.88.026001] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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18
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Wu B. On holographic thermalization and gravitational collapse of tachyonic scalar fields. JOURNAL OF HIGH ENERGY PHYSICS 2013; 2013:44. [DOI: 10.1007/jhep04(2013)044] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2013] [Accepted: 03/24/2013] [Indexed: 09/01/2023]
Abstract
Abstract
In this paper we study the thermalization of a spatially homogeneous system in a strongly coupled CFT. The non-equilibrium initial state is created by switching on a relevant perturbation in the CFT vacuum during Δt ≳ t ≳ − Δt. Via AdS/CFT, the thermalization process corresponds to the gravitational collapse of a tachyonic scalar field (m
2 = −3) in the Poincare patch of AdS
5. In the limit
$ \varDelta t<\frac{0.02 }{T} $
, the thermalization time t
T
is found to be quantitatively the same as that of a non-equilibrium state created by a marginal perturbation discussed in ref. [5]. In the case
$ \varDelta t\gtrsim \frac{1}{T} $
we also obtain double- collapse solutions but with a non-equilibrium intermediate state at t = 0. In all the cases our results show that the system thermalizes in a typical time
$ {t_T}\simeq \frac{O(1) }{T} $
. Besides, a conserved energy-moment current in the bulk is found, which helps understand the qualitative difference of the collapse process in the Poincare patch from that in global AdS [10, 11].
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Wu B. On holographic thermalization and gravitational collapse of massless scalar fields. JOURNAL OF HIGH ENERGY PHYSICS 2012; 2012:133. [DOI: 10.1007/jhep10(2012)133] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/01/2023]
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20
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Affiliation(s)
- Barbara V. Jacak
- Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794, USA
| | - Berndt Müller
- Department of Physics and Center for Theoretical and Mathematical Sciences, Duke University, Durham, NC 27708, USA
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22
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Heller MP, Mateos D, van der Schee W, Trancanelli D. Strong coupling isotropization of non-abelian plasmas simplified. PHYSICAL REVIEW LETTERS 2012; 108:191601. [PMID: 23003025 DOI: 10.1103/physrevlett.108.191601] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2012] [Indexed: 06/01/2023]
Abstract
We study the isotropization of a homogeneous, strongly coupled, non-abelian plasma by means of its gravity dual. We compare the time evolution of a large number of initially anisotropic states as determined, on the one hand, by the full nonlinear Einstein's equations and, on the other, by the Einstein's equations linearized around the final equilibrium state. The linear approximation works remarkably well even for states that exhibit large anisotropies. For example, it predicts with a 20% accuracy the isotropization time, which is of the order of t(iso)≲1/T, with T the final equilibrium temperature. We comment on possible extensions to less symmetric situations.
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Affiliation(s)
- Michal P Heller
- Instituut voor Theoretische Fysica, Universiteit van Amsterdam Science Park 904, 1090 GL Amsterdam, The Netherlands
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23
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Caron-Huot S, Chesler PM, Teaney D. Fluctuation, dissipation, and thermalization in nonequilibriumAdS5black hole geometries. Int J Clin Exp Med 2011. [DOI: 10.1103/physrevd.84.026012] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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