Introduction Elliptic Minimal surfaces in AdS4 Ongoing research Quantum Entanglement and Gravity as an Entropic Force Georgios Pastras NCSR “Demokritos” - INPP November 16 2017 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force Quantum Entanglement Consider a composite quantum system in a state of the form jΨiAB = jy1iA ⊗ j 1iB + jy2iA ⊗ j 2iB Then, I measurables of the two subsystems are correlated. I there is no state description for each subsystem. This is the phenomenon of quantum entanglement. It has no classical analogue. Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force Entanglement Entropy In general, subsystem A cannot be described by a state, but it is described by the density matrix 휌A = TrB휌AB: In the absence of entanglement, 휌A corresponds to a pure state as there is a state description for subsystem A. The more entangled the overall system state, the “more mixed” is the state that 휌A describes, i.e. the more dispersed the spectrum of 휌A. Therefore, a natural measure of quantum entanglement is Shannon entropy defined on the spectrum of 휌A, SEE := −Tr휌A ln 휌A; commonly known as the entanglement entropy. Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force Holographic Dualities Holographic dualities provide a broad framework allowing the description of gravitational theories with AdS asymptotics in d + 1 dimensions as emergent from strongly coupled conformal field theories in d dimensions. As they connect the strongly coupled regime of one theory to the weakly coupled regime of the other, holographic dualities have found many applications in the description of strongly coupled condensed matter systems through gravitational dynamics, as well as the understanding of quantum gravity as emerging from the boundary CFT. A relatively recent development in the dictionary of the holographic duality is the Ryu-Takayanagi conjecture. 1 1S. Ryu, T. Takayanagi, PRL 96, 181602 (2006), arxiv:hep-th/0603001 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force The RT conjecture The latter connects the entanglement entropy for a region A defined by the entangling surface @A in the boundary CFT to the area of an extremal co-dimension two open surface in the bulk gravitational dual theory with the same boundary @A. Area (Aextr) SA = : 4GN r Aextr @A Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force Gravity vs Thermodynamics I Black Hole Physics suggest Einstein equations could be effective thermodynamic relations for some underlying degrees of freedom. 2 I AdS/CFT correspondence could suggest that these underlying degrees of freedom are the boundary conformal field theory degrees of freedom. 3 I RT conjecture points out that the relation between gravity and thermodynamics should not be attributed to thermal statistics, but rather to quantum statistics related to quantum entanglement physics. 2J. M. Bardeen, B. Carter, S. W. Hawking, Com. Math. Phys. 31, 161 (1973) 3E. P. Verlinde, JHEP 1104, 029 (2011), arXiv:1001.0785 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force Gravity and Entanglement Thermodynamics The density matrix 휌A is hermitian and positive semidefinite, thus, one may define the modular Hamiltonian as −HA 휌A := e : If we assume a variation in the pure state of the overall system, the variation of the entanglement entropy and the expectation value of the modular Hamiltonian are related as 훿SA = 훿 hHAi : This is the direct analog of the first law of thermodynamics for entanglement physics. Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force Gravity and Entanglement Thermodynamics Variation of the state of the overall system is equivalent to the introduction of a metric perturbation in the bulk theory. Using the holographic dictionary to calculate 훿SA and 훿 hHAi we find that the first law of entanglement thermodynamics is holographically realized if and only if the metric perturbation obeys the Einstein equations. 45 4N. Lashkari, M. B. McDermott, M. Van Raamsdonk, JHEP 1404, 195 (2014), arXiv:1308.3716 5I. Bakas, G. Pastras, Nucl. Phys. B 896, 440 (2015) arXiv:1503.00627 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force Problems Unfortunately the non-linearity of the equations that minimal surfaces in hyperbolic spaces obey allow their solution in very few cases, namely region A is either a sphere or an infinite strip. This does not allow I the complete proof of the equivalence of linearized Einstein equations to the 1st law of entanglement thermodynamics I the study of the dependence of entanglement entropy on the geometric characteristics of the entangling surface Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Elliptic Minimal surfaces in AdS4 In order to construct new minimal surfaces in AdS4 we use tools usually applied to 2d Non-linear Sigma Models. This tool is the so called Pohlmeyer reduction which reduces the minimal surface equations to the equations of an integrable system of the family of the sine-Gordon equation. For this system one can find many interesting solutions, however it is difficult to invert Pohlmeyer reduction to find the corresponding minimal surfaces due to its non-local nature Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Elliptic Minimal surfaces in AdS4 We managed to invert Pohlmeyer reduction for a specific class of solutions of the reduced system (elliptic solutions) implementing an interesting relation between the corresponding minimal surfaces and the band structure of the analytically solvable n = 1 Lame periodic potential. 678 6I. Bakas and G. Pastras, JHEP 1607, 070 (2016), arXiv:1605.03920 7G. Pastras, arXiv:1612.03631 8G. Pastras, PoS CORFU 2016, 111 (2016), arXiv:1710.00545 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Entangling Curve The elliptic minimal surfaces correspond to entangling curves being the union of two logarithmic spirals, the one being the rotation of the other by 훿'. Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Interesting Limits The family of elliptic minimal surfaces contain as special limits the helicoids in hyperbolic spaces, Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Interesting Limits the catenoids Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Interesting Limits and the cusps. Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Geometric Phase Transitions It is possible that more than one of the minimal surfaces correspond to the same entangling curve. This phenomenon is related to phase transitions (confinement-deconfinement) of the boundary theory and the role of entanglement entropy as order parameter. In such cases there are three minimal surfaces with the same entagling curve I one globally stable I one locally stable, but globally unstable I one locally unstable with one unstable mode Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction