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Thermalization of isolated quantum systems (+ comments on black holes) M. Kruczenski Purdue University Based on arXiv:1312.4612 In collaboration w/ S.Khlebnikov (Purdue) Aspen 2014
● Introduction and motivation Thermalization of an isolated quantum system. ETH (EigenstateThermalization Hypothesis). Local observables and entropy generation. ●Numerical study of thermalization in an interacting boson gas. Numerical tests of ETH. Generation of entanglement entropy. Rapid initial growth: Summary
●Discussion of black holes. The metric is an ETH operator. No hair theorem is the GR equivalent of ETH. ● Conclusions
Motivation to study thermalization Cold atoms: thermalization (or not) of isolated systems with few atoms (~100). QGP formation in RHIC: tth ~ 0.5 fm/c Foundations of statistical mechanics: classical ergodic theorem QM Formation of black holes in quantum gravity. Au Au
Reminder of relation to quantum gravity Quantum gravity has two main problems: Strongly coupled at large energies (non-renormalizable strings) Information problem in black holes. (Through a process where an initial state forms a black hole and then evaporates, a pure state seems to evolve into a thermal density matrix) String theory and AdS/CFT give a partial answer.
Gauge theory String theory Strings (quantum gravity) in asymptotically AdS spaces is dual to a CFT (Maldacena, Gubser-Klebanov-Polyakov, Witten). e.g. AdS5xS5 S5 : X12+X22+…+X62 = 1AdS5: Y12+Y22+…-Y52-Y62 =-1 (sphere) (hyperbolic space) λsmall → gauge th. λ large → string th.
Since gravity in AdS is dual to a QFT all the rules of quantum mechanics apply. The theory is well defined, finite and unitary Black hole formation is a standard thermalization problem in the boundary. The problem is the dictionary: Where is information localized? Is there any information in the Hawking radiation?
Thermalization Statistical mechanics is based on the premise that a subsystem (of a larger macroscopic system) eventually reaches thermal equilibrium characterized by a density matrix with maximum entropy for given conserved quantities (energy, etc.). Local observables thermalize. Example: Expansion of a gas
Entropy • density matrix of the system. Von Neumann entropy of the system. In this case S(t)=0, for all t. • density matrix of subsystem A, . • Entanglement entropy SAB, SAB(t=0)=0 but increases with time. • thermal density matrix for system or subsystem • thermodynamical entropy S(E,N). • Maximum S for given E,N.
For thermalization, nothing is required from the state of the total system other than it has a fixed total energy, particle number, etc. In particular, in quantum mechanics, if it is initially in a pure state it should remain in a pure state and evolve according to This creates an apparent problem: Consider an observable associated with the subsystem. After a thermalization time it should become time independent and equal to its canonical average.
However we get: Should be (approximately) constant in time and independent of the coefficients ! Requiring thermalization of a subsystem for any initial state of the whole system leads to the ETH hypothesis (Deutsch and Srednicki).
are small, they determine the size of the time dependent fluctuations. In the initial state, the coefficients are chosen such that the off-diagonal elements add up, but after some time the coherence is lost (for most times) and the contribution is negligible giving: We still have to require that this is independent of the ‘s.
We now require that the diagonal elements are a smooth function of the total energy. Since the total energy is fixed (in a narrow band) then the mean value is fixed to a microcanonical average that agrees with the canonical average by standard arguments. The result is a non-trivial statement for the matrix elements of the local, or other ‘thermal’, operators of a macroscopic system.
Is ETH true? It can be tested numerically by exact diagonalization of small systems. Hard core lattice boson gas with near neighbors interactions up to ~20,000 states in the basis. “Thermalization and its mechanism for generic isolated quantum systems”, Rigol, Dunjko, Olshanii, Nature 452, 854 (2008). Later larger systems were (Rigol et al.) considered but such that the largest invariant subspace was of dim ~30,000.
Alternative check. (S. Khlebnikov, M.K.) +) Allows to consider systems up to dimension ~15,000,000 where ETH is more evident. -) Test is done on a subspace of states. Basic observation: For a bounded H and fixed t, the series is absolutely convergent and, for any given accuracy we need to keep only a finite number of terms ~ t. Truncate the Hilbert space to:
Projecting the state onto such subspace gives: where the (Ritz) vectors are the eigenvectors of the projected Hamiltonian: Therefore we can apply all the same reasoning to the Ritz vectors instead of the exact eigenstates. We call this KETH, Krylov ETH.
Since ETH KETH, checking KETH provides a test of the more stringent ETH. In the cases we study, a Krylov subspace of order 1,000 is enough to follow the evolution until thermalization. Given that, we can study much larger Hilbert spaces, up to order 15,000,000. (states still have 15,000,000 components). It is a different type of test because it does not imply that ETH is valid in the whole Hilbert space. However, we checked that increasing the size of the Krylov subspace (even up to the full Hilbert space if possible) does not change the results.
Comments on numerics: Time evolution is done by truncating the expansion to a finite number of terms. The Ritz vectors (eigenstates of H in the reduced subspace) are found using the Lanczos method with re-orthogonalization.
Results Thermalization of site occupation numbers: Nevertheless the evolution is completely reversible. (Plus checks with different methods).
KETH for 1-particle eigenstates occupation #’s For free particles they are integers instead
Off-diagonal elements for site occupation #’s as a function of energy difference
compared to Bose-Einstein distribution for two energy eigenstates. Agrees for the excited state
Why does this happen? (Deutsch) Consider the free case where are integers. The interaction mixes a large number of states that should be considered degenerate. If the are essentially random, then the average fluctuates little between i’s. It still fluctuates, as usual, between QM measurements.
Summary For a given Hamiltonian, ETH is a property obeyed by certain operators that is equivalent to the statement that their averages thermalize. For free particles, local operators and single particle occupation numbers do not obey ETH. However they do if a small an essentially random interaction is turned on. Mathematically this is a consequence of the central limit theorem. We tested these statement numerically for a particular system using the Krylov subspace method (allowing a large number of states in the basis).
Generation of entanglement entropy. For the system consider we can naturally study the entanglement entropy between systems A and B. Initially the entropy grows and then decays.
Derivation: implies analyticity of ρA Initially the eigenvalues of ρAare 1,0,0,0,0,0… So, using perturbation theory they are now 1 + tp ρ(p)00 , tpρ(p)aaimplying In our case p=2 and ρ00=-ΔE2 or
ETH property of entanglement entropy SAB as a function of eigenstate energy
Summary Entanglement entropy is generated by streaming particles into the vacuum and rises fast until it reaches the limit set by the thermodynamical entropy. In that waythe subsystem reaches a (quasi) thermal state. The growth of subsystem entanglement entropy can be thought as the physical reason for thermalization of local observables. After that, the subsystem keeps loosing energy and the entropy decreases until it reaches its true equilibrium value.
Application to black holes. (ETH = no-hair) A black hole has an entropy proportional to the horizon area in Planck units and therefore a large number of quantum microstates. From generic initial conditions, the metric evolves and after some time it becomes the black hole metric independently of the initial state (no hair theorem implies unique metric). ● For that reason it is natural to suggest that the metric is a “thermal” operator in the ETH sense. Therefore it should have the same expectation value in all microstates. ● This suggests that the no-hair theorem is the GR version of the ETH property. (no-hair = ETH). That is, both ETH and no-hair imply that the metric is completely determined by the total energy.
● Since the metric is well defined and the same for all microstates: ● It has no information on the microstate. The information is nowhere in space. ● Anything computed from the metric (e.g. Hawking radiation) ignores the microstate information. ● Seems to disagree with fuzzball proposal. ● The behavior of the entropy generation and decay is in agreement with the proposal of Page for the entanglement of the black hole with the Hawking radiation.
Conclusions ● We tested ETH for a particular system using a Krylov subspace method that allows us to study much larger systems than before. ● For the same system we studied the generation of entanglement entropy, derived an expression for the short time increase and showed that it is capped by the thermal entropy at large times. ● Regarding black holes, we proposed that the metric is a thermal operator in the ETH sense implying that all microstates have a well defined metric equal to the Schwarzschild metric.