Many-Body Quantum Dynamics in Closed Systems

Absence of thermalization 1 / 19 Absence of thermalization in non-integrable systems Christian Gogolin, Arnau Riera, Markus Müller, and Jens Eisert Dahlem Center for Complex Quantum Systems, Freie Universität Berlin Workshop “Many-Body Quantum Dynamics in Closed Systems” Barcelona September 7-9 2011 Absence of thermalization | Introductory words Old questions and new contributions How do quantum mechanics and statistical mechanics go together? 2 / 19 Absence of thermalization | Setup and terminology Many-Body Quantum Dynamics in Closed Systems [1] M. Cramer, C. Dawson, J. Eisert, and T. Osborne, PRL 100 (2008) 030602 [2] P. Reimann, PRL 101 (2008) 190403 3 / 19 Absence of thermalization | Setup and terminology 3 / 19 Many-Body Quantum Dynamics in Closed Systems H S ⊗1 1⊗H B System “Bath” [1] M. Cramer, C. Dawson, J. Eisert, and T. Osborne, PRL 100 (2008) 030602 [2] P. Reimann, PRL 101 (2008) 190403 Absence of thermalization | Setup and terminology 3 / 19 Many-Body Quantum Dynamics in Closed Systems H = H S ⊗1 + H SB + 1 ⊗ H B |ψt i = e− i H t |ψ0 i At = Tr[A|ψt ihψt |] ψtS = TrB [|ψt ihψt |] System “Bath” [1] M. Cramer, C. Dawson, J. Eisert, and T. Osborne, PRL 100 (2008) 030602 [2] P. Reimann, PRL 101 (2008) 190403 Absence of thermalization | Setup and terminology 3 / 19 Many-Body Quantum Dynamics in Closed Systems H = H S ⊗1 + H SB + 1 ⊗ H B |ψt i = e− i H t |ψ0 i At = Tr[A|ψt ihψt |] ψtS = TrB [|ψt ihψt |] Equilibration: t ? t System “Bath” strong: equilibrated between t1 and t2 [1] weak: equilibrated for most times [2] [1] M. Cramer, C. Dawson, J. Eisert, and T. Osborne, PRL 100 (2008) 030602 [2] P. Reimann, PRL 101 (2008) 190403 Absence of thermalization | Setup and terminology 3 / 19 Many-Body Quantum Dynamics in Closed Systems H = H S ⊗1 + H SB + 1 ⊗ H B |ψt i = e− i H t |ψ0 i At = Tr[A|ψt ihψt |] ψtS = TrB [|ψt ihψt |] Equilibration: t ? t System “Bath” strong: equilibrated between t1 and t2 [1] weak: equilibrated for most times [2] Thermalization: T ? T ψtS ≈ ρGibbs ∝ e−β H S [1] M. Cramer, C. Dawson, J. Eisert, and T. Osborne, PRL 100 (2008) 030602 [2] P. Reimann, PRL 101 (2008) 190403 Absence of thermalization | Equilibration and a maximum entropy principle Equilibration and a maximum entropy principle 4 / 19 Absence of thermalization | Equilibration and a maximum entropy principle Maximum entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average Tr[A ψt ] = Tr[A ψt ] = Tr[A ω], X where ω = πk ψ0 πk k (with πk the energy eigen projectors) is the dephased state that maximizes the von Neumann entropy, given all conserved quantities. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Equilibration and a maximum entropy principle averaging MaximumTime entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average ψ0 = Tr[A ψt ] = Tr[A ψt ] = Tr[A ω], X where ω = πk ψ0 πk k (with πk the energy eigen projectors) is the dephased state that maximizes the von Neumann entropy, given all conserved quantities. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Equilibration and a maximum entropy principle averaging MaximumTime entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average ψ0 = Tr[A ψt ] = Tr[A ψt ] = Tr[A ω], X where ω = πk ψ0 πk k (with πk the energy eigen projectors) is the dephased state that maximizes the von Neumann entropy, given all conserved quantities. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Equilibration and a maximum entropy principle averaging MaximumTime entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average ψt = Tr[A ψt ] = Tr[A ψt ] = Tr[A ω], X where ω = πk ψ0 πk k (with πk the energy eigen projectors) is the dephased state that maximizes the von Neumann entropy, given all conserved quantities. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Equilibration and a maximum entropy principle averaging MaximumTime entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average ω= Tr[A ψt ] = Tr[A ψt ] = Tr[A ω], X where ω = πk ψ0 πk k (with πk the energy eigen projectors) is the dephased state that maximizes the von Neumann entropy, given all conserved quantities. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Equilibration and a maximum entropy principle averaging MaximumTime entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average ω= Tr[A ψt ] = Tr[A ψt ] = Tr[A ω], X where ω = πk ψ0 πk k (with πk the energy eigen projectors) is the dephased state that maximizes the von Neumann entropy, given all conserved quantities. ψ0 → ω is a pinching ⇒ ω maximizes entropy. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Equilibration and a maximum entropy principle Maximum entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average Tr[A ψt ] = Tr[A ψt ] = Tr[A ω], X where ω = πk ψ0 πk k (with πk the energy eigen projectors) is the dephased state that maximizes the von Neumann entropy, given all conserved quantities. ⇒ Maximum entropy principle from pure quantum dynamics. Has nothing to do with (non)-integrability. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Equilibration and a maximum entropy principle Maximum entropy principle Theorem 1 (Maximum entropy principle [3]) If Tr[A ψt ] equilibrates, it equilibrates towards its time average Interesting open Tr[Aquestions: ψt ] = Tr[A ψt ] = Tr[A ω], Do we really need all (exponentially many) conserved X πk ψ0 πk quantities? where ω = k If not, then which? (with πk theDoes energy projectors) is the dephased state that thiseigen depend on integrability of the model? maximizes the von Neumann entropy, given all conserved quantities. What is the relation to the GGE? ⇒ Maximum entropy principle from pure quantum dynamics. Has nothing to do with (non)-integrability. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 5 / 19 Absence of thermalization | Thermalization and integrability Thermalization and integrability 6 / 19 Absence of thermalization | Thermalization and integrability 7 / 19 Thermalization is a complicated process Thermalization implies: 1 Equilibration [2, 4, 5] 2 Subsystem initial state independence [3] 3 Weak bath state dependence [6] 4 Diagonal form of the subsystem equilibrium state [7] 5 Gibbs state e−β H [5, 6] [2] [4] [5] [3] [6] [7] P. Reimann, PRL 101 (2008) 190403 N. Linden, S. Popescu, A. J. Short, and A. Winter, PRE 79 (2009) no. 6, 061103 J. Gemmer, M. Michel, and G. Mahler, Springer (2009) C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 A. Riera, C. Gogolin, and J. Eisert, 1102.2389 C. Gogolin, PRE 81 (2010) no. 5, 051127 Absence of thermalization | Thermalization and integrability 8 / 19 Thermalization and quantum integrability There is a common belief in the literature [8, 9, 10, 11, 12] . . . Non-integrable Integrable =⇒ =⇒ Thermalization No thermalization [8] C. Kollath et. al PRL 98, (2007) 180601 [9] S. Manmana, S. Wessel, R. Noack, and A. Muramatsu, ibid. 98 (2007) 210405 [10] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452 (2008) 854 [11] M. C. Banuls, J. I. Cirac, and M. B. Hastings, arXiv:1007.3957 [12] M. Rigol, PRL 103, (2009) 100403 Absence of thermalization | Thermalization and integrability 8 / 19 Thermalization and quantum integrability There is a common belief in the literature [8, 9, 10, 11, 12] . . . Non-integrable Integrable =⇒ =⇒ Thermalization No thermalization . . . but there are problems. [8] C. Kollath et. al PRL 98, (2007) 180601 [9] S. Manmana, S. Wessel, R. Noack, and A. Muramatsu, ibid. 98 (2007) 210405 [10] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452 (2008) 854 [11] M. C. Banuls, J. I. Cirac, and M. B. Hastings, arXiv:1007.3957 [12] M. Rigol, PRL 103, (2009) 100403 Absence of thermalization | Thermalization and integrability Notions of (non-)integrability A system is with n degrees of freedom is integrable if: There exist n (local) conserved mutually commuting linearly independent operators. There exist n (local) conserved mutually commuting algebraically independent operators. The system is integrable by the Bethe ansatz. The system exhibits nondiffractive scattering. The quantum many-body system is exactly solvable in any way. ... 9 / 19 Absence of thermalization | Thermalization and integrability Notions of (non-)integrability A system is with n degrees of freedom is integrable if: There exist n (local) conserved mutually commuting linearly independent operators. There exist n (local) conserved mutually commuting algebraically independent operators. The system is integrable by the Bethe ansatz. The system exhibits nondiffractive scattering. The quantum many-body system is exactly solvable in any way. ... And non-integrable otherwise? 9 / 19 Absence of thermalization | Thermalization and integrability Notions of (non-)integrability A system is with n degrees of freedom is integrable if: There exist n (local) conserved mutually commuting linearly independent operators. There exist n (local) conserved mutually commuting algebraically independent operators. The system is integrable by the Bethe ansatz. The system exhibits nondiffractive scattering. The quantum many-body system is exactly solvable in any way. ... And non-integrable otherwise? Lack of imagination? 9 / 19 Absence of thermalization | Thermalization and integrability 10 / 19 Reminder on integrability in classical mechanics Classical Liouville integrability A system with n degrees of freedom is called integrable if it entails a maximal set of n independent Poisson commuting constants of motion and is called non-integrable otherwise [13]. [13] V. I. Arnold, Mathematical Methods Of Classical Mechanics (1989) Absence of thermalization | Thermalization and integrability 10 / 19 Reminder on integrability in classical mechanics Classical Liouville integrability A system with n degrees of freedom is called integrable if it entails a maximal set of n independent Poisson commuting constants of motion and is called non-integrable otherwise [13]. Classical: integrability ⇒ systematic solvable and evolution on a n-torus Quantum: always systematic solvable and evolution on a d-torus [13] V. I. Arnold, Mathematical Methods Of Classical Mechanics (1989) Absence of thermalization | Thermalization and integrability 10 / 19 Reminder on integrability in classical mechanics Classical Liouville integrability A system with n degrees of freedom is called integrable if it entails a maximal set of n independent Poisson commuting constants of motion and is called non-integrable otherwise [13]. Classical: Quantum: integrability ⇒ systematic solvable and evolution on a n-torus always systematic solvable and evolution on a d-torus qualitative question quantitative question? [13] V. I. Arnold, Mathematical Methods Of Classical Mechanics (1989) Absence of thermalization | Thermalization and integrability 10 / 19 Reminder on integrability in classical mechanics Classical Liouville integrability A system with n degrees of freedom is called integrable if it entails a maximal set of n independent Poisson commuting constants of motion and is called non-integrable otherwise [13]. Classical: Quantum: integrability ⇒ systematic solvable and evolution on a n-torus always systematic solvable and evolution on a d-torus qualitative question quantitative question? thermalization ⇒ non-integrability thermalization ✟ ⇐ ✟ non-integrability thermalization ⇐ non-integrability ? [13] V. I. Arnold, Mathematical Methods Of Classical Mechanics (1989) Absence of thermalization | Thermalization and integrability Absence of thermalization in non integrable systems Result (Theorem 1 and 2 in [3]): Too little (geometric) entanglement in the energy eigenbasis prevents initial state independence. This can happen even in non-integrable systems. [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 11 / 19 Absence of thermalization | Thermalization and integrability 11 / 19 Absence of thermalization in non integrable systems Result (Theorem 1 and 2 in [3]): Too little (geometric) entanglement in the energy eigenbasis prevents initial state independence. This can happen even in non-integrable systems. S |ψ1 i B S B t |ψ2 i [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 Absence of thermalization | Thermalization and integrability 11 / 19 Absence of thermalization in non integrable systems The model: Result (Theorem 1 and 2 in [3]): Spin-1/2 XYZ (geometric) chain with random coupling on-site field. Too little entanglement in and the energy eigenbasis preventsn initial state independence. n−1 X X Z ~bi · ~σ NNsystems. This can happen H = evenhiinσinon-integrable + i i=1 S |ψ1 i i=1 B S B t |ψ2 i [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 Absence of thermalization | Thermalization and integrability 11 / 19 Absence of thermalization in non integrable systems The model: Result (Theorem 1 and 2 in [3]): Spin-1/2 XYZ (geometric) chain with random coupling on-site field. Too little entanglement in and the energy eigenbasis preventsn initial state independence. n−1 X X Z ~bi · ~σ NNsystems. This can happen H = evenhiinσinon-integrable + i i=1 S B Interesting open questions: |ψ1 i |ψ2 i i=1 S B What is the relation to Anderson localization? t Can this also happen in translation invariant systems? [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 Absence of thermalization | Thermalization and integrability 11 / 19 Absence of thermalization in non integrable systems Result (Theorem 1 and 2 in [3]): Too little (geometric) entanglement in the energy eigenbasis prevents initial state independence. This can happen even in non-integrable systems. S |ψ1 i B S B t |ψ2 i [3] C. Gogolin, M. P. Mueller, and J. Eisert, PRL 106 (2011) 040401 Absence of thermalization | Proving thermalization Proving thermalization 12 / 19 Absence of thermalization | Proving thermalization 13 / 19 Two ways to prove thermalization Thermalization ETH Our result |Ek i {hEk |ψ0 |Ek i} Assumptions about: Absence of thermalization | Proving thermalization 13 / 19 Two ways to prove thermalization Thermalization ETH Our result |Ek i {hEk |ψ0 |Ek i} Assumptions about: Absence of thermalization | Proving thermalization Structure of the argument [14] S. Goldstein, PRL 96 (2006) no. 5, 050403 [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 14 / 19 Absence of thermalization | Proving thermalization Structure of the argument Classical level counting à la Goldstein [14] with no interaction H 0 = H S ⊗1 + 1 ⊗ H B [14] S. Goldstein, PRL 96 (2006) no. 5, 050403 [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 14 / 19 Absence of thermalization | Proving thermalization Structure of the argument Classical level counting à la Goldstein [14] with no interaction H 0 = H S ⊗1 + 1 ⊗ H B + Perturbation theory for realistic weak coupling [6] k H SB k∞ ≪ kB T [14] S. Goldstein, PRL 96 (2006) no. 5, 050403 [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 14 / 19 Absence of thermalization | Proving thermalization 14 / 19 Structure of the argument Classical level counting à la Goldstein [14] with no interaction H 0 = H S ⊗1 + 1 ⊗ H B Typicality arguments + Perturbation theory for realistic weak coupling [6] k H SB k∞ ≪ kB T [14] S. Goldstein, PRL 96 (2006) no. 5, 050403 [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 Kinematic Absence of thermalization | Proving thermalization 14 / 19 Structure of the argument Classical level counting à la Goldstein [14] with no interaction H 0 = H S ⊗1 + 1 ⊗ H B Typicality arguments Kinematic Equilibration results Dynamic + Perturbation theory for realistic weak coupling [6] k H SB k∞ ≪ kB T [14] S. Goldstein, PRL 96 (2006) no. 5, 050403 [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 Absence of thermalization | Proving thermalization 15 / 19 The result k H SB k∞ ≫ gaps(H 0 ) k H SB k∞ ≪ kB T ≪ ∆ ΩB ∆ (E) hEk |ψ0 |Ek i [∆] E [ ] E =⇒ “Theorem” 2 (Theorem 2 in [6]) (Kinematic) Almost all pure states from a microcanonical subspace [E, E + ∆] are locally close to a Gibbs state. [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 Absence of thermalization | Proving thermalization 15 / 19 The result k H SB k∞ ≫ gaps(H 0 ) k H SB k∞ ≪ kB T ≪ ∆ ΩB ∆ (E) hEk |ψ0 |Ek i [∆] E [ ] E 6= =⇒ “Theorem” 2 (Theorem 2 in [6]) (Kinematic) Almost all pure states from a microcanonical subspace [E, E + ∆] are locally close to a Gibbs state. (Dynamic) All initial states ψ⊓,0 locally equilibrate towards a Gibbs state, even if they are initially far from equilibrium. [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 E Absence of thermalization | Proving thermalization 15 / 19 The result k H SB k∞ ≫ gaps(H 0 ) k H SB k∞ ≪ kB T ≪ ∆ ΩB ∆ (E) hEk |ψ0 |Ek i [∆] E [ ] E 6= =⇒ “Theorem” 2 (Theorem 2 in [6]) (Kinematic) Almost all pure states from a microcanonical subspace [E, E + ∆] are locally close to a Gibbs state. (Dynamic) All initial states ψ⊓,0 locally equilibrate towards a Gibbs state, even if they are initially far from equilibrium. [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 E Absence of thermalization | Proving thermalization 15 / 19 The result k H SB k∞ ≫ gaps(H 0 ) k H SB k∞ ≪ kB T ≪ ∆ ΩB ∆ (E) hEk |ψ0 |Ek i [∆] E [ ] E 6= =⇒ “Theorem” 2 (Theorem 2 in [6]) (Kinematic) Almost all pure states from a microcanonical subspace [E, E + ∆] are locally close to a Gibbs state. (Dynamic) All initial states ψ⊓,0 locally equilibrate towards a Gibbs state, even if they are initially far from equilibrium. [6] A. Riera, C. Gogolin, and J. Eisert, 1102.2389 E Absence of thermalization | Conclusions 16 / 19 Conclusions Absence of thermalization | Conclusions Conclusions There is equilibration in closed quantum systems. 17 / 19 Absence of thermalization | Conclusions Conclusions There is equilibration in closed quantum systems. We can prove thermalization under quite natural assumptions. 17 / 19 Absence of thermalization | Conclusions Conclusions There is equilibration in closed quantum systems. We can prove thermalization under quite natural assumptions. Quantum mechanics implies a maximum entropy principle. 17 / 19 Absence of thermalization | Conclusions Conclusions There is equilibration in closed quantum systems. We can prove thermalization under quite natural assumptions. Quantum mechanics implies a maximum entropy principle. How is this related to the GGE and ETH? 17 / 19 Absence of thermalization | Conclusions Conclusions There is equilibration in closed quantum systems. We can prove thermalization under quite natural assumptions. Quantum mechanics implies a maximum entropy principle. How is this related to the GGE and ETH? Can we capture the intuition behind non-integrability in a mathematically precise definition? 17 / 19 Absence of thermalization | Conclusions Conclusions There is equilibration in closed quantum systems. We can prove thermalization under quite natural assumptions. Quantum mechanics implies a maximum entropy principle. How is this related to the GGE and ETH? Can we capture the intuition behind non-integrability in a mathematically precise definition? How are non-integrability and thermalization related? 17 / 19 Absence of thermalization | Acknowledgements 18 / 19 Collaborators Arnau Riera Markus P. Müller Martin Kliesch Jens Eisert Absence of thermalization | References References Thank you for your attention! −→ slides: www.cgogolin.de [1] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, “Exact Relaxation in a Class of Nonequilibrium Quantum Lattice Systems”, Phys. Rev. Lett. 100 (2008) 030602. [2] P. Reimann, “Foundation of Statistical Mechanics under Experimentally Realistic Conditions”, Physical Review Letters 101 (2008) no. 19, 190403. [3] C. Gogolin, M. Müller, and J. Eisert, “Absence of Thermalization in Nonintegrable Systems”, Physical Review Letters 106 (2011) no. 4, 040401. [4] N. Linden, S. Popescu, A. J. Short, and A. Winter, “Quantum mechanical evolution towards thermal equilibrium”, Physical Review E 79 (2009) no. 6, 061103. [5] J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics, vol. 784. Springer, Berlin / Heidelberg, 2009. [6] A. Riera, C. Gogolin, and J. Eisert, “Thermalization in nature and on a quantum computer”, 1102.2389v1. [7] C. Gogolin, “Environment-induced super selection without pointer states”, Physical Review E 81 (2010) no. 5, 051127. [8] C. Kollath, A. Läuchli, and E. Altman, “Quench Dynamics and Nonequilibrium Phase Diagram of the Bose-Hubbard Model”, Physical Review Letters 98 (2007) no. 18, 180601. [9] S. Manmana, S. Wessel, R. Noack, and A. Muramatsu, “Strongly Correlated Fermions after a Quantum Quench”, Physical Review Letters 98 (2007) no. 21, 210405. [10] M. Rigol, V. Dunjko, and M. Olshanii, “Thermalization and its mechanism for generic isolated quantum systems”, Nature 452 (2008) no. 7189, 854. [11] M. C. Banuls, J. I. Cirac, and M. B. Hastings, “Strong and weak thermalization of infinite non-integrable quantum systems”, 1007.3957v1. http://www.citebase.org/abstract?id=oai:arXiv.org:1007.3957. [12] M. Rigol, “Breakdown of Thermalization in Finite One-Dimensional Systems”, Physical Review Letters 103 (2009) no. 10, 100403. [13] V. I. Arnold, Mathematical Methods Of Classical Mechanics. Spriunger-Verlag, 1989. [14] S. Goldstein, “Canonical Typicality”, Physical Review Letters 96 (2006) no. 5, 050403. 19 / 19