Many-Body Localized Phases and Phase Transitions

Many-Body Localized Phases and Phase Transitions Vedika Khemani Harvard University

Thermalization vs. Localization (t + t) = U (t)u Dynamics of Isolated Interacting Highly-excited Many-body systems undergoing unitary time evolution (Hamiltonian or Floquet)

Thermalization vs. Localization (t + t) = U (t)u Dynamics of isolated, interacting, highlyexcited many-body systems undergoing unitary time evolution (Hamiltonian or Floquet) Question: Can an isolated MB system act as it s own bath and bring its subsystems to thermal equilibrium? Yes: Thermal Phase New kind of phase transition No: MBL Phase

Emergent Integrability Generic models (not fine-tuned or explicitly integrable) H = X i h i z i + J X i ( x i x i+1 + y i y i+1 + z i z i+1) Detuned onsite fields Can be random/quasiperiodic/

Emergent Integrability Generic models (not fine-tuned or explicitly integrable) H = X i h i z i + J X i ( x i x i+1 + y i y i+1 + z i z i+1) Finite depth local unitary H = X i h i z i + X ij J ij z i z j + X ijk K ijk z i z j z k +... Extensively many local integrals of motion l-bits Oganesyan, Huse, Nandkishore (2014); Serbyn, Papic, Abanin (2013); Imbrie (2014); Bauer, Nayak (2013)

Phase structure with MBL MBL eigenstates can display frozen order (topological and non-topological) in highly-excited eigenstates. Orders might be forbidden in equilibrium (e.g. by Peierls-Mermin-Wagner type theorems) 1D transverse field Ising model Huse, Nandkishore, Oganesyan, Pal, Sondhi (2013); Pekker, Refael, Altman, Demler, Oganesyan (2013)

Phase structure with MBL MBL eigenstates can display frozen order (topological and non-topological) in highly-excited eigenstates. Orders might be forbidden in equilibrium (e.g. by Peierls-Mermin-Wagner type theorems) Interactions 1D transverse field Ising model Thermal MBL Spin-Glass MBL PM MBL Time-Crystal Parameter

This Talk Interactions Thermal Hybrid transition with continuous and discontinuous aspects VK, Lim, Sheng, Huse PRX (2016) MBL Spin-Glass MBL PM MBL Time-Crystal Parameter At least two different universality classes for the MBL PM- Thermal Transition VK, Sheng, Huse PRL (2017)

This Talk Interactions MBL Spin-Glass Thermal MBL PM MBL Time-Crystal New Dynamical Phase of matter previously thought to be disallowed in equilibrium. Rare example of eigenstate order that is experimentally observable VK, Lazarides, Moessner, Sondhi, PRL (2016) von Keyserlingk, VK, Sondhi, PRB (2016) Parameter VK, von Keyserlingk, Sondhi (2016) See also: Else, Bauer Nayak (2016) Choi, Choi, Landig, Kucsko, Zhou, Isoya, Jelezko, Onoda, Sumiya, VK, von Keyserlingk, Yao, Demler, Lukin, Nature (2017)

The MBL-Thermal Phase Transition All orders in perturbation theory (Basko, Aleiner, Altshuler 2006) Open questions: Exists in 1D lattice models with exponentially decaying interactions Almost proof including non-perturbative effects (Imbrie, 2014, 2016) Higher dimensions? Longer range interactions? Possible nonperturbative instabilities of the MBL phase (deroeck and Huveneers, 2016) Intermediate Phases? How do we distinguish an actual MBL-Thermal transition from a crossover between the Thermal and long-lived Thermal* phases? What is the structure of resonances driving the transition?

Dynamical Finite-Size Phase Diagram Transport dominated by rare Griffiths regions.

Discontinuity in entanglement entropy Grover (2014) showed that if the entanglement entropy of small subsystems varies continuously at a direct MBL-Thermal transition, then the entanglement of these subsystems looks thermal in the quantum critical regime. Careful numerical analysis finds the the QC regime looks localized. Either there is no direct transition, or entanglement is discontinuous at the transition. (VK, Lim, Sheng, Huse, PRX 2016)

Discontinuity in entanglement entropy Grover (2014) showed that if the entanglement entropy of small subsystems varies continuously at a direct MBL-Thermal transition, then the entanglement of these subsystems looks thermal in the quantum critical regime. Careful numerical analysis finds the critical regime actually looks localized. Thus, either there is no direct transition, or entanglement is discontinuous at the transition. (VK, Lim, Sheng, Huse, PRX 2016)

Discontinuity in entanglement entropy Transition driven by the proliferation of a sparse resonant backbone of entanglement. Just gains enough strength to thermalize the system on the thermal side of the crossover in the infinite size limit. Global discontinuity in presence of fully functional bath implies local discontinuity. Entanglement at the transition can show a strong non-local dependence on system size, since an infinite thermal system can act as a bath for any finite subsystem, no matter its local properties. S A = L A f(l 1/ (W W c ),L 1/ A (W W c)) Discontinuity was subsequently verified by a modified RG analysis (Dumitrescu, Vasseur, Potter 2017)

However, numerics show similar scaling for both random and quasiperiodic models at small sizes. But random models are beginning to show a crossover into the quenched-randomness-dominated universality class. VK, Sheng, Huse, PRL (2017) Finite-size Scaling For systems with quenched randomness, asymptotically at large L, 2/d (Harris 1974; Chayes, Chayes, Fisher, Spencer 1986; Chandran, Laumann, Oganesyan 2015) RG treatments find ' 3 Long standing mystery: all exact diagonalization numerics show scaling collapse, but with exponent ' 1 in violation of Harris

VK, Sheng, Huse, PRL (2017) Two Universality Classes for the MBL Transition External Randomness 1 Infinite Randomness Fixed Point ( 2/d) Thermal MBL 0 Non-Random Fixed Point Detuning

Summary Transition has discontinuous aspects: first-order like. What does first-order mean in this context? At least two universality classes (others?) non-random (quasiperiodic) infinite randomness Many open questions! Work with David Huse and Donna Sheng

VK, Lazarides, Moessner, Sondhi, PRL 116 (2016) Out of equilibrium phases: Floquet Eigenstate order in eigenstates of U(T) H(t) = ( H Z if 0 apple t<t 1 H X if T 1 apple t<t = T 1 + T 2 Floquet SPT H Z = X i H X = X i J i z i z i+1 + h i x i + JT 1 πsg/ Time Crystal P = Y i x i ht 2

VK, Lazarides, Moessner, Sondhi, PRL 116 (2016); von Keyserlingk, VK, Sondhi, PRB (2016) Floquet Phases - π SG Eigenstates are cat states with opposite parity eigenstates differing by quasienergy π/t U(T )=Pe it P i J i P = Y i x i z i z i+1 Long-range spin-glass eigenstate order in space: h z i z j i c = ±1 JT 1 πsg/ Time Crystal Order parameter oscillates with period 2T: z. i (nt )=( 1) n i z TTI breaking. ht 2 Correlated spatiotemporal order

VK, Lazarides, Moessner, Sondhi, PRL 116 (2016); von Keyserlingk, VK, Sondhi, PRB (2016) Spatio-temporal order Long-range spin order in space: h z i z j i c = ±1 Anti-ferromagnetic order in time: Discrete TTI breaking z i (nt )=( 1) n z i z (nt ) 1.0 0.5 Traditional MBL e.g., 0SG -0.5 5 10 15 20 Time crystal/πsg nt -1.0 2T

von Keyserlingk, VK, Sondhi (2016); Else Bauer Nayak (2016) π-sg is extremely robust 2 0 PM h gen. J z JT 1 0SG SG 0 Triv PM h x ht 2 2

von Keyserlingk, VK, Sondhi, PRB (2016) Temporal correlations: Generic initial states lim h 0 O(nT ) 0 i = c 0 + c ( 1) n + O L!1 apple 1 (nt ) 40 h x i (nt )i 20 0 10 h y i (nt )i 5 0 30 h z i (nt )i 15 0 /2T 0 /2T /T 3 /2T!

Experiments: Diamond NV centers Nature (2017)

Experiments: Trapped Ions Nature (2017)

Summary Discussed one example of a phase of matter which only exists in the out-of-equilibrium setting Novel form of correlated spatiotemporal order This is a case eigenstate order, but with clear experimental signatures! Lots of open questions and ongoing work on realizing new outof-equilibrium phases Nature of transitions between phases with different kinds of eigenstate order is also an active area of research Work with Shivaji Sondhi, Curt von Keyserlingk, Roderich Moessner, Achilleas Lazarides, Lukin Group