Ergodic and Non-Ergodic Dynamics -II Vedika Khemani Harvard - PowerPoint PPT Presentation

Ergodic and Non-Ergodic Dynamics -II Vedika Khemani Harvard University

Unitary Quantum Dynamics Dynamics of isolated, MB systems undergoing unitary time evolution: ρ ( t + δ t ) = spins/cold atom molecules/ black holes/… U ρ ( t ) U † strongly interacting, excited (no quasiparticles) Time-independent U ( t ) = e − iHt Hamiltonian: Floquet: U ( nT ) = [ U ( T )] n Random U ( t ) = unitary circuit:

Can reversible unitary time evolution bring a system to thermal equilibrium at late times? If so, how does the system reach thermal equilibrium? For local operators A, how does the system “hide” ⟨ A ⟩ t=0 ? What are the dynamics of quantum entanglement? How does hydrodynamics emerge from reversible reversible unitary dynamics?

Many-Body “Quantum Chaos” vs. Thermalization What is a precise formulation for many-body quantum chaos? Is there a useful definition for chaos that is distinct from thermalization? Are there distinct (universal) signatures of chaos at early/intermediate/late times? What are the most appropriate observables for probing these regimes?

For local operators A, how does the system “hide” ⟨ A ⟩ t=0 ? Look at the dynamics of “operator spreading” i.e. time evolution of operators in the Heisenberg picture A 0 ( t ) = U † ( t ) A 0 U ( t ) Operator generically spreads ballistically within t a “Lieb-Robinson” cone — getting highly entangled v LR t within the cone — for clean, thermalizing local quantum systems. x A 0

For local operators A, how does the system “hide” ⟨ A ⟩ t=0 ? • Spreading can be sub- ballistic ~ t a , a<1 for A 0 ( t ) = U † ( t ) A 0 U ( t ) disordered thermalizing systems due to Griffiths effects t • Spreading is logarithmic for v LR t MBL systems. • Spreading is also ballistic for integrable systems with x quasiparticles A 0

Setup L spin 1/2 qubit Local Hilbert space dimension: 2 (can also consider qudits with q) 4 operators per site: σ µ µ ∈ { 0 , 1 , 2 , 3 } i Orthonormal basis of operators: Y ⊗ σ µ i S = (4) L “Pauli strings” i i Tr[ S † S 0 ] / (2 L ) = δ SS 0 xIyz, IzII, xxxx · · · VK Vishwanath Huse (2017)

Operator Spreading t x O ( t ) = U † ( t ) O 0 U ( t ) X sum over (4) L O ( t ) = a S ( t ) S Pauli strings S

Operator Spreading: unitarity Unitarity preserves operator norm Tr[ O † 0 ( t ) O 0 ( t )] = Tr[ O † 0 O 0 ] = (2 q ) L 2 L = ⇒ | a S ( t ) | 2 = 1 X S X O ( t ) = a S ( t ) S S Tr[ S † S 0 ] / (2 L ) = δ SS 0

Operator shape: Right weight Right-Weight: “emergent” density following from unitarity ρ L ( x, t ) ρ R ( x, t ) Each string has right/left edges beyond which it is purely identity. ρ looks at the density distribution of the “right front” of the operator. As operator spreads, weight moves to longer Pauli strings. x

Dynamics with Random circuits • Unitary gates independent and random in space and time. 2 i 2 i + 1 t = 0 • Allows us to derive exact results about operator t = 1 spreading, building in only the requirements of unitarity and t = 2 locality. t = 3 • Hope (and numerically verify) t = 4 that results generalize to more realistic setting like time- independent Hamiltonians Nahum et. al., (2016, 2017), von Keyserlingk et. al (2017).

Operator shape: random circuit Front dynamics: biased diffusion t U † ( δ t ) S U ( δ t ) has amplitudes for making S leaving it same making S x shorter length longer But, biased towards making S longer. Example, only 3/15 non-identity two-site spin 1/2 operators have identity on the right site. Nahum et. al., (2017) von Keyserlingk et. al (2017)

Operator shape: unconstrained circuit t · · · S Nahum et. al., (2017) von Keyserlingk et. al (2017)

Operator shape: unconstrained circuit Probability: 12/15 t + 1 t · · · S Nahum et. al., (2017) von Keyserlingk et. al (2017)

Operator shape: unconstrained circuit Probability: 3/15 t + 1 t · · · S Nahum et. al., (2017) von Keyserlingk et. al (2017)

Operator shape: unconstrained circuit t Front dynamics: biased random-walk Emergent hydrodynamics: ∂ t ρ R ( x, t ) = v B ∂ x ρ R ( x, t ) + D ρ ∂ 2 x ρ R ( x, t ) x 4 π D ρ te − ( x − vBt )2 1 ρ R ( x, t ) ≈ 4 D ρ t p v B ∼ 1 − 2 q 2 ; D ρ ∼ 2 q 2 Nahum et. al., (2017) von Keyserlingk et. al (2017)

Operator shape: unconstrained circuit Figure from: von Keyserlingk et. al (2017)

Thermalization + Conservation Law Chaotic many-body system (ballistic information spreading) + locally conserved diffusive densities (energy/charge/..) VK Vishwanath Huse (2017)

Unitarity vs. Dissipation Chaotic many-body system (ballistic information spreading) + locally conserved diffusive densities (energy/charge/..) Q: How does unitary quantum dynamics, which is reversible, give rise to diffusive hydrodynamics, which is dissipative (increases entropy)? Unitary Dynamics: Reversible Diffusion: Irreversible/Dissipation VK Vishwanath Huse (2017)

Setup L spin 1/2 qubit z component of spin 1/2 qubits conserved L X S tot z i = z i [ U ( t ) , S tot z ] = 0 VK Vishwanath Huse (2017)

Setup: Random Conserving Circuit Model 2 i 2 i + 1 t = 0 ↑↑ t = 1 U ( q 2 ) t = 2 ↑↓ , ↓↑ t = 3 ↓↓ t = 4 VK Vishwanath Huse (2017) Builds on: Nahum et. al., (2016, 2017), von Keyserlingk et. al (2017).

Operator Spreading t Spreading constrained by: • Unitarity x • Conservation Law(s) O ( t ) = U † ( t ) O 0 U ( t ) X sum over (4) L O ( t ) = a S ( t ) S strings S

Operator Spreading: conservation law Separate operator into conserved and non-conserved pieces 0 ( t ) + O nc O 0 ( t ) = O c 0 ( t ) X O c a c 0 ( t ) = i ( t ) z i i X O ( t ) = a S ( t ) S S Tr[ S † S 0 ] / (2 L ) = δ SS 0

Operator Spreading: conservation law Separate operator into conserved and non-conserved pieces exp(L) mostly non-local 0 ( t ) + O nc O 0 ( t ) = O c 0 ( t ) strings, thus “hidden” X O c a c 0 ( t ) = i ( t ) z i i L local operator “strings”, conserved densities X O ( t ) = a S ( t ) S S Tr[ S † S 0 ] / (2 L ) = δ SS 0

Operator Spreading: conservation law Separate operator into conserved and non-conserved pieces exp(L) mostly non-local 0 ( t ) + O nc O 0 ( t ) = O c 0 ( t ) strings, thus “hidden” X O c a c 0 ( t ) = i ( t ) z i i L local operator “strings”, conserved densities Tr[ O 0 ( t ) S tot z ] = constant ⇒ = L X O ( t ) = a S ( t ) S X a c i ( t ) = constant S Tr[ S † S 0 ] / (2 L ) = δ SS 0 i =1

Operator Spreading Operator dynamics governed by the interplay between: | a S ( t ) | 2 = 1 X Unitarity: S L X a c Conservation law: i ( t ) = constant i =1 VK Vishwanath Huse (2017)

Spreading of conserved charges First, consider spreading of conserved density O 0 = z 0 a c i ( t = 0) = δ i 0 X a c i ( t ) = 1 i VK Vishwanath Huse (2017)

Diffusion & conserved amplitudes: intuition Initial state: Infinite temperature equilibrium t = 0 + local charge perturbation a c x ⇢ 0 = 1 2 L [ I + ✏ O 0 ] 1 O 0 = z 0 x

Diffusion & conserved amplitudes: intuition Initial state: Infinite temperature equilibrium t = 0 + local charge perturbation a c x ⇢ 0 = 1 2 L [ I + ✏ O 0 ] 1 O 0 = z 0 x Diffusive charge spreading (coarse grained): a c x t > 0 h z i ( x, t ) = Tr[ ⇢ ( t ) z x ] √ 1 / t = ✏ 2 L Tr[ ⇢ ( t ) z x ] √ t x ( t ) ⇠ 1 x 2 = ✏ a c p te − 4 Dct x

Diffusion & conserved amplitudes O 0 = z 0 Random conserving circuit model X a c i ( t ) = 1 i r 1 coarse grain+ 2 π te − x 2 a c ( x, t ) ≈ 2 t scaling limit D c = 1 independent of q 2 VK Vishwanath Huse (2017)

Diffusive Lump X a c i ( t ) = 1 i Total operator weight in the diffusive lump of conserved charges decreases as a power-law in time. Significant weight in a “diffusive cone” near the origin, even at late times. VK Vishwanath Huse (2017)

Slow emission of non-conserved operators • No net loss in operator weight (unitarity). • Conserved parts emit a steady flux of “non-conserved” operators. • The local production of non-conserved operators is proportional to the square of the diffusion current, as in Ohm’s law: δρ nc i ( t ) ∼ ( a c i ( t ) − a c i +1 ( t )) 2 ∼ ( ∂ x a c ( x, t )) 2 VK Vishwanath Huse (2017)

Emergence of dissipation The dissipative process is the conversion of operator weight from locally observable conserved parts to non-conserved, non-local (non-observable) parts at a slow hydrodynamic rate. Observable entropy increases , while total von Neumann entropy of the full system is conserved.

Increase in observable entropy ⇢ ( t ) = 1 2 L [ I + ✏ O 0 ( t )] S vn ( t ) = const X O c a c 0 ( t ) = i ( t ) z i i S c vn ( t ) = − Tr[ ρ c ( t ) log ρ c ( t )] = L log(2) − 1 i ( t ) | 2 + · · · X | a c 2 i 1 d Z dx | j c ( x ) | 2 vn ( t ) ∼ dtS c 2 D c VK Vishwanath Huse (2017)