Kinetically constrained models, from classical to quantum Juan P. Garrahan £

Kinetically constrained models, from classical to quantum Juan P. Garrahan (University of Nottingham) J. Hickey (Nottingham) S. Genway, B. Olmos, I. Lesanovsky £ 1.- Why & what of KCMs ? 2.- Generalising to quantum dissipative KCMs 3.- How KCM dynamics emerges in Rydberg gases 4.- KCMs and many-body localisation in quantum systems 5.- Outlook Basics of KCMs Competing perspectives on glass transition & how to model them Thermodynamic Statics Dynamics eg. RFOT {Parisi+Wolynes+many others} ideal models e.g. p-spin spin glass Dynamic Statics does not Dynamics metric → Dynamic facilitation ideal models KCMs {Anderson+Andersen+Jackle+many others} Basics of KCMs ∂t |P⟩ = W|P⟩ → W = � � (n�−1 + δ) � εσ�+ + σ�− � − ε(1 − n� ) − n� + (� ↔ � − 1) Basics of KCMs trivial statics heterogeneous & hierarchical dynamics East model ∂t |P⟩ = W|P⟩ → W = � � (n�−1 + δ) � εσ�+ + σ�− � − ε(1 − n� ) − n� + (� ↔ � − 1) Basics of KCMs trivial statics heterogeneous & hierarchical dynamics East model ∂t |P⟩ = W|P⟩ → W = � � (n�−1 + δ) FA model � εσ�+ + σ�− � − ε(1 − n� ) − n� + (� ↔ � − 1) Basics of KCMs trivial statics heterogeneous & hierarchical dynamics East model ∂t |P⟩ = W|P⟩ → W = � � (n�−1 + δ) FA model � εσ�+ + σ�− � − ε(1 − n� ) − n� + (� ↔ � − 1) constraint = operator valued rate Basics of KCMs trivial statics heterogeneous & hierarchical dynamics East model ∂t |P⟩ = W|P⟩ → W = � � (n�−1 + δ) FA model � εσ�+ + σ�− � − ε(1 − n� ) − n� + (� ↔ � − 1) constraint = operator valued rate Dynamics is more than statics Transitions between active (relaxing) and inactive (non-relaxing) dynamical phases Can extend beyond classical glasses? Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} Recap: Open quantum systems and Quantum Markov processes HT = H + HB + HBS ρ = TrB |ψT ⟩⟨ψT | quantum master � equation: ∂t ρ = −�[H, ρ] + L� ρL� † − 1/ 2{L� † L� , ρ} ≡ W(ρ) {Lindblad, Belavkin-Stratonovich} � Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} Recap: Open quantum systems and Quantum Markov processes HT = H + HB + HBS ρ = TrB |ψT ⟩⟨ψT | quantum master � equation: ∂t ρ = −�[H, ρ] + L� ρL� † − 1/ 2{L� † L� , ρ} ≡ W(ρ) {Lindblad, Belavkin-Stratonovich} stochastic Ct deterministic P(Ct ) → ∂t |P⟩ = � W|P⟩ stochastic |ψ(t)⟩ deterministic ρ(t) → ∂t ρ = W(ρ) Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} Recap: Open quantum systems and Quantum Markov processes HT = H + HB + HBS ρ = TrB |ψT ⟩⟨ψT | quantum master � equation: ∂t ρ = −�[H, ρ] + L� ρL� † − 1/ 2{L� † L� , ρ} ≡ W(ρ) {Lindblad, Belavkin-Stratonovich} � stochastic Ct deterministic P(Ct ) → ∂t |P⟩ = E.g. laser-driven 2-level system at T = 0: deterministic ρ(t) → ∂t ρ = W(ρ) W|P⟩ |1⟩ Ω stochastic |ψ(t)⟩ H = Ω σ� L1 = κ |0⟩ � κσ− quantum jump trajectory Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} Recap: Open quantum systems and Quantum Markov processes HT = H + HB + HBS ρ = TrB |ψT ⟩⟨ψT | quantum master � equation: ∂t ρ = −�[H, ρ] + L� ρL� † − 1/ 2{L� † L� , ρ} ≡ W(ρ) {Lindblad, Belavkin-Stratonovich} � stochastic Ct deterministic P(Ct ) → ∂t |P⟩ = E.g. laser-driven 2-level system at T = 0: deterministic ρ(t) → ∂t ρ = W(ρ) W|P⟩ |1⟩ Ω stochastic |ψ(t)⟩ κ γ |0⟩ H = Ω σ� L1 = L2 = � κσ− � quantum jump trajectory γσ+ NB: classical Master Eq. ⊂ Quantum Master Eq. (H = 0 & L� = rank-1 ⇒ ρdi�g ) Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} Quantum facilitated spin models: (*) constrained dynamics (**) trivial stationary state Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} ƒ�−1 , ƒ� Quantum facilitated spin models: (*) constrained dynamics (**) trivial stationary state ƒ� , ƒ�+1 |1⟩ Ω κ γ |1⟩ Ω κ |0⟩ �−1 ··· γ |1⟩ κ Ω |0⟩ � γ |0⟩ �+1 ···   Ω → ƒ� Ω    κ → ƒ� κ  (*) γ → ƒ� γ Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} ƒ�−1 , ƒ� Quantum facilitated spin models: (*) constrained dynamics (**) trivial stationary state |1⟩ Ω κ Ω � κ |0⟩ ∂t ρ = −�[H, ρ] + ··· H= |1⟩ γ �−1 � ƒ� , ƒ�+1 ƒ� Ω σ�� γ κ Ω |0⟩ �  Ω → ƒ� Ω    κ → ƒ� κ  (*) γ → ƒ� γ γ |0⟩ �+1 � �  |1⟩ L� ρL� † − 1/ 2{L� † L� , ρ}  � · · · κ σ� L = ƒ �↓ �  −  L�↑ = ƒ� � � γ σ+ � γ κ = e−1/ T � Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} ƒ�−1 , ƒ� Quantum facilitated spin models: (*) constrained dynamics (**) trivial stationary state |1⟩ κ Ω Ω � ƒ�−1 , ƒ� |1⟩ Ω κ γ |0⟩ γ κ Ω �  Ω → ƒ� Ω    κ → ƒ� κ  (*) γ → ƒ� γ γ |0⟩ �+1 � �  |1⟩ |0⟩ L� ρL� † − 1/ 2{L� † L� , ρ}  � · · · κ σ� L = ƒ �↓ �  − ƒ� Ω σ��  L�↑ = ƒ� � � � γ σ+ γ κ = e−1/ T � ƒ� , ƒ�+1 |1⟩ ρ�st�t. Ω κ |0⟩ ∂t ρ = −�[H, ρ] + ··· H= |1⟩ γ �−1 � ƒ� , ƒ�+1 |1⟩ γ �| κ � ⟩⟨e =κp�γ|�� ⟩⟨�� | Ω+ pe |e |0⟩ |0⟩ |e⟩ |1⟩ |�⟩ |0⟩ ρst�t. = � � ρ�st�t. (**) Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} ƒ�−1 , ƒ� Quantum facilitated spin models: (*) constrained dynamics (**) trivial stationary state |1⟩ κ Ω Ω � κ L� ρL� † − 1/ 2{L� † L� , ρ}  � · · · κ σ� L = ƒ �↓ �  − ƒ� Ω σ��  L�↑ = ƒ� † Ω κ |1⟩ ρ�st�t. Ω γ |0⟩ |1⟩ γ �| κ � ⟩⟨e =κp�γ|�� ⟩⟨�� | Ω+ pe |e |0⟩ γ �+1 � �  Ω → ƒ� Ω    κ → ƒ� κ  (*) γ → ƒ� γ |0⟩ ƒ�−1 ,−→ ƒ� (∗) & (∗∗) ƒ� = ƒƒ�� , ,ƒ�+1ƒ�2 = ƒ� , [ƒ� , ρst�t. ] = 0 |1⟩ κ Ω |0⟩ �  |1⟩ γ |0⟩ ∂t ρ = −�[H, ρ] + ··· H= |1⟩ γ �−1 � ƒ� , ƒ�+1 |0⟩ |e⟩ |1⟩ |�⟩ |0⟩ � � � γ σ+ γ κ = e−1/ T � eg. qEast ƒ�+1 ≡ |e� ⟩⟨e� | ρst�t. = � � ρ�st�t. (**) Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} ƒ�−1 , ƒ� Quantum facilitated spin models: (*) constrained dynamics (**) trivial stationary state |1⟩ Ω κ qFA space Ω κ γ |1⟩ κ Ω |0⟩ � γ |0⟩  unconstrained ···  Ω → ƒ� Ω    κ → ƒ� κ  (*) γ → ƒ� γ �+1 qEast ··· time γ |1⟩ |0⟩ �−1 γ=0 ƒ� , ƒ�+1 Quantum KCM models of dissipative quantum glasses {Olmos-Lesanovsky-JPG, PRL 2012 + New 2014} ƒ�−1 , ƒ� Quantum facilitated spin models: (*) constrained dynamics (**) trivial stationary state ƒ� , ƒ�+1 |1⟩ κ Ω γ |1⟩ Ω γ κ |0⟩ |0⟩ �+1 � qFA qEast γ=0 ··· space  Ω → ƒ� Ω    κ → ƒ� κ  (*) γ → ƒ� γ γ κ Ω |0⟩ �−1  |1⟩ unconstrained ··· time 0.25 0.5 1 200 2 QJMC theory 0 0.25 2 0 {cf. Markland+ 2011} 4 0.5 !/" 1 2 0.5 1 !/" 2 1.0 1.0 N=5 N=5 exactexact Eq.(6) 0.5 0.5 Eq.(6) 0.0 0.0 D/D0-1 (x1e-2) 0 quantum (a) relaxation reentrance " D/D0-1 (x1e-2) 400 qFA classical vs unconstrained 6 500 -2 " = 10 " =#10 # D/D0-1 (x 1e-2) 600 8 qEast qFA 1000 #relax purely quantum relaxation slowdown at small Ω 1500 QMandel 800 -2 (b) D/D0-1 (x 1e-2) (a)0 γ= 5 0 -5 -0.5 -0.5 4 -1.0 -1.0 0 02 24 46 ! / "! / " 68 8 0 - Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} γ Rydberg atoms s-states van-der-Waals interaction: |1⟩ Ω {cf. Bloch/Kuhr} |0⟩ 1 atom/site no hopping strong dephasing Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} γ Rydberg atoms s-states van-der-Waals interaction: |1⟩ Ω {cf. Bloch/Kuhr}   � �  �  H = � n� + V�j n� nj  + Ω σ�� j � laser slow classical energy fast dephasing fast ∂t ρ = −� [H, ρ] + γ � � (n� ρn� − n� ρ − ρn� ) = Wf�st (ρ) + Wslow (ρ) |0⟩ 1 atom/site no hopping strong dephasing Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} γ Rydberg atoms s-states van-der-Waals interaction: |1⟩ Ω |0⟩ {cf. Bloch/Kuhr}  1 atom/site no hopping strong dephasing  � �j � classical energy fast dephasing fast ∂t ρ = −� [H, ρ] + γ � � (n� ρn� − n� ρ − ρn� ) = Wf�st (ρ) + Wslow (ρ) γ + �(� + V) Weak coupling Ω Ω laser slow Coherent superpositions decay fast Weak coupling � �  �  H = � n� + V�j n� nj  + Ω σ�� Classical subspace of interest Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126}  �  ∂t P =  �  � 1 � 1 + � + R2α nm ��=j |r̂� −r̂j |α � � �  �2  σ � − 1 P interactions → kinetic constraint trivial Pst�t. = 1 Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126}  �  ∂t P =  �  � 1 � 1 + � + R2α nm ��=j |r̂� −r̂j |α � � �  �2  σ � − 1 P trivial interactions → kinetic constraint Pst�t. = 1 (1) (2) ON resonance Δ=0 cf. Rydberg “blockade” OFF resonance Δ<0 “facilitation radius” Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} (1) ON resonance Δ=0  �  ∂t P =  �  � 1 � 1 + � + R2α nm ��=j |r̂� −r̂j |α � � �  �2  σ � − 1 P Trajectories (1D): dynamically heterogeneous & glassy Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} (1) ON resonance Δ=0  �  ∂t P =  �  1 � � 1 + � + R2α relaxation towards equilibrium (2D) nm ��=j |r̂� −r̂j |α � � �  �2  σ � − 1 P NB: trivial at t→∞ but via hyperuniform t 1/ 7 states ⟨N2 ⟩ − ⟨N⟩2 ≈0 theory: � n(t) ∼ t R−12 � d d+12 {cf . Torquato+} Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} (2) OFF resonance Δ<0  �  ∂t P =  �  � 1 � 1 + � + R2α nm ��=j |r̂� −r̂j |α � � �  �2  σ � − 1 P Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} (2) OFF resonance Δ<0  �  ∂t P =  �  � 1 � 1 + � + R2α n(t) Q(t) nm ��=j |r̂� −r̂j |α � � �  �2  σ � − 1 P Emergence of KCM dynamics in Rydberg systems {Lesanovsky-JPG, PRL 2013 + arXiv:1402.2126} (2) OFF resonance Δ<0  �  ∂t P =  �  � 1 � 1 + � + R2α nm ��=j |r̂� −r̂j |α � � �  �2  σ � − 1 P n(t) Q(t) uncorrelated excitation correlated growth saturation (h/u) KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Many-body localisation (MBL) transition: {Altshuler+, Huse+, many others} ‣ Like Anderson localisation but for interacting system ‣ Singular change throughout spectrum ‣ Eigenstates change from “thermal” (ETH {Deutsch, Sdrenicki}) to MBL ‣ Observables do not relax in MBL phase ‣ Often thought of as “glass transition” but modelled with disorder KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Many-body localisation (MBL) transition: {Altshuler+, Huse+, many others} ‣ Like Anderson localisation but for interacting system ‣ Singular change throughout spectrum ‣ Eigenstates change from “thermal” (ETH {Deutsch, Sdrenicki}) to MBL ‣ Observables do not relax in MBL phase ‣ Often thought of as “glass transition” but modelled with disorder Can KCMs (good models for classical glasses) say anything about quantum MBL? KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Recap: active-inactive “space-time” transitions in KCMs (eg. East/FA) 0.08 0.08 � � � � � −s + − W → Ws = n�−1 e !(s) εσ� + σ� − ε(1 − n� ) − !(s) n� + (� ↔ � − 1) 0.04 0.04 � Largest e/value = cumulant G.F. for activity 0.00 → 1st order phase transition 0.00 0.08 0.6 0.6 -0.2 -0.1 0.6 0.4 0.4 !(s) K(s) 0.4 0.04 0.2 0.00 0 0 s 0.1 0.2 0.2 K(s) 0.2 0 -0.2 -0.1 0 s 0.1 K(s) 0.2 0 -0.2 -0.1 0 s 0.1 0.2 KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Recap: active-inactive “space-time” transitions in KCMs (eg. East/FA) 0.08 0.08 � � � � � −s + − W → Ws = n�−1 e !(s) εσ� + σ� − ε(1 − n� ) − !(s) n� + (� ↔ � − 1) 0.04 0.04 � Largest e/value = cumulant G.F. for activity 0.00 → 1st order phase transition 0.00 0.08 0.6 0.6 -0.2 -0.1 0.6 0.4 0.4 !(s) K(s) 0.4 0.04 0.2 0.00 0 0 s 0.1 K(s) 0.2 0.2 0 -0.2 -0.1 0 s 0.1 0.2 K(s) Can 0.2 transform into Hermitian operator through equilibrium distribution Hs ≡ − P −10 � � −s � Ws P = − n�−1 e -0.2 -0.1 �0 0.1 0.2 εσ�� − ε(1 − n� ) − n� + (� ↔ � − 1) −�t Hs s |ψ0 ⟩ Consider as Hamiltonian and corresponding quantum unitary dynamics |ψt ⟩ = e KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Hs = − � � � n�−1 e −s � εσ�� − ε(1 − n� ) − n� + (� ↔ � − 1) |ψt ⟩ = e−�t Hs |ψ0 ⟩ Signatures of MBL transition: (i) relaxation / non-relaxation of observables time evolution of magnetisation � ⟨M⟩t = ⟨ψt |σ�z |ψt ⟩ � does not relax on inactive side s>0 relaxes on active side s<0 KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Hs = − � � � n�−1 e −s � εσ�� − ε(1 − n� ) − n� + (� ↔ � − 1) |ψt ⟩ = e−�t Hs |ψ0 ⟩ Signatures of MBL transition: (ii) transitions throughout spectrum no ETH on inactive side s>0 N = 16 N=9 ETH on active side s<0 KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Hs = − � � � n�−1 e −s � εσ�� − ε(1 − n� ) − n� + (� ↔ � − 1) |ψt ⟩ = e−�t Hs |ψ0 ⟩ Signatures of MBL transition: (iii) localisation onto classical basis widely spread active s<0 e/states concentrated inactive s>0 KCMs and many-body localisation in closed quantum systems {Hickey-Genway-JPG, arXiv:1405.5780} Hs = − � � � n�−1 e −s � εσ�� − ε(1 − n� ) − n� + (� ↔ � − 1) |ψt ⟩ = e−�t Hs |ψ0 ⟩ Signatures of MBL transition: (iii) localisation onto classical basis widely spread active s<0 e/states concentrated inactive s>0 active-inactive transition if 1st order MBL transition in whole spectrum first model with MBL transition without disorder SUMMARY KCMs as open quantum glasses qFA & qEast interplay between classical & quantum fluctuations KCMs emergent in atomic Rydberg gases correlated non-equilibrium structures (e.g. hyperuniform) recent experimental evidence KCMs and many-body localisation in quantum systems s-ensemble active-inactive transition throughout spectrum MBL models without disorder

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