Topology & Entanglement in Driven (Floquet) Many Body Quantum Systems Ashvin Vishwanath Harvard University Adrian Po Lukasz Fidkowski Andrew Potter Harvard Seattle Texas Overview 1 ⌫ = log 2 ⌫ = log p log q 2 − Chiral phases in driven systems. The Driven Toric code. arXiv:1701.01440. Chiral Floquet arXiv:1701.01440. `Radical’ chiral phases. Po, Fidkowski, Morimoto, Potter, Floquet phases in a driven Kitaev AV. Phys. Rev. X 6, 041070 (2016) model. Po, Fidkowski, AV, Potter Overview 1 ⌫ = log 2 ⌫ = log p log q 2 − Chiral phases in driven systems. The Driven Toric code. arXiv:1701.01440. Chiral Floquet arXiv:1701.01440. `Radical’ chiral phases. Po, Fidkowski, Morimoto, Potter, Floquet phases in a driven Kitaev AV. Phys. Rev. X 6, 041070 (2016) model. Po, Fidkowski, AV, Potter Introduction: A gapped Hamiltonian J Volume = sites i i ⇠ · H ⌦ H energy H = Hr r X J = maxr Hr | r| excited states ∆ ground state low energy physics 4 Introduction: A gapped Hamiltonian J Volume = sites i i ⇠ · H ⌦ H energy H = Hr r X J = maxr Hr | r| excited states ∆ ground state low energy physics gapless 4 Introduction: A gapped Hamiltonian J Volume = sites i i ⇠ · H ⌦ H energy H = Hr r X J = maxr Hr | r| excited states ∆ ground state low energy physics gapless Need cooling (T=0) 4 Introduction: Entanglement Signatures Gapped Phases Gapless Phases Thermal Phases 1 S(⇢ ) ↵ @R log D + Area Law*log @R S Volume(R) R ⇠ | | − 2 ··· ⇠ Area Law R Volume Law Introduction: Entanglement Signatures Gapped Phases Gapless Phases Thermal Phases 1 S(⇢ ) ↵ @R log D + Area Law*log @R S Volume(R) R ⇠ | | − 2 ··· ⇠ Area Law R Volume Law Introduction: Classifying Gapped Quantum Phases Ground states of gapped local Hamiltonians How do we classify gapped ground states? No symmetry except t -> t+a Introduction: Classifying Gapped Quantum Phases Ground states of gapped local Hamiltonians How do we classify gapped ground states? No symmetry except t -> t+a Example 1:Quantum Hall Phases n=1$ h 1 Von$Klitzing+ n=2$ ⇢xy = e2 n n=3$ Bulk%Gapped% Edge%State% • Different integers - different phases. • Need to cool to low temperatures - protected by energy gap. • Also differentiated by Thermal-Hall. Integer in the right units. Example 2:Topological Order • Topological order (Witten, Wen). eg. Fractional Quantum Hall & Toric Code/Z2 Gauge theory 1 S(⇢ ) ↵ @R log D + R ⇠ | | − 2 ··· Example 2:Topological Order • Topological order (Witten, Wen). eg. Fractional Quantum Hall & Toric Code/Z2 Gauge theory topological ground state degeneracy on a torus anyonic excitations topological entanglement entropy 1 S(⇢ ) ↵ @R log D + R ⇠ | | − 2 ··· Example 2:Topological Order • Topological order (Witten, Wen). eg. Fractional Quantum Hall & Toric Code/Z2 Gauge theory topological ground state degeneracy on a torus anyonic excitations topological entanglement entropy 1 S(⇢ ) ↵ @R log D + R ⇠ | | − 2 ··· Half quantized thermal Hall c=5/2 Banerjee et al.1710.00492 Can we observe topological properties at finite energy density? Conventional answer: No energy J Volume A ⇠ · finite energy density Deutsch 91, Srednicki 94 Eigenstate Thermalization ground state Hypothesis (ETH) Can we observe topological properties at finite energy density? Conventional answer: No energy J Volume A ⇠ · finite energy density | i Deutsch 91, Srednicki 94 Eigenstate Thermalization ground state s=0 Hypothesis (ETH) Can we observe topological properties at finite energy density? Conventional answer: No energy J Volume A ⇠ · H/kB T ⇢a = e− finite energy density | i s = SA/NA Deutsch 91, Srednicki 94 Eigenstate Thermalization ground state s=0 Hypothesis (ETH) Avoiding thermalization - Hamiltonian made of commuting terms: z H0 = h σj Pauli z matrix on site j j X all eigenstates of H are area law entangled (could also do e.g. toric code) - BUT unstable to small perturbations due to translation symmetry: H = h σz + J σxσx + j j j+1 ··· j j X X finite energy density eigenstates are volume law entangled Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X H = H0 + H1 Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X H = H0 + H1 U = Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X H = H0 + H1 U = - can make H diagonal in z-basis using finite depth local unitary U Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X H = H0 + H1 U = - can make H diagonal in z-basis using finite depth local unitary U (Imbrie) Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X H = H0 + H1 U = - can make H diagonal in z-basis using finite depth local unitary U (Imbrie) U †XU Avoiding thermalization - Hamiltonian made of commuting terms with disordered coefficients: disordered couplings z z z H0 = hiσi + Ji,jσi σj + ... i i,j X X x - perturb with H1 = ciσi i X H = H0 + H1 U = - can make H diagonal in z-basis using finite depth local unitary U (Imbrie) if X is a local operator, then U † XU ‘dressed’ quasi-local operator. Many-body localization (Anderson, Mirlin et al., Basko, Aleiner, Altschuler; Oganesyan, Huse, Pal) - can make H diagonal in z-basis using finite depth unitary U: z z z U †HU = hi0 ⌧i + Jij0 ⌧i ⌧j + ... i ij X X - eigenstates of H are all of the form U ; product state in z-basis | i in particular, area law entangled - excited states `like’ ground states! - z forms complete set of quasi-local conserved quantities U⌧i U † (Huse, Nandkishore, Oganesyan `l-bits’ Serbyn, Papic, Abanin, Vosk&Altman ) Many-body localization and topological phases - replace ‘gapped’ with ‘many-body localized’ (MBL) generic Hamiltonians MBL Hamiltonians - topological order / symmetry protected topological phases at finite energy density (Bahri, Vosk, Altman, AV; Chandran, Khemani, Laumann, Sondhi, Huse, Nandkishore, Oganesyan, Pal) - BUT no chiral (quantum hall) phases allowed in MBL excited states. Commuting projectors incompatible with thermal Hall - (Kitaev, Levin, Potter-AV) Floquet driving - periodic time dependent Hamiltonian: T H(t + T )=H(t) UF = T exp i H(t)dt Z0 ! - diagonalize the ‘Floquet unitary’ UF Energy not conserved. h Only `quasi-energy’ mod ~! = T 14 Floquet systems: heating problem - generically, system will absorb energy until it is at infinite temperature: ⇢ (t)=Tr¯ (t) (t) A A | ih | A 1 βH ⇢A(t) 1 = e− ! ( Z ) as β 0 A¯ ! - entropy has been maximized, - no energy constraint. - Temperature ->∞ 15 MBL in Floquet systems: - MBL can be stable upon turning on a time dependent periodic perturbation: Ponte, Papic, Huveneers, Abanin; Lazarides, Das, Moessner iT Heff UF = e− with z z z U †He↵ U = hi0 σi + Ji,j0 σi σj i i,j X X - schematically, quasi-local commuting unitaries UF = U↵ ↵ Y - Can we find MBL Floquet phases that have no equilibrium analogue? What distinguishes them? No symmetries. Floquet SPTs in 1D & 3D: Else, Bauer, Nayak. Kayserlingk, Sondhi, Potter, Morimoto, AV. Potter, AV, Fidkowski. Free fermions (here bosons/Spins): Kitagawa, Demler, Rudner,Lindner, Berg, Levin. Rafael. Harper, Roy. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin. MODEL of a CHIRAL Floquet phase: U = U1 U2 U3U4 SWAP operation along different links Bulk: & Returns-to-starting- position-after-every- Floquet period EDGE: Advances one unit in each period One$Floquet period Rudner,Lindner, Berg, Levin.