Many-Body Localization. Geoffrey Ji

Transcription

1 Many-Body Localization Geoffrey Ji

2 Outline Aside: Quantum thermalization; ETH Single-particle (Anderson) localization Many-body localization Some phenomenology (l-bit model) Numerics & Experiments

3 Thermalization Classic Thermalization: System in contact with reservoir energy (and particle) flow Can describe system by few parameters after long time P, T, V, S, mu, etc. Quantum systems? Also can use classical reservoir But is concept of reservoir necessary? No! Classical Quantum?? Reservoir

4 Closed-System Quantum Thermalization In quantum system, initial state information perfectly preserved Unitary evolution System at time t depends on exact initial state no loss of information Obvious in eigenstate basis Directly contradicts thermalization! But does subsystem look thermal at long times? Can lose information about initial state to other subsystems!

5 Subsystem Thermalization Given a subsystem!: Needs to be local measure with few-body operators. E.g. compact in real-space Given some global initial state ": must have vanishing relative uncertainty in E, m, N, etc., as the reservoir! size (no cat states) Reduced density matrix is thermal at long times for large system! lim ) *,, * ", - = " / (1) A. M. Kaufman, et. al. Science 353, 6301 (2016)

6 Eigenstate Thermalization Hypothesis (ETH) Eigenstates don t time evolve ETH says all many-body eigenstates are already thermal (ETH) The exact statement is more precise (equivalence of So subsystem thermalization happens because eigenstates already are thermal! Where s the proof?? (of either subsystem thermalization or ETH) NONE (at least in general) Strong numerical evidence (exact diagonalization)

7 Breakdown of Thermalization Some systems fail to thermalize in general Integrable systems Conserved quantities ~4 Localizing systems Anderson localization single particles Many-body localization (interactions)

8 Anderson Localization Discussed briefly in class Reminder: Free spinless fermions in disorder Quantum interference from coherent backscattering 5 2: always localizing (i.e. wavefunction decays with length dependent on disorder) 5 > 2: localizing transition for some critical disorder strength

9 Anderson Localization Anderson (1958): Perturbative expansion in potential (not diagrammatics but essentially the same as in class) Coherent backscattering from higher order expansion terms causes localization (divergence of resistivity) Localization means failure to thermalize!

10 What about interactions? Do they matter? No general answer! Basko, Aleiner, Altshuler (2006): electron electron interaction alone cannot cause finite conductivity even when temperature is finite, but small enough MBL! Perturbative approach Mobility edge (unbound systems) Imbrie (2014): In 1D spin chains, non-perturbative effects bounded resonances rare Not 100% foolproof

11 Intuitive Picture (Perturbative Approach) Weak interactions should only modify the eigenstates! Start from localized model, dress with interactions New wavefunctions have weak hybridization Must do calculation to see if enough to thermalize! E 9 = 0 9 > 0

12 Phenomenology In general, localizing phases of models with interactions differ from non-interacting ones. But statements difficult to prove. Thermal Single-Particle Localizing Many-Body Localizing Conducting Insulating Insulating Initial conditions hidden Some initial conditions persist ETH holds ETH fails ETH fails Some initial conditions persist Power-law entanglement No entanglement Logarithmic growth of entanglement Dephasing & dissipation No dephasing or dissipation Dephasing (from entanglement), no dissipation R. Nandkishore, et. al., Annu. Rev. Condens. Matter Phys :15-38

13 Phenomenological Model In general, hard to get analytic results. Heavy dependence on ED Assuming the single-particle wavefunctions weakly hybridize (i.e. localization), we can construct effective Hamiltonian! Consider some Hamiltonian with 2-state local DOF (spin or charge) and local interactions, e.g.: Heisenberg w/ random z-field (also known as random-field XXZ) Single-spin Hubbard w/ NN interactions and random offsets Transverse Ising w/ weak (integrability-breaking) x-coupling Kitaev model (maps to above)

14 Phenomenological Model (Huse, 2014) Local DOF: N p-bits (physical bit), i.e. pseudospins Strong disorder: all eigenstates localized (finite # of eigenstates) So can also define N l-bits (localized bit) that are also pseudospins l-bit operators are ; <, and for non-interacting case look like, ; = ABCA D E In general, if eigenstates are localized (no transport), I = J h < L < M < + J O <,P L < M L P M <,P G F G + J J Q <,P,{S} L M < L M S1 L M M SW L P W=1 <,P,{S}

15 Making l-bits Perturbatively Could go perturbative route Intuitively, gives us all the right things! ; = composed of F G with exponential falloff Trouble when Δ =,=YG = 0! (resonances) Somewhat resolvable by arguing sufficient rarity of resonances not easy and not general (Imbrie 2014) Formulation is painful (Basko 2005) Let s not use perturbation theory

16 Making l-bits Non-Perturbatively Huse (2014): In MBL phase, should be possible to construct {; = } Each L = [ is (trivially) composed of sums of tensor products of \ = ] (including identity spin operators) Tensor product =,] _ =,] \ = ] has maximum distance between nonidentity operators Choose {; = } to minimize average of average distance of products in ; = Have to do this since possible that some ; = look non-local, but are exponentially rare Result should be finite!

17 Phenomenological Model I = J h = L = [ = + J O =,G L = [ L G [ =,G * + J J Q =,G,{`} L [ = L`a [ L`b [ [ L G cde =,G,{`} For 9 = 0, O =,G = Q =,G,{`} = 0 (no dephasing) Ergodicity breaking: all local L = [ are constants of motion! Long-range entanglement through phase information

18 Short-Time Growth of Entanglement The l-bits are exponentially localized in real space (p-bits) p-bits interact locally l-bit interaction suppressed exponentially vs. distance Start in product state of l-bits Short time dynamics: entanglement with nearby l-bits (localization length) Area-law entanglement particle entanglement grows as the coordination number, subsystem entanglement proportional to border size, not volume Area-law entanglement generally true for non-interacting systems as well

19 Long-Time Growth of Entanglement (No MBL) Thermalizing system: Consider spin-chain length f, interactions of O Entanglement from end-to-end should take g~hi Essentially a Lieb-Robinson velocity Intuition: site < gets entangled with site < + 1, site < + 1 gets entangled with site < + 2 and causes < and < + 2 to become entangled Aside: Sites <, P entangled if you trace over site < and site P is now mixed Localization, no interactions: Effective l-bit Hamiltonian isolates l-bits from one another Phases of l-bits evolve independently

20 Long-Time Growth of Entanglement (MBL) Without transport, entanglement growth not power-law Phase of ; = depends on L [ G, not ; G! l-bits cannot transfer entanglement amongst one another Entanglement only through direct terms in Hamiltonian So, rate of entanglement growth goes like: * O jkk =,G = O =,G + J J Q =,G,{`} L`a [ [ L`b cde p-bits exponentially local in l-bits => couplings die off exponentially => entanglement grows logarithmically {`}

21 Numerics Much of previous discussion not proven need numerics to verify DMRG on XX-model with static magnetic field and variable O z interaction (Bardarson, 2012), L=10 (and 20) Start with spins randomly aligned in z-axis, quench interactions on DMRG reduces complexity of problem by truncating unnecessary phase space Aligning in z-axis reduces entanglement, making computation feasible

22 Numerics Results Saturation of entanglement entropy for no O z (for subsystem) XX-model maps onto noninteracting fermions For O m > 0, unbounded logarithmic growth in entanglement entropy Even for very small interactions! Universal behavior Eventually saturates (finite system limits max entropy)

23 Numerics Results & Discussion Observation of volume-law entanglement entropy growth But saturated entanglement entropy per site much lower than thermal Perhaps limited thermalization with restricted ensemble Particle number fluctuations do not show phase transition Growth in entanglement is qualitiative difference between Anderson localization and MBL!

24 Experimental Work MBL difficult to observe in solid-state system Depends on isolated quantum system! Bath connection destroys MBL signatures (Johri 2015) Advances in AMO have made experiments possible Trapped ions (Smith 2016) Log growth of quantum fisher information (lower bound of entanglement entropy) Cold atoms (Schrieber 2015) Future experiments? Direct measurement of entanglement entropy possible! (Islam 2015)

25 References Theoretical Reviews: Nandkishore, R. & Huse, D. A. Many-Body Localization and Thermalization in Quantum Statistical Mechanics. Annual Review of Condensed Matter Physics 6, (2015). Theoretical Results: Anderson, P. W. Absence of Diffusion in Certain Random Lattices. Physical Review 109, (1958). Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal insulator transition in a weakly interacting many-electron system with localized single-particle states. Annals of Physics 321, (2006). Imbrie, J. Z. On Many-Body Localization for Quantum Spin Chains. Journal of Statistical Physics 163, (2016). Phenomenological Models: Huse, D. A., Nandkishore, R. & Oganesyan, V. Phenomenology of fully many-body-localized systems. Physical Review B 90, (2014). Numerics: Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded Growth of Entanglement in Models of Many-Body Localization. Physical Review Letters 109, (2012). Experiments: Smith, J. et al. Many-body localization in a quantum simulator with programmable random disorder. Nature Physics 12, (2016). Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, (2015).