Measuring entanglement entropy of a generic many-body system. Dima Abanin (Harvard) Eugene Demler (Harvard)

Transcription

1 Measuring entanglement entropy of a generic many-body system Dima Abanin (Harvard) Eugene Demler (Harvard) CIFAR Summer School, Toronto May 17, 2012

2 Entanglement Entropy: Definition -Many-body system in a pure state -Divide into two parts, L, R L ψ l R -Reduced density matrix for left part (effectively mixed state) ρ L = Tr R ψ ψ -Entanglement entropy: S = Trρ L logρ L -Characterizes the degree of entanglement in ψ

3 Entanglement entropy across different fields -Many-body quantum systems: universal way to characterize quantum phases -Guide for numerical simulations of 1D quantum systems (e.g., spin chains) -Topological entanglement entropy: measure of topological order -Black hole entropy -Quantum field theories

4 Scaling law for entanglement entropy -1D system,? -Gapped systems: -Fermi gas S(l) S(l) const S(l) lnl l l -Any critical system (conformal field theory): c -- central charge S(l) = c 3 lnl +O(1) Wilczek et al 94 Vidal et al 03 Cardy, Calabrese 04 IMPLICATIONS: -Measure of the phase transition location and central charge -Independent of the nature of the order parameter!

5 Topological entanglement entropy Topological order -no symmetry breaking or order parameter -degeneracy of the ground state on a torus -anyonic excitations -gapless edge states (in some cases) Physical realizations: -Fractional quantum Hall states -Z2 spin liquids (simulations) -Kitaev model and its variations DIFFICULT TO DETECT

6 Topological entanglement entropy -Three finite regions, A, B, C -Define topological entanglement entropy: (Kitaev Preskill 06 ;Levin, Wen 06) S Topo = S A + S B + S C S A B S A C S B C + S A B C -In a topologically non-trivial phase, S Topo = γ -A unique way to detect top. order -Proved useful in numerical studies invariant characterizing the kind of top. order Isakov, Melko, Hasting 11 Grover, Vishwanath 11

7 Existing proposals to measure entanglement entropy experimentally -Free fermions in 1D (e.g., quantum point contact) -Relate entanglement entropy to particle number fluctuations in left region in the ground state (Physical reason: particle number fluctuations in a Fermi gas grow as log(l)) Klich, Levitov 06 Song, Rachel, Le Hur et al 10, 12 -Limited to the case of free particles -Breaks down when interactions are introduced (e.g., for a Luttinger liquid) Hsu, Grosfield, Fradkin 09 Song, Rachel, Le Hur 10

8 Is it possible to measure entanglement in a generic interacting many-body system? (such that the measurement complexity would not grow exponentially with system size) Challenging nonlocal quantity, involves exponentially many degrees of freedom..

9 Proposed solution: entangle (a specially designed) composite many-body system with a qubit. Will show that Entanglement Entropy can be measured by studying just the dynamics of the qubit

10 Renyi Entanglement Entropy -Many-body system in a pure state L ψ R -Reduced density matrix -n-th Renyi entropy: PROPERTIES: -Obeys scaling laws ρ L = Tr R ψ -Analytic continuation nà 1 gives von Neumann entropy -Knowing all Renyi entropies à reconstruct full entanglement spectrum -As useful as the von Neumann entropy ψ ( ) S n = 1 1 n log Tr ρ n L l

11 System of interest L R l -Finite many-body system Δ -short-range interactions and hopping (e.g., Hubbard model) -Ground state separated from excited states by a gap Δ Gapped phase: Gapless phase Δ Δ 0,l >> ξ ξ Δ hv F / l v F Correlation length Fermi velocity

12 Useful fact: relation of entanglement and overlap of a composite many-body system -Consider two identical copies of the many-body system 2 Different ways of connecting 4 sub-systems: Way 1: Way 2: L 1 R 1 L 1 R 1 L 2 R 2 L 2 R 2 Ground state 0 Ground state 0' -Overlap gives second Renyi entropy: 0 0' = e S 2

13 Derivation L R Schmidt decomposition of ground state for a single system l Ψ 0 = λ i ψ ϕ i L i R i ψ i ϕ i L R Orthogonal sets of vectors in L and R sub-systems Represent ground states of the composite system using Schmidt decomposition: # & # & 0 = % λ i ψ i ϕ L i 1 R $ 1 ( % λ j ψ j ϕ L j i ' 2 R $ 2 ( j ' # & # & 0' = % λ k ψ k L1 ϕ k R $ 2 ( % λ m ψ m L2 ϕ m R1 ( ' $ ' k m 0 0' = λ 4 i = Tr(ρ 2 L ) = e S 2 i Zanadri, Zolka, Faoro 00, Horodecki, Ekert 02; others

14 Main idea of the present work -Quantum switch coupled to composite system (a two-level system) -Controls connection of 4 sub-systems depending on its state R 2 R 2 L 1 L 1 L 2 L 2 R 1 R 1 Ground state 0 Ground state 0'

15 Spectrum of the composite system R 2 Switch has no own dynamics (for now); Two decoupled sectors R 2 L 1 L 1 L 2 L 2 R 1 R 1 Eigenstates of a single system Δ E i + E j Ψ i L 1 R 1 Ψ j L 2 R 2 Ψ i L 1 R 2 Ψ j L 2 R 1 Energy eigenfunction

16 Introduce switch dynamics -Turn on H T = T + h.c. T -Require: T << Δ (gap is finite) -For our composite many-body system, such a term couples two sectors, and the two ground states T ' Δ

17 Effective Hamiltonian Restrict to two lowest energy states T ' Δ H eff = T ' GS GS' + h.c. Renormalized tunneling; GS = 0 GS' = 0' T ' = T 0 0'

18 Rabi oscillations: a way to measure the Renyi entanglement entropy T ' Δ Slowdown of the Rabi oscillations due to the coupling to many-body system Bare Rabi frequency (switch uncoupled from many-body system) Renormalized Rabi frequency: Ω' = T ' = Ω = / T Ω 0 0' = Ωe S 2 Directly related to the second Renyi entropy

19 Generalization for n>2 Renyi entropies -n copies of the many-body system -Two ways to connect them L 1 R 1 L 1 R 1 L 2 R 2 L 2 R 2 L n R n L n R n Ground state 0 n Ground state 0 n ' Overlap gives n-th Renyi entropy S n 0 n 0 n ' = e (n 1)S n

20 Proposed setup for measuring n>2 Renyi entropies L 2 L 2 R 2 R 2 R 3 R 3 L 1 L 1 L 3 L 3 R 1 R 1 L 4 R 4 L 4 R 4 -Quantum switch controls the way in which 2n subsystem are connected -Renormalization of the switch Rabi frequency à overlap à n-th Renyi entropy

21 A possible design of the quantum switch in cold atomic systems R 2 a b a b a b L 1 L 2 c d c d c d R 1 -quantum well -polar molecule: *forbids tunneling of blue particles -particle that constitutes many-body system tunneling

22 A possible design of the quantum switch in cold atomic systems a b a b a b c d c d c d -Doubly degenerate ground state that controls connection of the composite many-body system -Q-switch dynamics can be induced by controlling the barriers between four wells -Rabi oscillations by monitoring the population of the wells

23 Generalization to the 2D case copies of the system, engineer double connections across the boundary

24 Generalization to the 2D case Couple to an extended qubit living along the boundary -Depending on the qubit state, tunneling either within or between layers is blocked -Measure n=2 Renyi entropy, and detect top. order

25 Summary Details: DA, Demler, arxiv: A method to measure entanglement entropy in a generic many-body systems -Difficulty of measurement does not grow with the system size APPLICATIONS -Test scaling laws; detect location of critical points without measuring order parameter -Extensions to 2D detect top. order MESSAGE: ENTANGLEMENT ENTROPY IS MEASURABLE

26 PART 2: Time-dependent impurity in cold fermions: orthogonality catastrophe and beyond In collaboration with: Michael Knap (Graz) Yusuke Nishida (Los Alamos) Eugene Demler (Harvard)

27 Single impurity problems in condensed Rich many-body physics matter physics -Edge singularities in the X-ray absorption spectra (exact solution of non-equilibrium many-body problem) -Kondo effect: entangled state of impurity spin and fermions Influential area, both for methods (renormalization group) and for strongly correlated materials

28 Probing impurity physics in solids is limited -Many unknowns; Simple models hard to test (complicated band structure, unknown impurity parameters, coupling to phonons, hole recoil) -Limited probes (usually only absorption spectra) -Dynamics beyond linear response out of reach (relevant time scales GHz-THz, experimentally difficult) X-ray absorption in Na

29 Cold atoms: new opportunities for studying impurity physics -Parameters known à fully universal Properties -Impurity properties tunable by the Feshbach resonance (magnetic field controls scatt. length) à new regimes -Fast control of microscopic parameters (compared to many-body scales) -Rich toolbox for probing many-body states

30 Introduction to Anderson orthogonality catastrophe (OC) -Overlap S = FS FS' S 0 L - as system size, orthogonality catastrophe -Infinitely many low-energy electron-hole pairs produced Fundamental property of the Fermi gas

31 Orthogonality catastrophe and X-ray absorption spectra Without impurity With impurity -Relevant overlap: δ S(t) = FS e ih 0t e ih f t FS t δ2 /π 2 -- scattering phase shift at Fermi energy -Manifestation: a power-law singularity in the X-ray absorption spectrum A(ω) exp(iωt)s(t)dt 1 ω 1 δ2 /π 2 Nozieres, DeDominicis; Anderson

32 Universal OC in cold atoms Previously: S(t) t δ2 /π 2,t A(ω) 1 ω 1 δ2 /π 2 (very long times) (very small energies) -This work: exact solution for (all times and energies); S(t), A(ω) -Determined solely by imp. scattering length -Time domain: new sub-leading oscillating contribution to overlap S(t) C 1 t δ2 /π 2 + C 2 e ie Ft t α -Energy domain: cusp singularities with new exponent at energy E F above absorption threshold

33 PLAN -Manifestations of orthogonality catastrophe in cold atomic systems -Exact solutions for OC and RF absorption spectra: new singularities, new exponents -Extensions: multi-component Fermi gas, simulating quantum transport

34 Setup -Fermi gas+single localized impurity -Two pseudospin states of impurity, and - -state scatters fermions -state does not -Scattering length, -Fermion Hamiltonian for pseudospin -- a H 0, H f -Such systems have been realized experimentally (R. Grimm, M. Zwierlein, C. Salomon, others)

35 Ramsey interferometry new manifestation of OC -Entangle pseudospin and Fermi gas; utilize optical control over pseudospin à study Fermi gas dynamics -Ramsey interferometry 1) pi/2 pulse FS 1 2 FS FS 2) Evolution 12 e ih 0 t FS e ih f t FS 3) pi/2 pulse, measure < S z >= Re[S(t)] S(t) = FS e ih 0t e ih f t FS Direct measurement of OC overlap in the time domain

36 RF spectroscopy of impurity atom Free atom

37 RF spectroscopy of impurity atom Free atom Atom in a Fermi sea many-body effects completely change absorption function

38 RF spectroscopy of impurity atom Free atom Atom in a Fermi sea many-body effects completely change absorption function 2 ω E 1+δ π 2 new singularity

39 Edge singularities in the RF spectra: Intuitive picture Low-energy e-h pairs e-h pairs + One excitations from Band bottom Threshold singularity Singularity at Ef -Photon à pseudospin flipped+massive production of excitations -Threshold singularity: multiple low-energy e-h pairs -Singularity at E F : one electron promoted from band bottom to Fermi surface + multiple low-energy e-h pairs à power-law singularity with a new exponent

40 Functional determinant approach to orthogonality catastrophe -Solution in the long-time limit is known (Nozieres- DeDominicis 69); based on solving singular integral equation OUR GOAL: full solution at all times -Approach 1: write down an integral equation with exact Greens functions; solve numerically (possible, but hard) -Approach 2: reduce to calculating functional determinants (easy) Combescout, Nozieres 71; Klich 03, Muzykantskii 03; Abanin, Levitov 04; Ivanov 10; Gutman, Mirlin

41 Functional determinant approach to orthogonality catastrophe Represent H 0 = w/o impurity S(t) = FS e ih 0t e ih f t FS k ε k c + k c k H f = ε k c + k c k +V S(t) = Tr(eiH 0t e ih f t ρ) Tr(ρ) with impurity ρ = e λ kc k + c k k,k' c k + c k' Density matrix Trace is over the full many-body state; dimensionality N -number of singleparticle states 2 N

42 A useful relation Consider quadratic many-body operators A = a ij c + i c j B = b ij c + i c j Then Tr(e A e B ) = det(1+ e a e b ) Trace over many-body space (dimensionality 2 N ) à Determinant in the single-particle space (dimensionality N ) -Holds for an arbitrary number of exponential operators -Derivation: step1 prove for a single exponential (easy) step2 for two or more exponentials, use Baker-Hausdorf formula reduce to step 1 e A e B = e A+B+1 2 [ A,B]+...

43 Functional determinant approach to orthogonality catastrophe S(t) = Tr(eiH 0t e ih f t ρ) Tr(ρ) Represent as a determinant in single-particle space S(t) = det(1 n + Rn) n(ε) =θ(e F ε) R = e ih0t e ih f t Fermi distribution function Time-dep. scattering operator -Long-time behavior: determinant has been calculated analytically, reduces to Riemann-Hilbert problem Muzykantskii, Adamov 03, Abanin, Levitov 04, -Arbitrary times: consider finite system and evaluate the determinant numerically (this work)

44 Results: overlap S(t) S(t) k a= 0.5 F k F a= 1.5 k F a= 3.0 k F a= t a<0; no bound state Weak oscillations from van Hove singularity at band bottom 10 1 k F a=0.5 k F a=1.5 k F a=3.0 k F a= t a>0; bound state Strong oscillations (bound state either filled or empty) DA, Knap, Nishida, Demler, in preparation

45 Exact RF spectra a<0; no impurity bound state Single threshold in absorption Kink at Fermi energy a>0; bound state Two thresholds -bound state filled after absorption -bound state empty

46 Spin echo: probing non-trivial dynamics of the Fermi gas -Response of Fermi gas to process in which impurity switches between different states several times S 1 (t) = FS e ih 0t e ih f t e ih 0t e ih f t FS -Advantage: insensitive to slowly fluctuating magnetic fields (unlike Ramsey) -Such responses cannot be probed in solid state systems

47 Spin echo response: features -Power-law decay at long times with an enhanced exponent S 1 (t) t 3δ2 /π 2 -Unlike the usual situation, where spin-echo decays slower than Ramsey! -Universal -Generalize to n-spin-echo; yet faster decay S n (t) t (4n 1)δ2 /π 2

48 Seeing OC in the state of fermions -So far, concentrated on measuring impurity properties -Measurable property of the Fermi gas which reveals OC physics?

49 Seeing OC in the state of fermions -Yes, distribution of energy fluctuations following a quench 1) Flip pseudospin starting with interacting state 2) Measure distribution of total energy of fermions with new Hamiltonian χ(λ) = P(E)e iλe = FS' e ih 0λ FS' = S(λ) -Measurable in time-of-flight experiments FS FS Overlap function Also: Silva 09; Cardy 11

50 Generalizations: non-equilibrium OC, non-commuting Riemann-Hilbert problem -Impurity coupled to several Fermi seas at different chemical potentials -Theoretical works in the context of quantum transport Muzykantskii et al 03 Abanin, Levitov 05 -Mathematically, reduces to noncommuting Riemann-Hilbert problem (general solution not known) -Experiments lacking

51 Multi-component Fermi gas: access to nonequilibrium OC and quantum transport in cold atomic system -Fermions with two hyperfine states, u and d, +impurity -Imbalance, µ u > µ d -pi/2 pulse on fermions u + d 2 u d 2 play the role of fermions in two leads -Impurity scattering creates both reflection and transmission - Simulator of the non-equilibrium OC and quantum transport DA, Knap, Nishida Demler, in preparation

52 Other directions -OC for interacting fermions (e.g., Luttinger liquid) -Dynamics: many-body effects in Rabi oscillations of impurity spin -Very different physics for an impurity inside BEC

53 Summary -New manifestations of OC in atomic physics experiments and in energy counting statistics -Exact solutions for Fermi gas response and RF spectra obtained; New singularities at Fermi energy -Extensions to multi-component cold atomic gases à simulate quantum transport and more Knap, Nishida, DA, Demler, in preparation

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56 Spectrum of the composite system R 2 Switch has no own dynamics; Two decoupled sectors R 2 L 1 L 1 L 2 L 2 R 1 R 1 L L 1, R 2, R 1 1 entangled L 2, R L 1, R 2 2 Eigenstates of a single system entangled Δ E i + E j Ψ i L 1 R 1 Ψ j L 2 R 2 Ψ i L 1 R 2 Ψ j L 2 R 1 Energy eigenfunction

57 Multi-component Fermi gas: access to nonequilibrium OC and quantum transport in cold atomic system Specie 1 Specie 2 -Imbalance different species -Mix them by pi/2 pulse on -Realization of non-equilibrium OC problem - Simulator of quantum transport and non-abelian Riemann-Hilbert problem -Charge full counting statistics can be probed DA, Knap, Demler, in preparation