Dynamics in one dimension: integrability in and out of equilibrium

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1 Dynamics in one dimension: integrability in and out of equilibrium Jean-Sébastien Caux Universiteit van Amsterdam Work done in collaboration with: P. Calabrese, A. Faribault, A. Klauser, J. Mossel, M. Panfil, G. Palacios L. Glazman, A. Imambekov, A. Shashi, H. Konno, M. Sorrell, R. Weston,...

2 Plan of the talk Introduction Part 1: equilibrium dynamics Lieb-Liniger, Heisenberg Applications Part 2: quench dynamics Richardson, Heisenberg, Lieb-Liniger Geometric quench Interaction quench Conclusions

3 Why is 1d special? Particles can t avoid each other : interaction effects are automatically saturated Perturbation theory fails: Fermi surface fully nested, divergent susceptibilities (Fermi liquid jams up in 1d) What can be done? Brute-force resummations (Dzyaloshinskii-Larkin 1974) Low-energy phenomenology: the Luttinger Liquid (Haldane 1981) Exact methods: integrability (Bethe 1931)

4 The idea (always the same): Start with your favourite quantum state (expressed in terms of Bethe states) O Apply some operator on it {λ} Reexpress the result in the basis of Bethe states: O {λ} = {µ} F O {µ},{λ} {µ} using matrix elements F O {µ},{λ} = {µ} O {λ}

5 Elements of bare Bethe liquid theory Integrable Hamiltonians and their deformations H = H 0 + a Q a Their exact wavefunctions H {λ} = E {λ} {λ} Matrix elements of physical operators {µ} O {λ} = F O ({µ} {λ}) Efficient Hilbert space resummation methods {λ} = 1ph + 2ph + 3ph +...

6 Features of bare Bethe liquid theory First-principles, from microscopic model Use of exact wavefunctions: results are not restricted to low energies Detailed understanding of eigenstates and matrix elements: allows great optimizations Since the Schrödinger equation is directly solved, allows for in- as well as out-ofequilibrium situations

7 Part I: Equilibrium dynamics

8 Models which we treat: Heisenberg spin-1/2 chain H = N j=1 [ J(S x j S x j+1+s y j Sy j+1 + Sz j S z j+1) H z S z j ] Interacting Bose gas (Lieb-Liniger) H N = N j=1 2 x 2 j +2c 1 j<l N δ(x j x l ) Richardson model (+ Gaudin magnets) H BCS = N ε α 2 c ασc ασ g N c α+c α c β c β+ α=1 σ=+, α,β=1

9 What we can calculate: DYNAMICAL STRUCTURE FACTOR S aā (q, ω) = 1 N N j,j =1 e iq(j j ) dte iωt S a j (t)sāj (0) c inelastic neutron scattering DENSITY-DENSITY FUNCTION S(k, ω) = dx dte ikx+iωt ρ(x, t)ρ(0, 0) ONE-BODY FN G 2 (x, t) = Ψ (x, t)ψ(0, 0 Bragg spectroscopy, interference experiments,... (zero temperature only (for now!))

10 Building correlation functions piece by piece Our needed building blocks are: S a,ā (q, ω) =2π µ 0 O a q µ 2 δ(ω E µ + E 0 ) 1) A basis of eigenstates Bethe Ansatz; quantum groups 2) The matrix elements of interesting operators in this basis 3) A way to sum over intermediate states Algebraic Bethe Ansatz; q. groups Numerics (ABACUS); analytics

11 The Lieb-Liniger Bose gas (the simplest nontrivial integrable model) H N = N j=1 2 x 2 j +2c 1 j<l N δ(x j x l ) Exact many-body wavefunctions: Ψ N (x 1,..., x N )= P N ( 1) [P ] e i N j=1 λ P j x j + i 2 N j>k 1 φ(λ P j λ Pk ) where the 2-body phase shift is φ(λ) =2atan λ c

12 Lieb-Liniger Bose gas Density-density (dynamical SF) S(k, ω) = 2π L 0 ρ k α 2 δ(ω E α + E 0 ) α (J-S C & P Calabrese, PRA 2006)

13 Correspondence with excitations Particle-like Hole-like Umklapp

14 Connection with Field Theory Singularity structure (Khodas, Pustilnik, Kamenev, Glazman; Imambekov & Glazman) S 1 ω ɛ I µ 1 (θ(ɛ I ω)+ν 1 θ(ω ɛ I )) Singularity at upper 2p threshold High-energy tail Lower threshold S (ω II ) µ 2

15 Beyond CFT (I): Nonlinear Luttinger Liquid Theory Glazman, Imambekov, Khodas, Kamenev, Cheianov, Pustilnik, Affleck, Pereira, Sirker, JSC,... Three subband model From Imambekov & Glazman, SCIENCE 323 (2009) H l,r = iv H d = Effective mass: dx : Ψ l (x) Ψ l (x) : : Ψ r(x) Ψ r (x) : dx d (x) vp + p2 2m i v + p m d(x) 1 m = v v K 1/2 h + v2 K 2K 3/2 h

16 Beyond CFT (II): Correlation prefactors from BA Asymptotics from Luttinger theory: ˆρ(x)ˆρ(0) ρ K 2(πρ 0 x) 2 + m 1 A m cos(2mπρ 0 x) (ρ 0 x) 2m2 K ˆψ (x) ˆψ(0) ρ 0 m 0 B m cos(2mπρ 0 x) (ρ 0 x) 2m2 K+1/2K Infinite sets of nonuniversal prefactors These can be computed exactly by considering size scaling of form factors from integrability (A. Shashi, L. Glazman, J.-S. Caux and A. Imambekov, arxiv ) (Kozlowski and Terras 2011)

17 Heisenberg chains S(k, ω), =1, h =0

18 Neutron scattering neutrons from reactor time & target direction: energy & momentum scattered neutrons new particles: spinons (quantum solitons) detectors

19 Neutron scattering (HMI, Berlin) NEAT time-of-flight spectrometer

20 Hahn-Meitner-Institut Berlin in der Helmholtz-Gemeinschaft S(Q,w) Bethe Ansatz Spinons in KCuF3

21 Sr CuO : XXX 2 3 Walters, Perring, JSC, Savici, Gu, Lee, Ku, Zaliznyak, NatPhys 2009

22 (C D N) CuBr XXZ AFM at anisotropy =1/2 B. Thielemann, Ch. Rüegg, H. M. Rønnow, A. M. Läuchli, J.-S. Caux, B. Normand, D. Biner, K. W. Krämer, H.-U. Güdel, J. Stahn, K. Habicht, K. Kiefer, M. Boehm, D. F. McMorrow, J. Mesot, PRL, 2009

23 New experimental method: RIXS (Resonant Inelastic X-ray Scattering) Synchrotron X-ray induces a 1s-4p transition on copper, modifying exchange term

24 RIXS response from ABACUS A. Klauser, J. Mossel, JSC and J. van den Brink, Phys. Rev. Lett. 106, (2011) Given by the dynamical correlation function of neighbouring exchange operators: S exch (q, ω) =2π α 0 X q α 2 δ(ω ω α ) in which X q 1 N e iqj (S j 1 S j + S j S j+1 ) j Matrix elements: obtained from ABA Kitanine & al.; A. Klauser, J. Mossel and JSC, to be published Resummed by ABACUS

25 Using SU(2) symmetry: S exch (q, ω) = cos 2 (q/2) 72π N α S tot =0 j e iqj 0 Sj z Sj+1 z α 2 δ(ω ω α ) Crucial prefactor (vanishes at pi) Only total spin zero sector contributes Sum rules: Integrated intensity f-sumrule dω 2π dω 2π ωsexch (q, ω) =6sin 2 (q) 1 N q S exch (q, ω) = 1 4 ln(2) ζ(3) (x 1 x 2 ) 1 4 cos 2 (q/2) + in which x i S z j S z j+i 3ζ(3) 4 ln(2) 8

26 RIXS response from ABACUS No signal at low energy!!! Instead found at high energy Waiting for experimental confirmation (ongoing)

27 1 Intensity [A.U.] q = 1/4! q = 1/2! S exch. S zz q = 3/4! " [J] 3

28 states! Other RIXS-like response functions: Dynamical correlation of z-exchange S z j S z j+1 reveals 4-spinon

29 SzSz response: intuitive picture Two-step process: α S z j S z j+1 GS GS S z p GS S z q p S z p GS

30 Gapless XXZ AFM: vertex operator approach JSC, H. Konno, M. Sorrell and R. Weston, arxiv: We consider the XXZ in zero field, H = J N S x j S x j+1 + S y j Sy j+1 + Sz j S z j j=1 Longitudinal structure factor: S zz (k, ω)= 1 N ) e ik(j j dte iωt Sj z (t)sj z (0) j,j

31 Spinons in XXZ Dispersion relation e(p) =v F sin p p [ π, 0] Fermi velocity 1 2 (anisotropy-dependent) v F ( ) = πj 2 acos Two-spinon continuum k = p 1 p 2 ω = e(p 1 )+e(p 2 ) ω 2,l (k) =v F sin k ω 2,u (k) =2v F sin k/2

32 Longitudinal structure factor Separates into S zz (k, ω) = m=1 S(2m) zz (k, ω) Matrix elements: from vertex operator approach Jimbo, Miwa, Lashkevich, Pugai, Kojima, Konno, Weston 2 (k, ω) = Θ(ω 2,u(k) ω)θ(ω ω 2,l (k)) (1 + 1/ξ) 2 e Iξ(ρ(k,ω)) ω2,u 2 (k) ω2 cosh 2πρ(k,ω) ξ + cos π ξ S zz where ξ = π acos 1 cosh(πρ(k, ω)) = ω 2 2,u (k) ω 2 2,l (k) ω 2 ω 2 2,l (k) I ξ (ρ) 0 dt t sinh(ξ + 1)t sinh ξt cosh(2t) cos(4ρt) 1 cosh t sinh(2t)

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36 Sum rule saturations from two-spinon states Integrated intensity I zz = 2π 0 dk 2π 0 dω 2π S(k, ω) =1/4 f-sumrule I zz 1 (k) = 2π 0 dω 2π ωs(k, ω) = 2Xx (1 cos k) X x S x j S x j+1

37 Threshold behaviour Near upper threshold: For 0 < 1: S zz 2 (k, ω) ω ω 2,u (k) f u (ξ)(sin k 2 ) 7/2 ω 2,u (k) ω For 0: S zz 2 (k, ω) ω ω 2,u (k) f u (1) (sin k 2 ) 1/2 ω2,u (k) ω Agrees with Nonlinear LL predictions (Pereira, Affleck,...) Adds exact momentum-dependent prefactors

38 Threshold behaviour Near lower threshold: For 0 < 1: S2 zz (k, ω) ω ω 2,l (k) f l(ξ) sin 1 k 2 (1 1 ξ ) (sin k 2 2 ) ξ [ω ω 2,l (k)] 1 2 (1 1 ξ ) For 0: S zz 2 (k, ω) ω ω 2,l (k) O(1)

39 Integrability for correlations: generic features Exact realization of ground state, taking all correlations and entanglement into account Exact realization of excited states (spinons, Lieb types I, II, Gaudinos,...), irrespective of their energy Action of local operators: accurately captured by using only a handful of excitations incredibly efficient basis for many physically relevant correlations

40 Part 2: Out-ofequilibrium dynamics

41 Quenches: some trivialities Sudden change of Single quench interaction parameter (Barouch & McCoy,..., Calabrese & Cardy,... Cazalilla, Lamacraft, Klich, Lannert & Refael, Barmettler & al,...) t = 0 At quench time: Ψ 0 g = α Ψ α g Ψ α g Ψ 0 g α M α0 g g Ψ α g Subsequent time evolution: Ψ(t) = α M α0 g ge iωα g t Ψ α g Crucial building block: Ψ α g Ψ β g M αβ g g We know how to calculate the quench matrix for the Richardson model!!

42 Quench matrix elements

43 Time dependence of observables order parameter Ψ OD (t) α,β ψ(t) S + α S β ψ(t) Plotted against mean-field prediction (Barankov & Levitov, PRL 2006) g0 g gap for initial g gap for final g asymptotic gap

44 Domain wall quenched into XXZ J. Mossel and JSC, NJP 2010 Initial state: φ =... M.... N M Time evolution dictated by H XXZ = J N j=1 1 2 S j S+ j+1 + S+ j j+1 S + S z j Sj+1 z Solution to Schrödinger eqn: φ(t) = n e ie nt Q n Ψ n Quench vector elements: Q n Ψ n φ n Q n 2 =1

45 Dominant total M overlaps: with string states Qn , 1, M 2 2, M 2 1, M E 0 1 particle E particles 3 particles Excitation continua for various state families

46 Work probability distribution P (W )= n φ Ψ n 2 δ(w E n + E 0 ) N250 M N250 M M M PW , M 1 2, M 2 W PW , M 1 2, M 2 W , M W W

47 Loschmidt echo L(t) = φ e ih 0t e iht φ 2 Φ0Φt t t Φ0Φt relaxation : e tτ long time average t `Eigenstate thermalization hypothesis (Deutsch, Srednicki) does not apply here Initial state is `remembered at all times

48 Geometric quenches J. Mossel, G. Palacios and JSC, 2010 t<0:x i [0,L 1 [ t>0:x i [0,L 2 [ Initial wavefunction: nonlinear mapping Ψ (1) c ({x} {λ} L1 )= Ψ (2) c ({x} {λ} L1 ), 0 x i <L 1, 0 otherwise

49 Geometric quenches The overlap can in fact be calculated using Slavnov! It s just the overlap in the original space domain: {λ L 1 c } {µ L 2 c } = d N x(ψ L 1 c ({x i } {λ i })) ψ L 2 c 0 x 1 <x 2 <... L 1 ({x i } {µ i })=F ({λ} {µ}) This works for any model for which Slavnov is available.

50 Interaction quench in Lieb-Liniger J. Mossel and JSC, unpublished For all t<0:h = H c=0 For all t>0:h = H c=0 Simplest time-dependent correlation: g 2 (x; t) = φ(t) Ψ (x)ψ (0)Ψ(x)Ψ(0) φ(t) φ(t) Ψ (0)Ψ(0) φ(t) 2 g 2 x φ(t) Ψ (x)ψ (0)Ψ(x)Ψ(0) φ(t) = 1 L k 1,k 2,k 3 e ik13(x 2k23t) φ Ψ k 1 Ψ k 2 Ψ k3 Ψ k1 +k 23 φ

51 Conclusions Correlations from integrability: real usefulness in physics (relevant systems; real observables; measurable energy scales) Intimate relationship with new field theory developments (Nonlinear Luttinger Liquid Theory) Out-of-equilibrium from integrability: reliable test-cases for various scenarios

52 Challenges for the future: Correlations in nested systems Finite temperatures Applications in nanostructures Quenches from integrability: other cases Driven systems Renormalization from integrable points New results for field theory Phenomenological extensions: Bethe liquid theory