Dynamics and relaxation in integrable quantum systems

Transcription

1 Dynamics and relaxation in integrable quantum systems WEH Seminar Isolated Quantum Many-Body Quantum Systems Out Of Equilibrium Bad Honnef, 30 November 2015 Jean-Sébastien Caux Universiteit van Amsterdam Work done in collaboration with (among others): A dam gang: R. van den Berg, R. Vlijm, S. Eliens, J. De Nardis, B. Wouters, M. Brockmann, D. Fioretto, O. El Araby, F.H.L. Essler, R. Konik, N. Robinson, M. Haque, E. Ilievski, T. Prosen,

2 Plan of the talk Out-of-equilibrium dynamics Quasisoliton dynamics in XXZ Quantum Newton s cradle: TG limit The Interaction quench in Lieb-Liniger Quench Action Anisotropy quench in XXZ Summary & perspectives

3 Applications of integrability in many-body physics Quantum magnetism Ultracold atoms Quantum dots, NV centers Atomic nuclei

4 Models discussed in this talk: Interacting Bose gas (Lieb-Liniger) H N = N j=1 2 x 2 j +2c 1 j<l N δ(x j x l ) Heisenberg spin-1/2 chain H = N [ J(S x j Sj+1+S x y j Sy j+1 + Sz j Sj+1) H z z Sj z ] j=1

5 Bethe Ansatz (1931) Integrable Hamiltonian: H = Z L 0 dx H(x) Reference state : vacuum, FM state,... July 2, 1906 March 6, 2005 Particles : atoms, down spins,... Exact many-body wavefunctions (in N-particle sector): N({x} { }) = X P ( 1) [P ] A P ({ })e ix jk( Pj )... parametrized... with by rapidities... specified relative made up amplitudes... of free waves and obeying some form of Pauli principle

6 The Bethe Wavefunction Michel Gaudin s book La fonction d onde de Bethe is a uniquely influential masterpiece on exactly solvable models of quantum mechanics and statistical physics. Available in English for the first time, this translation brings his classic work to a new generation of graduate students and researchers in physics. It presents a mixture of mathematics interspersed with powerful physical intuition, retaining the author s unmistakably honest tone. The book begins with the Heisenberg spin chain, starting from the coordinate Bethe Ansatz and culminating in a discussion of its thermodynamic properties. Delta-interacting bosons (the Lieb-Liniger model) are then explored, and extended to exactly solvable models associated with a reflection group. After discussing the continuum limit of spin chains, the book covers six- and eight-vertex models in extensive detail, from their lattice definition to their thermodynamics. Later chapters examine advanced topics such as multicomponent delta-interacting systems, Gaudin magnets and the Toda chain. MICHEL GAUDIN is recognized as one of the foremost experts in this field, and has worked at Commissariat à l énergie atomique (CEA) and the Service de Physique Théorique, Saclay. His numerous scientific contributions to the theory of exactly solvable models are well known, including his famous formula for the norm of Bethe wavefunctions. JEAN-SÉBASTIEN CAUX is a Professor in the theory of low-dimensional quantum condensed matter at the University of Amsterdam. He has made significant contributions to the calculation of experimentally observable dynamical properties of these systems. Gaudin and Caux The Bethe Wavefunction The Bethe Wavefunction Michel Gaudin Translated by Jean-Sébastien Caux Cover illustration: a representation of the Yang-Baxter relation by John Collingwood. Cover designed by Hart McLeod Ltd

7 The general idea, simply stated: Start with your favourite quantum state (expressed in terms of Bethe states) O Apply some operator on it { }i Reexpress the result in the basis of Bethe states: O { }i = X {µ} F O {µ},{ } {µ}i using matrix elements F O {µ},{ } = h{µ} O { }i

8 Quantum spin chains Correlations, experiments (INS, RIXS), prefactors,... (C D N) CuBr KCuF 3 Sr CuO 2 3 Thielemann, Rüegg, Rønnow, Läuchli, Caux, Normand, Biner, Krämer, Güdel, Stahn, Habicht, Kiefer, Boehm, McMorrow, Mesot, PRL 2009 Lake, Tennant, Caux, Barthel, Schollwöck, Nagler, Frost, PRL 2013 Walters, Perring, Caux, Savici, Gu, Lee, Ku, Zaliznyak, NATURE PHYSICS 2009 Sr CuO 3 2 (RIXS) Schlappa, Wohlfeld, Zho, Mourigal, Haverkort, Strocov, Hozoi, Monney, Nishimoto, Singh, Revcolevschi, Caux, Patthey Rønnow, van den Brink, Schmitt, NATURE 2012

9 Out-ofequilibrium dynamics from integrability

10 The simple pendulum on its head Kapitza pendulum, 1951 Pyotr L. Kapitza (8/7/1894-8/4/1984)

11 The Kapitza pendulum

12 Out-of-equilibrium using integrability Highly excited initial (eigen)states: The super Tonks-Girardeau gas Split Fermi sea in Lieb-Liniger Quasisolitons Interaction quench in Richardson Domain wall release in Heisenberg Quenched states: Geometric quench Interaction turnoff in Lieb-Liniger Release of trapped Lieb-Liniger BEC to Lieb-Liniger quench Quantum Newton s cradle: TG Néel to XXZ quench Driven systems: Spin echo in quantum dots

13 Quasisoliton dynamics in spin chains

14 Solitons (classical) John Scott Russell: wave of translation (1834) (Herriot-Watt University)

15 Solitons (classical) (Boussinesq) Korteweg-de Vries t u + u@ x u + 3 xu =0 First simulations: Fermi-Pasta-Ulam-(Tsingou) absence of ergodicity Further simulations: Zabusky & Kruskal 1965 concept of a soliton Classical inverse scattering

16 Particle content of XXZ: nontrivial Solution of Bethe equations: rapidities + strings } iζ Single particle (bound state of magnons) λ j,a α = λ j α + i ζ 2 (n j +1 2a)+iδ j,a α O(e (cst)n ) Classification of strings: Bethe, Takahashi, Suzuki,...

17 String wavepackets Vlijm, Ganahl, Fioretto, Brockmann, Haque, Evertz and Caux, 2015 In the eigenbasis, time evolution of a generic state: simple! (t)i = e iht (0)i = X { } e ie { }t C { } { }i Localized wavepacket: (0)i = N 0 X e ipx 2 4 (p p)2 (n) (p)i p Dispersion: width ~ t: tj =0.9 Two-string 1 Three-string j 0.3 hs z j (t)i x(t) = 2 n = J 2 r 2 2(0) 2n(0) 4 + nt function of anisotropy: 2 cos 2 (p)

18 Quasisoliton scattering (quantum) 0 1str-1str Vlijm, Ganahl, Fioretto, Brockmann, Haque, Evertz and Caux, str-2str 0.5 Worldlines of colliding wavepackets: =2 tj Displacement as a function of anisotropy (fixed incoming momenta) j tj Measured av. Linear fits j j Displacement Single magnons Bound magnons (1,1) (2,2) hs z j (t)i

19 Quantum quenches

20 Quenches (more generally: prepare and release ) The problem: considering a generic initial state, what is the time evolution of the system? (t)i = e iht (t = 0)i Hamiltonian driving time evolution { initial state is NOT eigenstate of H Observable expectation values depend on time: Ō(t) h (t) O (t)i h (t) (t)i

21 First question: what is the steady state long after the quench? Fundamental issue: does the system relax? thermalize? Crucial point: time evolution in the presence of myriads of constraints (due to integrability) is special Conjecture: steady state is described by a generalized Gibbs ensemble (GGE) lim t!1 Ō(t) =hoi GGE = Tr{Oe P n n Q n } Tr{e P n nq n} Rigol, Dunjko, Yurovsky, Olshanii, PRL 2007 see also Jaynes, Phys. Rev. 1957

22 GGE implementation Generalized inverse temperatures to be set using the initial conditions on conserved charges n ˆQ P m =Tr ˆQm e n n ˆQ o n /Z GGE m =0, 1, 2,... where Z GGE =Tre P n n ˆQ n In reality, three major difficulties: Conserved charges are generically nontrivial Finding a complete set of charges is an issue Self-consistency problem difficult to solve In practice: implementable for free theories only (charges: momentum occupation modes) For interacting cases: not understood in general.

23 Quantum quenches: Quantum Newton s cradle

24 David Weiss s quantum Newton s cradle experiment Ergodicity in interacting quantum systems close to an integrable model

25 Atoms do not thermalize during the experimental time scale (about 50 cycles) Experimentally possible to break integrability in different ways, to test relaxation and ergodicity Does there exist a quantum KAM theorem? Nonequilibrium & quench physics experimentally accessible

26 Quantum Newton s cradle: strongly-interacting limit van den Berg, Wouters, Eliëns, De Nardis, Konik and Caux, 2015 Kapitza-Dirac pulse: Z! Û B (q, A) =exp ia dx cos(qx) ˆ (x) ˆ (x) Tonks-Girardeau limit: bosonic wavefns from fermionic ones B(x; t) = Y 1applei<jappleN sgn(x i x j ) F (x; t) Slater determinant of single-particle states

27 Observables: Local density hˆ (x, t)i = h ˆ (x, t) ˆ (x, t)i Momentum dist fn hˆn(k, t)i = 1 2 Z dxdye i(x y)k h ˆ (x, t) ˆ (y, t)i Two geometries: Single-particle wavefunctions after pulse of momentum q: Ring (periodic): j(x; t) = 1X = 1 I ( ia) 1 p L e i( j+ q)x e i( j+ q) 2 t/2m Harmonic trap: j(x; t) = 1X = 1 I ( ia)e i q cos(!t) (x+ q 2m! sin(!t) ) j(x + q m! sin(!t))e i!(j+ 1 2 )t

28 Overlaps of post-bragg pulse states with eigenstates: from matrix elements of Bragg pulse operator hµ ÛB(q, A) i L N =det N apple I j q µ k ( ia) (q) j,µ k Stationary state distribution: (Quench Action and GGE give same) sp q,a ( )= 1 2 X 2Z ( q + F ) ( q F ) I (ia) 2 Local density: limh q,a (t) ˆ (x) q,a (t) = nm th q F t 1X J ( 2A sin(q 2 t/2m)) cos(xq ) sin(q F t/m) = 1

29 Quantum Newton s cradle: TG limit, trap geometry Local density q =3, A =1.5,! = 10/N (N = 50)

30 Quantum Newton s cradle: TG limit, trap geometry Momentum distn fn extremely rapid relaxation (much faster than trap oscillation) q =3, A =1.5,! = 10/N (N = 50)

31 Impurity in a Lieb-Liniger gas: arrested expansion/quantum stutter Neil Robinson, J.-S. C. and Robert Konik

32 Quantum quenches: BEC to repulsive Lieb-Liniger quench

33 Quench from BEC to repulsive gas Start from GS of noninteracting theory, 0 N i 1 p LN N! k=0 N 0i Turn repulsive interactions on from t=0 onwards: particles repel away from each other, system heats up, momentum distribution broadens,...

34 This is a difficult problem to treat... 1) Generalized Gibbs ensemble logic Kormos, Shashi, Chou and Imambekov, arxiv: Conserved charges: ˆQ n : ˆQn { } N i = Q n { } N i Q n ({ } N )= NX j=1 n j Davies 1990; Davies and Korepin GGE inapplicable, charges take infinite values! J-S C + J. Mossel, unpublished 2) GGE on lattice, q-deformed model Works, partial results only (using a few charges) Kormos, Shashi, Chou, JSC, Imambekov, PRA 2014

35 The quench action approach J-SC & F.H.L. Essler, PRL 2013 in pictures... Initial state: in pre-quench Hilbert space basis in post-quench Hilbert space basis

36 The quench action approach J-SC & F.H.L. Essler, PRL 2013 Quench action landscape: determined by overlaps & entropy Variational approach, implemented by a Generalized thermodynamic Bethe Ansatz J. Mossel and J-SC, JPA 2012; J-SC & R. Konik, PRL 2012, see also Fioretto & Mussardo NJP 2010, Pozsgay JSTAT 2011

37 The quench action approach J-SC & F.H.L. Essler, PRL 2013 Saddle-point state: determines steady state States around saddle-point: determine relaxation towards steady state

38 The quench action approach Generic time-dependent expectation values: J-SC & F.H.L. Essler, PRL 2013 lim Th 1 Ō(t) =lim Th 2 X {e} h e +e S {e} [ sp ] i! {e} [ sp ]t h sp O sp ; {e}i i S {e} [ sp]+i! {e} [ sp ]t h sp ; {e} O sp i Main message: the *full* time dependence is recoverable using a minimal amount of data saddle-point distribution (from GTBA) excitations in vicinity of sp state (easy) differential overlaps selected matrix elements

39 Back to BEC-LL quench Explicit result: J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014 M. Brockmann JPA 2014 h{ j } N/2 j=1, { j} N/2 j=1 0i = s (cl) N N! det N j,k=1 G jk N/2 Y j=1 N/2 detj,k=1 GQ jk c j s 2 j c with matrix (reminiscent of Gaudin formula) G Q jk = jk L + N/2 X l=1 K Q (l j,l l ) K Q (,µ)=k( µ)+k( + µ) K( )= 2c K Q (l j,l k ) 2 + c 2

40 Quench action approach to BEC-LL quench J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014 We are now in position to apply the quench action logic! Need thermodynamic limit form of overlaps: L 2 n log c n +1 lim Th h, exp 0i =exp L 2 Z 1 0 d ( ) log apple 2 c 2 2 c O(L 0 ) Quench action now defined, saddle-point solution via generalized thermodynamic Bethe ansatz

41 Quench action solution to BEC-LL quench J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014 It is in fact possible to give a closed form solution of the GTBA for the saddle-point state, for any value of the interaction: ( )= (1 + a( )) 1 a( )= 2 / c sinh 2 c I 1 2i c 4 p I 1+2i c 4 p

42 Quench action solution to BEC-LL quench J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014 sp( ) = 1 =8 =0.8 =0.08 = Asymptotics as from q-bosons: /n s (x) th s (x) x Subplot: scaled fn s (x) = p (c p x/2)/2 Large c: 2 ( ) n4 2 ( )= 1 2 ( ) 1 p s 4n n 2 Small c: semicircle n6 3 (24 ) n 2 Tail explains divergences of evalues of conserved charges

43 Quantum quenches: Néel to XXZ quench

44 Quench from Néel to XXZ Start from Néel state: From t=0 onwards, evolve with XXZ Hamiltonian H = N [ J(S x j Sj+1+S x y j Sy j+1 + Sz j Sj+1) H z z Sj z ] j=1 Can one treat this problem exactly?

45 Quench action approach to Néel-XXZ quench First step: exact overlaps of Néel state with XXZ eigenstates Tsuchiya JMP1998; Kozlowski & Pozsgay JSTAT 2012 Gaudin-like form again! M. Brockmann, J. De Nardis, B. Wouters & J-SC JPA 2014 h 0 {± j } M/2 j=1 i k{± j } M/2 j=1 k = 2 M/2 p Y 2 4 j=1 p tan( j + i /2) tan( j 2 sin(2 j ) 3 i /2) 5 v u t det M/2(G + jk ) det M/2 (G jk ) G ± jk = jk /2 ( j ) M/2 X 1 K + ( j, l) A + K ± ( j, k) l=1 K ± (,µ)=k ( µ) ± K ( + µ) K ( )= sinh(2 ) sin( + i )sin( i )

46 Quench action approach to Néel-XXZ quench Second step: generalized TBA B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M.Rigol & J-SC, PRL 2014 ln n ( )= 2 hn ln W n ( )+ 1X m=1 a nm ln 1+ 1 m ( ) where n ( ) n,h ( )/ n ( ) a n ( )= 1 sin n cosh n cos 2 and the effective driving terms (pseudo-energies) are W n ( )= 8 >< >: 1 2 n+1 sin 2 2 cosh n cos 2 coshn +cos 2 tan 2 cosh n cos 2 2 n coshn +cos 2 Q n 2 Q n 1 2 j=1 cosh(2j 1) cos 2 (cosh(2j 1) +cos 2 )(cosh 4 j cos 4 ) 2 if n odd, Q 1 n 2 2 j=1 (cosh 2(2j 1) cos 4 ) 2 j=1 cosh 2j cos 2 cosh2j +cos 2 Solution of this GTBA gives steady-state (analytically!) 2 if n even.

47 Quench action approach to Néel-XXZ quench Equivalent form of generalized TBA: ln( n )=d n + s ln(1 + n 1 )+ln(1+ n+1 ) with driving terms d n ( )= X k2z e 2ik tanh( k) k ( 1) n ( 1) k GGE with local charges: same form of coupled equations, but driving term only for n=1: d 1 ( )= 1 1X m=1 2m X k2z e 2ik k 2m 2 cosh k unknown

48 Néel-XXZ quench: conserved charges Initial expectation value of local charges: lim N!1 Remarkable correspondence between 1-string Fagotti & Essler JSTAT N hnéel Q n+1 Néeli = n n cosh[ p 1 2 x] 2 x=0 hole density and local charges: X k2z k 2m 2 e k ˆ h 1(k) 2 cosh k B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol & J-SC, PRL 2014 = hq 2m i m 2 N which fixes Néel 1,h ( )= 2 a 3 1( )sin 2 (2 ) 2 a 2 1 ( )sin2 (2 ) + cosh 2 ( ) Quench action nontrivially reproduces this; GGE also of course, but only by definition

49 The steady state: Néel to XXZ sp 1 ( ) (a) sp 2 ( ) (b) Solid lines: quench action =1.2 =1.5 =2 =5 = Dashed lines: GGE (local charges) QA and (local)gge 0.2 GGE 1 ( ) sp 1 ( ) (c) have different saddlepoint densities Large Delta expansion: GGE 1 sp 1 = O( 3 ), GGE 2 sp 2 = 1 3 sin2 ( ) O( 3 ).

50 Difference in distribution: impact on correlations (a) h z 1 z 3 i h z z 3 i (b) h z 1 h z 1 h 1 z 3 zi 3 zi sp z 3 i GGE h z 1 z 3 i NLCE Numerical verification using NLCE (M. Rigol) Large Delta expansions: h z 1 z 2i QA = h z 1 z 2i GGE =

51 Not convinced? Look at other results by Budapest group B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, G. Takács, PRL 2014 reobtain our Néel results also consider initial dimer state obtain numerical (itebd) evidence for correlations being different in dimer case There remains no doubt about the correctness of the quench action results because

52 Revalidating the GGE for Néel to XXZ

53 Quasilocal charges in XXZ Previously discovered in XXX, XXZ(gapless) Prosen 2011; Prosen and Ilievski 2013; Ilievski and Prosen 2013; Prosen 2014 Pereira, Pasquier, Sirker and Affleck, JSTAT 2014 Mierzejewski, Prelovšek and Prosen 2015 Here : need generalization to XXZ(gapped) Ilievski, Medenjak and Prosen, arxiv: Ilievski, De Nardis, Wouters, Caux, Essler, Prosen 2015 Starting point: q-deformed L-operator L(z,s) = 1 sinh sinh(z) cosh ( s z s) 0 + cos(z)sinh( s z s) z +sinh( )(s s + + s + s ) Auxiliary spins obey q-deformed su(2) [s + s,s s ]=[2s z s] q [s z s,s ± s ]=±s ± s [x] q =sinh( x)/ sinh( ) in 2s+1-dim irrep s z s ki = k ki, s ± s ki = q [s +1± k] q [s k] q k ± 1i

54 Quasilocal charges in XXZ(gpd) Higher-spin transfer matrices: lead to spin-s conserved charges T s (z) =Tr a [L a,1 (z,s)...l a,n (z,s)] X s ( )= 1 s ( )T s (z )T 0 s(z + ), z ± = ± 2 + i in which s ( )=f( (s ) + i )f((s ) + i ) f(z) =(sinh(z)/ sinh ( )) N More convenient for ThLim: bx s ( ):=T ( ) s (z )T (+)0 s (z + ) built from transfer matrix with L (±) (z,s) =L(z,s)sinh( )/[sinh (z ± s )] Families of quasilocal charges: H (n+1) s = 1 b Xs ( ) =0 s = 1 2, 1, 3 2,...

55 A complete GGE for XXZ Ilievski, De Nardis, Wouters, Caux, Essler, Prosen 2015 Throughout the gapped regime (including XXX limit), the GGE density matrix is given by ˆ% GGE = 1 Z exp " 1X n,s=1 nh s (n) s/2 Steady state: fixed by initial conditions through the generalized remarkable string-charge correspondence # 0 2s,h ( )=a 2s( ) s ( + i 2 )+ 0 s ( i 2 ) 0 s h 0 X b s ( ) 0i ( )=lim th N s = 1 2, 1, 3 2,...

56 Fixing the Néel-to-XXZ GGE Ilievski, De Nardis, Wouters, Caux, Essler, Prosen PRL 2015 Implementing the construction for the Néel-to-XXZ quench makes the GGE converge to correct QA answer Effect on some simple steady-state correlations: h z 1 z 3i h z 1 z 3 i QA GGE 1/2 GGE 1 GGE 3/2 GGE

57 Summary & perspectives Integrability out of equilibrium real-time dynamics in experimentally accessible setups quasisoliton scattering pulsed systems Quench action logic new approach to out-of-equilibrium problems gives access to full time evolution with minimal data Food for thought for GGE users BEC to LL: exact solution from QA (inaccessible to GGE) Néel to XXZ: exact solution from QA GGE with local charges gives different steady state! GGE needs to include quasilocal charges to reproduce QA Take-home message: there is more to equilibration than meets the eye