Quantum quenches in the non-integrable Ising model

Transcription

1 Quantum quenches in the non-integrable Ising model Márton Kormos Momentum Statistical Field Theory Group, Hungarian Academy of Sciences Budapest University of Technology and Economics in collaboration with Tibor Rakovszky, Márton Mestyán, Gábor Takács, Mario Collura, Pasquale Calabrese ELTE Particle Physics Seminar 15/02/2017

2 Relaxation & thermalisation

3 Relaxation & thermalisation Do isolated quantum systems relax? Do they thermalise?

4 Relaxation & thermalisation Do isolated quantum systems relax? Do they thermalise? What does this even mean? we can t distinguish the global state of our system from thermal equilibrium by performing local measurements. (The reduced density matrices of subsystems are mixed due to entanglement, the system acts as its own bath.)

5 Relaxation & thermalisation Do isolated quantum systems relax? Do they thermalise? What does this even mean? we can t distinguish the global state of our system from thermal equilibrium by performing local measurements. (The reduced density matrices of subsystems are mixed due to entanglement, the system acts as its own bath.) Integrable systems can t thermalise: conserved quantities!

6 Relaxation & thermalisation Do isolated quantum systems relax? Do they thermalise? What does this even mean? we can t distinguish the global state of our system from thermal equilibrium by performing local measurements. (The reduced density matrices of subsystems are mixed due to entanglement, the system acts as its own bath.) Integrable systems can t thermalise: conserved quantities! Generalised Gibbs Ensemble: ˆ GGE = e P m m ˆQ m Tr e P m m ˆQ m M. Rigol et al., PRL 98, (2007)

7 Relaxation & thermalisation Do isolated quantum systems relax? Do they thermalise? What does this even mean? we can t distinguish the global state of our system from thermal equilibrium by performing local measurements. (The reduced density matrices of subsystems are mixed due to entanglement, the system acts as its own bath.) Integrable systems can t thermalise: conserved quantities! Generalised Gibbs Ensemble: ˆ GGE = e P m m ˆQ m Tr e P m m ˆQ m M. Rigol et al., PRL 98, (2007) What if integrability is broken? Is there a quantum KAM theorem? Prethermalisation G. Brandino, J.-S. Caux, R. Konik, PRX 5, (2015)

8 Relaxation & thermalisation Do isolated quantum systems relax? Do they thermalise? What does this even mean? we can t distinguish the global state of our system from thermal equilibrium by performing local measurements. (The reduced density matrices of subsystems are mixed due to entanglement, the system acts as its own bath.) Integrable systems can t thermalise: conserved quantities! Generalised Gibbs Ensemble: ˆ GGE = e P m m ˆQ m Tr e P m m ˆQ m M. Rigol et al., PRL 98, (2007) What if integrability is broken? Is there a quantum KAM theorem? Prethermalisation Are there universal features of the time evolution? G. Brandino, J.-S. Caux, R. Konik, PRX 5, (2015)

9 Relaxation & thermalisation Do isolated quantum systems relax? Do they thermalise? What does this even mean? we can t distinguish the global state of our system from thermal equilibrium by performing local measurements. (The reduced density matrices of subsystems are mixed due to entanglement, the system acts as its own bath.) Integrable systems can t thermalise: conserved quantities! Generalised Gibbs Ensemble: ˆ GGE = e P m m ˆQ m Tr e P m m ˆQ m M. Rigol et al., PRL 98, (2007) What if integrability is broken? Is there a quantum KAM theorem? Prethermalisation Are there universal features of the time evolution? Cold atom experiments! G. Brandino, J.-S. Caux, R. Konik, PRX 5, (2015) A. Kaufman et al., Science 353, 794 (2016) T. Kinoshita et al., Nature 440, 900 (2006)

10 What can we do?

11 focus on 1D systems What can we do?

12 What can we do? focus on 1D systems free systems (bosons, fermions, anyons)

13 What can we do? focus on 1D systems free systems (bosons, fermions, anyons) systems that can be mapped to free models (Ising or XY spin chain, hard core bosons, bosonisation) P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012)

14 What can we do? focus on 1D systems free systems (bosons, fermions, anyons) systems that can be mapped to free models (Ising or XY spin chain, hard core bosons, bosonisation) P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012) integrable systems: in some models the asymptotic state is understood via GGE or the Quench Action method (CFT, XXZ, Lieb-Liniger).

15 What can we do? focus on 1D systems free systems (bosons, fermions, anyons) systems that can be mapped to free models (Ising or XY spin chain, hard core bosons, bosonisation) P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012) integrable systems: in some models the asymptotic state is understood via GGE or the Quench Action method (CFT, XXZ, Lieb-Liniger). Time evolution is hard.

16 What can we do? focus on 1D systems free systems (bosons, fermions, anyons) systems that can be mapped to free models (Ising or XY spin chain, hard core bosons, bosonisation) P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012) integrable systems: in some models the asymptotic state is understood via GGE or the Quench Action method (CFT, XXZ, Lieb-Liniger). Time evolution is hard. lattice systems: efficient numerical methods based on MPS representations (tdmrg, itebd)

17 What can we do? focus on 1D systems free systems (bosons, fermions, anyons) systems that can be mapped to free models (Ising or XY spin chain, hard core bosons, bosonisation) P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012) integrable systems: in some models the asymptotic state is understood via GGE or the Quench Action method (CFT, XXZ, Lieb-Liniger). Time evolution is hard. lattice systems: efficient numerical methods based on MPS representations (tdmrg, itebd) continuum systems, field theory? universality out of equilibrium?

18 Outline of the talk

19 Outline of the talk Part I: Truncated Hilbert space approach to non-equilibrium dynamics with Tibor Rakovszky, Márton Mestyán, Mario Collura, Gábor Takács, Nucl. Phys. B 911, 805 (2016)

20 Outline of the talk Part I: Truncated Hilbert space approach to non-equilibrium dynamics with Tibor Rakovszky, Márton Mestyán, Mario Collura, Gábor Takács, Nucl. Phys. B 911, 805 (2016) Part II: Space-time structure of out of equilibrium evolution, dynamical confinement with Mario Collura, Pasquale Calabrese, Gábor Takács, Nature Physics, Advanced Online Publication

21 Part I Truncated Hilbert space method Nucl. Phys. B 911, 805 (2016)

22 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990)

23 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H = H 0 + H pert

24 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly

25 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated

26 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated finite size and cutoff effects are well understood RG for TCSA P. Giokas, G. Watts, arxiv , G. Feverati et al., J. Stat. Mech. P03011 (2008); R.M. Konik, Y. Adamov, Phys. Rev. Lett. 98, (2007); M. Lencsés, G. Takács, JHEP 09, 052 (2014)

27 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated finite size and cutoff effects are well understood RG for TCSA P. Giokas, G. Watts, arxiv , G. Feverati et al., J. Stat. Mech. P03011 (2008); R.M. Konik, Y. Adamov, Phys. Rev. Lett. 98, (2007); M. Lencsés, G. Takács, JHEP 09, 052 (2014) recent revival of the method: S. Rychkov, L. G. Vitale, Phys. Rev. D 91, (2015), Phys. Rev. D 93, (2016); Z. Bajnok, M. Lájer, JHEP 10, 050 (2016)

28 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated finite size and cutoff effects are well understood RG for TCSA P. Giokas, G. Watts, arxiv , G. Feverati et al., J. Stat. Mech. P03011 (2008); R.M. Konik, Y. Adamov, Phys. Rev. Lett. 98, (2007); M. Lencsés, G. Takács, JHEP 09, 052 (2014) recent revival of the method: S. Rychkov, L. G. Vitale, Phys. Rev. D 91, (2015), Phys. Rev. D 93, (2016); Z. Bajnok, M. Lájer, JHEP 10, 050 (2016) Potential for studying real time dynamics

29 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated finite size and cutoff effects are well understood RG for TCSA P. Giokas, G. Watts, arxiv , G. Feverati et al., J. Stat. Mech. P03011 (2008); R.M. Konik, Y. Adamov, Phys. Rev. Lett. 98, (2007); M. Lencsés, G. Takács, JHEP 09, 052 (2014) recent revival of the method: S. Rychkov, L. G. Vitale, Phys. Rev. D 91, (2015), Phys. Rev. D 93, (2016); Z. Bajnok, M. Lájer, JHEP 10, 050 (2016) Potential for studying real time dynamics non-perturbative

30 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated finite size and cutoff effects are well understood RG for TCSA P. Giokas, G. Watts, arxiv , G. Feverati et al., J. Stat. Mech. P03011 (2008); R.M. Konik, Y. Adamov, Phys. Rev. Lett. 98, (2007); M. Lencsés, G. Takács, JHEP 09, 052 (2014) recent revival of the method: S. Rychkov, L. G. Vitale, Phys. Rev. D 91, (2015), Phys. Rev. D 93, (2016); Z. Bajnok, M. Lájer, JHEP 10, 050 (2016) Potential for studying real time dynamics non-perturbative does not rely on integrability: effects of integrability breaking!

31 Truncated Hilbert Space Approach V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990) H 0 H = H 0 + H pert is solvable (CFT, free field theory) in a finite volume the matrix elements of H pert can be calculated in the basis of H 0 truncated at some energy the matrix of H can be computed exactly diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated finite size and cutoff effects are well understood RG for TCSA P. Giokas, G. Watts, arxiv , G. Feverati et al., J. Stat. Mech. P03011 (2008); R.M. Konik, Y. Adamov, Phys. Rev. Lett. 98, (2007); M. Lencsés, G. Takács, JHEP 09, 052 (2014) recent revival of the method: S. Rychkov, L. G. Vitale, Phys. Rev. D 91, (2015), Phys. Rev. D 93, (2016); Z. Bajnok, M. Lájer, JHEP 10, 050 (2016) Potential for studying real time dynamics non-perturbative does not rely on integrability: effects of integrability breaking! access to microscopic data (spectrum, overlaps)

32 Quantum Ising chain and Ising field theory H QIM = J NX i=1 x i x i+1 + h z NX i=1 z i + h x NX i=1 x i!

33 Quantum Ising chain and Ising field theory H QIM = J NX i=1 x i x i+1 + h z NX i=1 z i + h x NX i=1 x i! For h x =0 it can be mapped to free spinless fermions (Jordan-Wigner trf.)

34 Quantum Ising chain and Ising field theory H QIM = J NX i=1 x i x i+1 + h z NX i=1 z i + h x NX i=1 x i! For h x =0 it can be mapped to free spinless fermions (Jordan-Wigner trf.) Quantum critical point at h z =1,h x =0 separating the ferromagnetic h z < 1 and paramagnetic h z > 1 phases

35 Quantum Ising chain and Ising field theory H QIM = J NX i=1 x i x i+1 + h z NX i=1 z i + h x NX i=1 x i! For h x =0 it can be mapped to free spinless fermions (Jordan-Wigner trf.) Quantum critical point at h z =1,h x =0 separating the ferromagnetic h z < 1 and paramagnetic h z > 1 phases Scaling limit: J!1,h z! 1, M=2J 1 h z,h/ J 15/8 h x fixed

36 Quantum Ising chain and Ising field theory H QIM = J NX i=1 x i x i+1 + h z NX i=1 z i + h x NX i=1 x i! For h x =0 it can be mapped to free spinless fermions (Jordan-Wigner trf.) Quantum critical point at h z =1,h x =0 separating the ferromagnetic h z < 1 and paramagnetic h z > 1 phases Scaling limit: J!1,h z! 1, M=2J 1 h z,h/ J 15/8 h x fixed H IFT = H c=1/2 + H IFT = Z dx 1 2 Z apple i 2 Z dx "(x)+h dx (x) (x)@ x (x) (x)@x (x) im (x) (x) + h (x)

37 Quantum Ising chain and Ising field theory H QIM = J NX i=1 x i x i+1 + h z NX i=1 z i + h x NX i=1 x i! For h x =0 it can be mapped to free spinless fermions (Jordan-Wigner trf.) Quantum critical point at h z =1,h x =0 separating the ferromagnetic h z < 1 and paramagnetic h z > 1 phases Scaling limit: J!1,h z! 1, M=2J 1 h z,h/ J 15/8 h x fixed H IFT = H c=1/2 + H IFT = Z dx 1 2 Z apple i 2 Z dx "(x)+h dx (x) (x)@ x (x) (x)@x (x) im (x) (x) + h (x) measure everything in M h = hm 15/8 ` = ML

38 Truncated fermionic space The mode expansion r X (x, t) = L n r X (x, t) = L diagonalises the Hamiltonian n e n/2 p cosh n!a( n )e ip nx ie n t +!a ( n )e ip nx+ie n t P. Fonseca, A. Zamolodchikov, J. Stat. Phys. 110, 527 (2003) e n/2 p cosh n!a( n )e ip nx ie n t +!a ( n )e ip nx+ie n t, a( n ),a ( n 0) = n,n 0 acting on the Fock space H FF = M X n cosh( n )a ( n )a( n ) 1, 2,..., n i = a ( 1 )a ( 2 )...a ( n ) 0i p n = M sinh n = 2 n L, E n = M cosh n

39 Truncated fermionic space The mode expansion r X (x, t) = L n r X (x, t) = L diagonalises the Hamiltonian n e n/2 p cosh n!a( n )e ip nx ie n t +!a ( n )e ip nx+ie n t P. Fonseca, A. Zamolodchikov, J. Stat. Phys. 110, 527 (2003) e n/2 p cosh n!a( n )e ip nx ie n t +!a ( n )e ip nx+ie n t, a( n ),a ( n 0) = n,n 0 acting on the Fock space H FF = M X n cosh( n )a ( n )a( n ) Truncation of the Hilbert space 1, 2,..., n i = a ( 1 )a ( 2 )...a ( n ) 0i p n = M sinh n = 2 n L, E n = M cosh n M X n cosh( n ) apple

40 Quantum quenches in the IFT

41 Quantum quenches in the IFT Integrable quenches: H(M 0, 0)! H(M,0)

42 Quantum quenches in the IFT Integrable quenches: H(M 0, 0)! H(M,0) Non-integrable quenches: H(M 0, 0)! H(M,h 6= 0)

43 Quantum quenches in the IFT Integrable quenches: H(M 0, 0)! H(M,0) Non-integrable quenches: H(M 0, 0)! H(M,h 6= 0) Use the truncated basis of H(M,0)!

44 Quantum quenches in the IFT Integrable quenches: H(M 0, 0)! H(M,0) Non-integrable quenches: H(M 0, 0)! H(M,h 6= 0) Use the truncated basis of H(M,0)! Time evolution: Chebyshev polynomials: e iht 0 i = J 0 (t) 0 i +2 1X ( i) n J n (t)t n (H) 0 i n=1 T n+1 (x) =2xT n (x) T n 1 (x), T 0 (x) =1, T 1 (x) =x

45 Quantum quenches in the IFT Integrable quenches: H(M 0, 0)! H(M,0) Non-integrable quenches: H(M 0, 0)! H(M,h 6= 0) Use the truncated basis of H(M,0)! Time evolution: Chebyshev polynomials: e iht 0 i = J 0 (t) 0 i +2 1X ( i) n J n (t)t n (H) 0 i n=1 Maximal time: t max L/2 T n+1 (x) =2xT n (x) T n 1 (x), T 0 (x) =1, T 1 (x) =x

46 Benchmarking the method: Integrable quenches in the ferromagnetic phase H(M 0, 0)! H(M,0)

47 Statistics of work P (W )= X [W (E E 0 )] h 0 i 2 ML = 40, = 14M

48 Statistics of work P (W )= X [W (E E 0 )] h 0 i 2 0i = N exp ( i X n K ( n,m,m 0 ) a ( n ) a ( n ) 0i ) ML = 40, = 14M N = Y n K (,M,M 0 ) = tan 1+ K( n,m,m 0 ) 2 1/2 apple 1 2 arctan (sinh ) 1 M 2 arctan sinh M 0

49 Loschmidt echo L (t) = h 0 e iht 0i 2 = Z dw P (W )e iw t 2

50 Loschmidt echo L (t) = h 0 e iht 0i 2 = Z dw P (W )e iw t 2 exact result: L(t) = exp X n apple 1+ K( n,m,m 0 ) 2 e 2iE nt log 1+ K( n,m,m 0 ) 2! 2

51 Decay of the magnetisation analytic result for large time: h (t)i = e t/ for large t, apple Z 2M 1 = d K( ) 2 sinh O(K 6 ) D. Schuricht, F. Essler, J. Stat. Mech. P04017 (2012) (P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012))

52 Decay of the magnetisation analytic result for large time: h (t)i = e t/ for large t, apple Z 2M 1 = d K( ) 2 sinh O(K 6 ) 1 1 TFSA A TFSA M 0 =0.5M M M M 0 =1.5M M M D. Schuricht, F. Essler, J. Stat. Mech. P04017 (2012) (P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012))

53 Non-integrable quenches in the ferromagnetic phase H(M 0, 0)! H(M,h 6= 0)

54 h = 0.1, =8M Spectrum at finite magnetic field

55 Spectrum at finite magnetic field McCoy-Wu scenario of weak confinement B. McCoy, T. Wu, PRD 18, 1259 (1978) Free domain wall Bound state = meson h = 0.1, =8M

56 Spectrum at finite magnetic field McCoy-Wu scenario of weak confinement B. McCoy, T. Wu, PRD 18, 1259 (1978) Free domain wall Bound state = meson h = 0.1, =8M V (d) =h d

57 Spectrum at finite magnetic field McCoy-Wu scenario of weak confinement B. McCoy, T. Wu, PRD 18, 1259 (1978) Free domain wall Bound state = meson h = 0.1, =8M V (d) =h d approximations for the meson spectrum m Ai n = M(2 + 2/3 z n ) Ai( z n )=0 sinh (2# n ) 2# n =2 (n 1/4), m WKB n =2M cosh (# n )

58 M 0 =1.5M Time evolution of the magnetisation

59 M 0 =1.5M Time evolution of the magnetisation

60 Time evolution of the magnetisation Diagonal ensemble hoi DE = X C 2 h O i M 0 =1.5M

61 Time evolution of the magnetisation Diagonal ensemble hoi DE = X C 2 h O i M 0 =1.5M

62 Time evolution of the magnetisation Diagonal ensemble hoi DE = X C 2 h O i M 0 =1.5M Oscillations around infinite time average with frequencies set by the meson masses.

63 Time evolution of the magnetisation Diagonal ensemble hoi DE = X C 2 h O i M 0 =1.5M Oscillations around infinite time average with frequencies set by the meson masses. see also M.K., M. Collura, P. Calabrese, G. Takács, Nat. Phys. '16

64 Statistics of work and Loschmidt echo M 0 =1.5M, ML = 40, h = 0.1, =8M

65 Non-integrable quenches in the paramagnetic phase H(M 0, 0)! H(M,h 6= 0)

66 Spectrum at finite magnetic field h = 0.1, =8M

67 Spectrum at finite magnetic field h = 0.1, =8M The spectrum changes perturbatively, there is a single massive particle.

68 Time evolution of the magnetisation

69 Time evolution of the magnetisation First order FF perturbation theory for pure h (t)i h 2 M 2 0 h quench: F 1,0 (0 ) 2 [cos(mt) + 1] G. Delfino, J. Phys. A 47, (2014) J. Viti, G. Delfino, arxiv:

70 Time evolution of the magnetisation First order FF perturbation theory for pure h (t)i h 2 M 2 0 h quench: F 1,0 (0 ) 2 [cos(mt) + 1] G. Delfino, J. Phys. A 47, (2014) J. Viti, G. Delfino, arxiv: For the combined quench it is damped! Let s fit it with h (t)i = Ae t/ cos(ft)+c

71 Time evolution of the magnetisation First order FF perturbation theory for pure h (t)i h 2 M 2 0 h quench: F 1,0 (0 ) 2 [cos(mt) + 1] G. Delfino, J. Phys. A 47, (2014) J. Viti, G. Delfino, arxiv: For the combined quench it is damped! Let s fit it with h (t)i = Ae t/ cos(ft)+c

72 Time evolution of the magnetisation First order FF perturbation theory for pure h (t)i h 2 M 2 0 h quench: F 1,0 (0 ) 2 [cos(mt) + 1] G. Delfino, J. Phys. A 47, (2014) J. Viti, G. Delfino, arxiv: For the combined quench it is damped! Let s fit it with h (t)i = Ae t/ cos(ft)+c f = M(h)

73 Time evolution of the magnetisation First order FF perturbation theory for pure h (t)i h 2 M 2 0 h quench: F 1,0 (0 ) 2 [cos(mt) + 1] G. Delfino, J. Phys. A 47, (2014) J. Viti, G. Delfino, arxiv: For the combined quench it is damped! Let s fit it with h (t)i = Ae t/ cos(ft)+c f = M(h)

74 Time evolution of the magnetisation First order FF perturbation theory for pure h (t)i h 2 M 2 0 h quench: F 1,0 (0 ) 2 [cos(mt) + 1] G. Delfino, J. Phys. A 47, (2014) J. Viti, G. Delfino, arxiv: For the combined quench it is damped! Let s fit it with h (t)i = Ae t/ cos(ft)+c f = M(h) apple 2M Z 1 0 d K( ) 2 sinh 1

75 Statistics of work and Loschmidt echo M 0 =1.5M, ML = 40, h = 0.1, =8M

76 Final check: comparison with itebd simulation on the lattice H QIM = J NX i=1 x i x i+1 + h z NX i=1 z i + h x NX i=1 x i!

77 Conclusions (Part I) Nucl. Phys. B 911, 805 (2016)

78 Conclusions (Part I) Nucl. Phys. B 911, 805 (2016) demonstrated that Hamiltonian truncation methods can be used effectively to study non-equilibrium dynamics of field theories

79 Conclusions (Part I) Nucl. Phys. B 911, 805 (2016) demonstrated that Hamiltonian truncation methods can be used effectively to study non-equilibrium dynamics of field theories non-integrable quench in the ferromagnetic phase: the low-lying mesons play an important role

80 Conclusions (Part I) Nucl. Phys. B 911, 805 (2016) demonstrated that Hamiltonian truncation methods can be used effectively to study non-equilibrium dynamics of field theories non-integrable quench in the ferromagnetic phase: the low-lying mesons play an important role non-integrable quench in the paramagnetic phase: oscillations, analytic formula

81 Conclusions (Part I) Nucl. Phys. B 911, 805 (2016) demonstrated that Hamiltonian truncation methods can be used effectively to study non-equilibrium dynamics of field theories non-integrable quench in the ferromagnetic phase: the low-lying mesons play an important role non-integrable quench in the paramagnetic phase: oscillations, analytic formula no prethermalisation

82 Conclusions (Part I) Nucl. Phys. B 911, 805 (2016) demonstrated that Hamiltonian truncation methods can be used effectively to study non-equilibrium dynamics of field theories non-integrable quench in the ferromagnetic phase: the low-lying mesons play an important role non-integrable quench in the paramagnetic phase: oscillations, analytic formula no prethermalisation rare states? G. Biroli, C. Kollath, A. Läuchli, Phys. Rev. Lett. 105, (2010)

83 Conclusions (Part I) Nucl. Phys. B 911, 805 (2016) demonstrated that Hamiltonian truncation methods can be used effectively to study non-equilibrium dynamics of field theories non-integrable quench in the ferromagnetic phase: the low-lying mesons play an important role non-integrable quench in the paramagnetic phase: oscillations, analytic formula no prethermalisation rare states? G. Biroli, C. Kollath, A. Läuchli, Phys. Rev. Lett. 105, (2010) Outlook: quenches in the sine-gordon model

84 Part II Real time confinement after a quench in the quantum Ising chain Nature Physics, Advanced online publication arxiv:

85 Part II Real time confinement after a quench in the quantum Ising chain Nature Physics, Advanced online publication arxiv: Small perturbations can have dramatic effects on the dynamics.

86 Light cone in interacting models and entanglement after a quench 08: in thebose-hubbard Bose-Hubbard model Kollath-Lauechli time t <b0br>(t) low pass <n0nr>(t) rescaled t 0 2 Carleo et al., 14: Bose-Hubbard (a) time t [J ] (a) V = vinst (c) V = (b) V = i j i j (d) V = FIG. 1: (color online) Time evolution of the equal-time density correlation function Ci,j (t) of spinless fermions after a quench from the CDW ground state of H(V0 ) with V0 = 10, evolved by the Hamiltonian H(V ), with (a) V = 0, (b) V = 2, (c) V = 5, (d) and V = 20. Bonnes, Essler, Lauchli 14: XXZ spin chain to u(v = 0) = 2vF = 4th, as expected, where vf denotes the Fermi velocity for V = 0. In addition to the light cone, additional propagation fronts at later times can be identified in Fig. 1(a), which, however, possess a lower velocity. This signals that slower quasiparticles stemming from regions without linear dispersion also participate in spreading information. Figure 1(c) shows the evolution of the correlation function for a quench within the CDW phase, i.e., a case which should not be describable i by conformal field theory. Interestingly, we nevertheless find a pronounced light-cone behavior in the correlation function. Although the conformal field theory underlying 0 ofrcalabrese and Cardy is not valid in this the treatment region, the physical picture that ballistically propagating quasiparticles are generated by the quench seems to hold. r However, in contrast to the case of the quench to the LL displayed in Figs. 1(a) and (b), we see that a strong alternating pattern forms in the density correlation function and remains present and qualitatively unchanged after the onset of the light cone from U = 5 2J to Figure after a quench 6 3. Time-evolution of correlation functions R Uf = 40J. The upper panel shows the single particle correlation functions0 b b for different distances r. The correlations show partial revivals up to a time t when they start to reach a quasi-steady state. This time tr grows approximately linearly 5 with2the distance r as marked by the vertical lines. The central panel shows the same correlations functions after filtering out the high frequencies, see text for details density correlations function n0 nr after shifting The 0lowest panel shows the density and rescaling their amplitude forrbetter visibility. The common vertical dashed lines denote the arrival of the minima as determined from the density-density correlations. 1.4 t x r=2 r=3 r=4 r=5 r= time t time t time t <b0br>(t) et al 08: interacting fermions Manmana A more detailed view of the temporal evolution of the correlation functions is shown in Fig. 2, in which we plot the values of Ci,j (t) as a function of time for increasing distance i j for V = 0 and V = 2, the two extremes of the Luttinger-liquid phase. After the arrival of the first signal, oscillatory behavior as a function of time can be observed at each distance. However, as V is increased, the observed oscillations both decrease in magnitude and are damped out more rapidly. Comparing the results for the free case to the ones obtained for V = 2 in Fig. 2, it can be seen that the incoming front travels with a higher velocity when V is larger, as can also be seen in Fig. 1. In contrast to the oscillatory behavior in the Luttingerliquid phase, a steady increase of the correlations is observed when the quench occurs within the CDW phase, as can be seen in Fig. 3. The alternating pattern imprinted at the onset of the light cone is preserved. Presumably, the correlation functions saturate at some time that is significantly longer than the maximum time reached here. While results for both V < V0 and V > V0 show the same

87 Light cone in experiments M. Cheneau et al., Nature 481, 484 (2012) a quench position b time ~ ~ d=vt FIG. 1. Spreading of correlations in a quenched atomic Mott insulator. a, A1d ultracold gas of bosonic atoms (black balls) in an optical lattice is initially prepared deep in the Mott-insulating phase with unity filling. The lattice depth is then abruptly lowered, bringing the system out of equilibrium. b, Followingthequench,entangledquasiparticle pairs emerge at all sites. Each of these pairs consists of a doublon (red ball) and a holon (blue ball) on top of the unityfilling background, which propagate ballistically in opposite directions. It follows that a correlation in the parity of the site occupancy builds up at time t between any pair of sites separated by a distance d = vt, wherev is the relative velocity of the doublons and holons. FIG pro 9.0. obt

88 Light cone spreading of entanglement entropy P. Calabrese, J. Cardy 2005

89 Light cone spreading of entanglement entropy P. Calabrese, J. Cardy 2005 After a global quench, the initial state 0 i has an extensive excess of energy

90 Light cone spreading of entanglement entropy P. Calabrese, J. Cardy 2005 After a global quench, the initial state 0 i has an extensive excess of energy It acts as a source of quasi-particles at t =0. A particle of momentum p has energy and velocity v p =de p /dp E p B 2t A 2t B t

91 Light cone spreading of entanglement entropy P. Calabrese, J. Cardy 2005 After a global quench, the initial state 0 i has an extensive excess of energy It acts as a source of quasi-particles at t =0. A particle of momentum p has energy and velocity v p =de p /dp E p For t>0 the particles move semiclassically with velocity v p B 2t A 2t B t

92 Light cone spreading of entanglement entropy P. Calabrese, J. Cardy 2005 After a global quench, the initial state 0 i has an extensive excess of energy It acts as a source of quasi-particles at t =0. A particle of momentum p has energy and velocity v p =de p /dp E p For t>0 the particles move semiclassically with velocity v p Particles emitted from regions of size of the initial correlation length are correlated and entangled, particles from points far away are incoherent B 2t A 2t B t

93 Light cone spreading of entanglement entropy P. Calabrese, J. Cardy 2005 After a global quench, the initial state 0 i has an extensive excess of energy It acts as a source of quasi-particles at t =0. A particle of momentum p has energy and velocity v p =de p /dp E p For t>0 the particles move semiclassically with velocity v p Particles emitted from regions of size of the initial correlation length are correlated and entangled, particles from points far away are incoherent When v p is bounded (e.g. Lieb-Robinson bounds) v p <v max, the entanglement entropy grows linearly with time up to a value linear in ` B 2t A 2t B t

94 Example: Transverse Field Ising chain P. Calabrese, J. Cardy 2005

95 Example: Transverse Field Ising chain P. Calabrese, J. Cardy 2005 S(t) =t q Analytically for q t, l 1 with t/l constant Z 2 0 t<` d' H(cos locality ha 1 cos '(h + h 0 )+hh 0 cos ' = ' 0 ' contains ')+ ` Z 2 0 t>` d' 2 H(cos H(x) = 1+x 2 M. Fagotti, P. Calabrese, 2008 ') (2) log 1+x 2 1 x 2 log 1 2 x

96 Light cone spreading of correlations The same scenario is valid for correlations:

97 Light cone spreading of correlations The same scenario is valid for correlations: Horizon: points at separation r become correlated when left- and right-moving particles originating from the same point first reach them

98 Light cone spreading of correlations The same scenario is valid for correlations: Horizon: points at separation r become correlated when left- and right-moving particles originating from the same point first reach them If v p <v max, connected correlations are then frozen for t<r/2v max

99 Light cone spreading of correlations The same scenario is valid for correlations: Horizon: points at separation r become correlated when left- and right-moving particles originating from the same point first reach them If v p <v max, connected correlations are then frozen for t<r/2v max Example: Ising model within ferromagnetic phase P. Calabrese, F. Essler, M. Fagotti 2011/12

100 Suppression of the light cone Starting from the ferromagnetic state (all spins up) and evolving with H = LX J xj x j+1 + h z z j + h x x j j=1 with h z =0.25

101 Suppression of the light cone Starting from the ferromagnetic state (all spins up) and evolving with H = LX J xj x j+1 + h z z j + h x x j j=1 with h z =0.25 h x 1 x m+1 i c

102 Suppression of the light cone Starting from the ferromagnetic state (all spins up) and evolving with H = LX J xj x j+1 + h z z j + h x x j j=1 with h z =0.25 h x 1 x m+1 i c Entanglement entropy S h = 0/0.5 h = 0.25 z h x = 0 h x = h x = 0.05 h x = 0.1 h x = 0.2 h = 0.4 x z t

103 Confinement in the Ising model H = J LX xj x j+1 + h z z j + h x x j McCoy & Wu 78 j=1

104 Confinement in the Ising model X H = J LX xj x j+1 + h z z j + h x x j McCoy & Wu 78 j=1 For h x =0free fermions with dispersion p "(k) =2J 1 2h z cos k + h z2.

105 Confinement in the Ising model X H = J LX xj x j+1 + h z z j + h x x j McCoy & Wu 78 j=1 For h x =0free fermions with dispersion p "(k) =2J 1 2h z cos k + h z2. h z =1separates two massive phases

106 Confinement in the Ising model X H = J LX j=1 xj x j+1 + h z z j + h x x j For h x =0free fermions with dispersion h z =1separates two massive phases McCoy & Wu 78 p "(k) =2J 1 2h z cos k + h z2. For h z < 1 (ferro phase), the massive fermions can be seen as domain walls separating domains of magnetization =(1 h z ) 1/8 Free DW

107 Confinement in the Ising model X H = J LX j=1 xj x j+1 + h z z j + h x x j For h x =0free fermions with dispersion h z =1separates two massive phases McCoy & Wu 78 p "(k) =2J 1 2h z cos k + h z2. For h z < 1 (ferro phase), the massive fermions can be seen as domain walls separating domains of magnetization =(1 h z ) 1/8 h x induces an attractive interaction between DW that for small enough h x can be approximated as a linear potential V (x) =2Jh x x Free DW

108 Confinement in the Ising model X H = J LX j=1 xj x j+1 + h z z j + h x x j For h x =0free fermions with dispersion h z =1separates two massive phases For h z < 1 (ferro phase), the massive fermions can be seen as domain walls separating domains of magnetization =(1 h z ) 1/8 h x induces an attractive interaction between DW that for small enough h x can be approximated as a linear potential V (x) =2Jh x x DW do not propagate freely but get confined into mesons McCoy & Wu 78 p "(k) =2J 1 2h z cos k + h z2. Free DW Bound state = meson

109 Back to quenches What happens if there are mesons in the spectrum of the postquench Hamiltonian in the quasi-particle picture? domain walls get confined into mesons domain wall time domain wall space

110 Back to quenches What happens if there are mesons in the spectrum of the postquench Hamiltonian in the quasi-particle picture? 0 i acts as a source of quasi-particles at t =0 domain walls get confined into mesons domain wall time domain wall space

111 Back to quenches What happens if there are mesons in the spectrum of the postquench Hamiltonian in the quasi-particle picture? 0 i acts as a source of quasi-particles at t =0 pairs of particles move in opposite directions with velocity v p domain walls get confined into mesons domain wall time domain wall space

112 Back to quenches What happens if there are mesons in the spectrum of the postquench Hamiltonian in the quasi-particle picture? 0 i acts as a source of quasi-particles at t =0 pairs of particles move in opposite directions with velocity v p moving away the quasi-particles feel the attractive interaction domain walls get confined into mesons domain wall time domain wall space

113 Back to quenches What happens if there are mesons in the spectrum of the postquench Hamiltonian in the quasi-particle picture? 0 i acts as a source of quasi-particles at t =0 pairs of particles move in opposite directions with velocity v p moving away the quasi-particles feel the attractive interaction The interaction will eventually turn the particles back domain walls get confined into mesons domain wall time domain wall space

114 1-point function h x i

115 1-point function h x i Quenches from ferro to ferro hz=0.5, hx =0, hz =0.25, hx =0.1 hz =0.5, hx =0, hz =0.25, hx =0.2 hz =0.25, hx =0, hz =0.25, hx = Σ x Σ x Σ x t t t itebd vs. ED with L=8,10,12

116 1-point function h x i Quenches from ferro to ferro hz=0.5, hx =0, hz =0.25, hx =0.1 hz =0.5, hx =0, hz =0.25, hx =0.2 hz =0.25, hx =0, hz =0.25, hx = Σ x Σ x Σ x t Power spectrum of h x i t t itebd vs. ED with L=8,10, Σ x Ω Σ x Ω Σ x Ω Ω Ω Ω m2-m1 =0.46, m1 =3.7, m2 =4.1, m3 =4.5 m2-m1 =0.68, m1 =4.0, m2 =4.7 m2-m1 =0.46, m1 =3.7, m2 =4.1, m3 =4.5

117 Zooming in: escaping correlations H = J LX xj x z x j+1 + h z j + h x j j=1 h x 1 x m+1 i c

118 Zooming in: escaping correlations H = J LX xj x z x j+1 + h z j + h x j j=1 h x 1 x m+1 i c

119 Zooming in: escaping correlations H = J LX xj x z x j+1 + h z j + h x j j=1 h x 1 x m+1 i c Σ x 1Σ x 23 c t

120 Zooming in: escaping correlations H = J LX xj x z x j+1 + h z j + h x j j=1 h x 1 x m+1 i c Σ x 1Σ x 23 c Σ x 1Σ x 23 c t t

121 Quench in the paramagnetic phase h x 1 x m+1 i c

122 Quench in the paramagnetic phase Ferro h x 1 x m+1 i c

123 Quench in the paramagnetic phase Ferro h x 1 x m+1 i c Para

124 Quench in the paramagnetic phase Ferro h x 1 x m+1 i c Para Entanglement entropy 1.2 h = 2 h = 1.75 z z S h x = 0 h x = 0.1 h x = 0.2 h = 0.4 x Change for small hx is perturbative. For large hx new fast excitations appear(?) No confinement t

125 Conclusions (Part II) In the Ising chain, confinement changes the light cone spreading of correlations and entanglement

126 1 Conclusions (Part II) In the Ising chain, confinement changes the light cone spreading of correlations and entanglement time space domain walls get confined into mesons domain wall domain wall

127 1 Conclusions (Part II) In the Ising chain, confinement changes the light cone spreading of correlations and entanglement time space domain walls get confined into mesons domain wall domain wall Questions:

128 1 Conclusions (Part II) In the Ising chain, confinement changes the light cone spreading of correlations and entanglement time space domain walls get confined into mesons domain wall domain wall Questions: Is it a general property of other cond-mat models featuring confinement? Presumably yes, possible to check numerically

129 1 Conclusions (Part II) In the Ising chain, confinement changes the light cone spreading of correlations and entanglement time space domain walls get confined into mesons domain wall domain wall Questions: Is it a general property of other cond-mat models featuring confinement? Presumably yes, possible to check numerically Is it true in higher dimensions? e.g. in QCD? maybe holography can offer some hints