quantum systems Pasquale Calabrese University of Pisa Cargese, - PowerPoint PPT Presentation

quantum systems Pasquale Calabrese University of Pisa Cargese, September 2014 Based on joint works with: John Cardy, Mario Collura, Fabian Essler, Maurizio Fagotti, Marton Kormos, Spyr

• A many-body quantum system is prepared in the ground-state | Ψ • At t=0, H 0 ➔ H, i.e. an Hamiltonian parameter is quenched • Isolated: No contact with an external world • For t>0, it evolves unitarily: | Ψ ( t ) ⟩ = e -iHt | Ψ 0 ⟩ • How can we describe the dynamics? • What about a “stationary state”? von Neumann in 1929 posed these questi It stayed a purely academic question: for condensed matter sys the coupling to the environment is unavoidable

few hundreds 87 Rb atoms in a 1D trap Essentially unitary time evolution

- 2D and 3D systems relax quickly and thermalize: 9 τ 0 τ 4 τ 2 τ

Non-equilibrium new states of matter - 2D and 3D systems relax quickly and thermalize: When and why a steady state is thermal?? 9 τ 0 τ 4 τ 2 τ The 1D case is special because the system is almost integra

a 0.6 t .".". 0.4 U / J "="2.44(2) U 0.2 ( i )"Preparation ( ii )"Evolution ( iii )"Readout K / J$=$ 5·10 .3 b 0 n odd n odd c 0.6 n even c d " 0.4 1.0 Position"( hk ) 4 U / J "="5.16(7) 2 0.2 n odd 0.5 0 K / J$=$ 9·10 .3 K " >2 >4 0 0.0 0 1 2 3 4 5 0 1 2 0 1 2 3 4 0 1 2 3 8 10 12 14 16 18 20 t "(ms) t "(ms) 4 Jt$ /" h • Numerics and experiment agree perfectly • The stationary state looks thermal

a 0.6 t .".". 0.4 U / J "="2.44(2) U 0.2 ( i )"Preparation ( ii )"Evolution ( iii )"Readout K / J$=$ 5·10 .3 b 0 n odd n odd c 0.6 n even c d " 0.4 1.0 Position"( hk ) 4 U / J "="5.16(7) 2 0.2 n odd 0.5 0 K / J$=$ 9·10 .3 K " >2 >4 0 0.0 0 1 2 3 4 5 0 1 2 0 1 2 3 4 0 1 2 3 8 10 12 14 16 18 20 t "(ms) t "(ms) 4 Jt$ /" h • Numerics and experiment agree perfectly • The stationary state looks thermal Common Belief: - Generic systems “thermalizes” - Integrable systems are different But the system is always in a pure state!

| Ψ ( t ) ⟩ time dependent pure state A B ρ (t) = | Ψ ( t ) ⟩⟨ Ψ ( t )| density matrix of Reduced density matrix: ρ A (t)=Tr B ρ The expectation values of all local observables in A are ⟨ Ψ (t)|O A (x) | Ψ (t) ⟩ = Tr[ ρ A (t) O A (x)]

| Ψ ( t ) ⟩ time dependent pure state A B ρ (t) = | Ψ ( t ) ⟩⟨ Ψ ( t )| density matrix of Reduced density matrix: ρ A (t)=Tr B ρ The expectation values of all local observables in A are ⟨ Ψ (t)|O A (x) | Ψ (t) ⟩ = Tr[ ρ A (t) O A (x)] Stationary state: if for any finite subsystem A of an infinite system, it exi lim ρ A (t) = ρ A ( ∞ ) t →∞ Note: in a finite system, the stationary state is the regime |A|<< v t <<

Consider the Gibbs ensemble for the entire system A U B ρ T = e - H/T eff /Z ⟨ Ψ 0 | H | Ψ 0 ⟩ = Tr[ ρ T H ] with Teff is fixed by the energy in the initial state: no fre Reduced density matrix for subsystem A: ρ A,T =Tr B ρ T The system thermalizes if for any finite subsystem A ρ A,T = ρ A ( ∞ ) The infinite part B of the system “acts as an heat bath for A”

Proposal by Rigol et al 2007: The GGE density matrix ρ GGE = e - ∑ λ m Im /Z with λ m fixed by ⟨ Ψ 0 | I m | Ψ 0 ⟩ = Tr[ ρ GGE Again no fre I m are the integrals of motion of H, i.e. [ I m ,H ] = 0

Proposal by Rigol et al 2007: The GGE density matrix ρ GGE = e - ∑ λ m Im /Z with λ m fixed by ⟨ Ψ 0 | I m | Ψ 0 ⟩ = Tr[ ρ GGE Again no fre I m are the integrals of motion of H, i.e. [ I m ,H ] = 0 Reduced density matrix for subsystem A: ρ A,GGE =Tr B ρ GGE The system is described by GGE if for any finite subsystem A of a infinite system [Barthel-Schollwock ’08] ρ A,GGE = ρ A ( ∞ ) [Cramer, Eisert, et al ’08] + ........ [PC, Essler, Fagotti ’12]

Any quantum system has too many integrals of motion, regardless of integrability, e.g. O m = |E m ⟩⟨ E m |

Any quantum system has too many integrals of motion, regardless of integrability, e.g. O m = |E m ⟩⟨ E m | New proposal: [PC, Essler, Fagotti ’12] ρ GGE = e - ∑ λ m Im /Z where I m is a complete set of local (in space) integrals of moti [ I m ,I n ] = 0 [ I m ,H ] = 0 I m = ∑ O m ( x ) x In this case B is not a standard heat bath for A: infinite information on the initial state is retained!

Quenching the frequency in one harmonic oscillator H 0 = p 2 2 + ω 2 H = p 2 2 + ω 2 2 x 2 0 2 x 2 H 0 ⇤ Solving Heisenberg equation of motion ⌃ x 2 ( t ) ⌥ = ω 2 + ω 2 4 ω 0 ω 2 + ω 2 � ω 2 0 0 4 ω 0 ω 2 cos 2 ω t � ⇥ Not surprisingly, the harmonic oscillator oscillates

� � � H ( m ) = 2 a π n + am ϕ n + a ( ϕ n +1 − ϕ n ) , n =0 k = m 2 + Each momentum mode is a free oscillator Ω 2 e ikr ( Ω 2 e ikr ( Ω 2 0 k � Ω 2 ⇧ ⇧ k )(1 � cos(2 Ω k t )) 0 k � ⇤ ⌃ φ r ( t ) φ 0 ( t ) ⌥ � ⌃ φ r (0) φ 0 (0) ⌥ = dk Ω 2 Ω 2 k Ω 0 k t ⇤ ⇧ BZ BZ k This compatible with the GGE ρ GGE = e − P k λ k n k ! 4 Ω k Ω 0 k n k = a † k a k λ k = ln 1 + Z ( Ω k � Ω 0 k ) 2

� � � H ( m ) = 2 a π n + am ϕ n + a ( ϕ n +1 − ϕ n ) , n =0 k = m 2 + Each momentum mode is a free oscillator Ω 2 e ikr ( Ω 2 e ikr ( Ω 2 0 k � Ω 2 ⇧ ⇧ k )(1 � cos(2 Ω k t )) 0 k � ⇤ ⌃ φ r ( t ) φ 0 ( t ) ⌥ � ⌃ φ r (0) φ 0 (0) ⌥ = dk Ω 2 Ω 2 k Ω 0 k t ⇤ ⇧ BZ BZ k This compatible with the GGE ρ GGE = e − P k λ k n k ! 4 Ω k Ω 0 k n k = a † k a k λ k = ln 1 + Z ( Ω k � Ω 0 k ) 2 Non local... but linear combinations of loca The GGE built with n k and X X λ k n k = γ m I m with I m with are equivalent! m k

• Mass quenches in (lattice) field theories PC-Cardy ’07, Barthel-Schollwock ’08, Cramer, Eisert, et al ’08, Sotiriadi • Luttinger model quartic term quench Cazalilla ’06, Cazalilla-Iucci ’09, Mitra-Giamarchi ’10.... • Transverse field quench in Ising/XY model Barouch-McCoy ’70, Igloi-Rieger ’00-13, Sengupta et al ’04, Rossini et al. ’10, PC, Ess Foini-Gambassi-Cugliandolo’12, Bucciantini, Kormos, PC ’14........ • Quench to the Tonks-Girardeau model Rostunov, Gritsev, Demler ’10, Collura, Sotiriadis, PC ’13, Kormos, Collura • Few more..... The GGE always turned out to work

If we take a linear superposition of a finite number of eigenstates system will obviously oscillate forever Can we find some conditions for the initial state/Hamiltonian gua steady state and GGE/thermalization?

If we take a linear superposition of a finite number of eigenstates system will obviously oscillate forever Can we find some conditions for the initial state/Hamiltonian gua steady state and GGE/thermalization? A simple general condition Sotiriadis, PC 2014 For a free theory, the steady state is described by the GGE if the state satisfy the cluster decomposition property D Y E D Y ED Y E Y lim φ ( x i ) φ ( x j + R ) = φ ( x i ) φ ( x j ) . R →∞ i j i j see also Crame

Lieb-Liniger gas XXZ Spin chain N ∂ 2 H LL = � 1 X X + c δ ( x i � x j ) , ∂ x 2 2 L j j =1 i 6 = j X i +1 + S y i S y S x i S x ⇥ H = J i +1 i =1 The calculations become immensely more complicated

We developed a method to calculate expectation values in the G Fagotti, Collura, Es Analytics vs Numerics for the Neel →Δ quench: 1 | Neel, θ ⟩ ⟶ Δ = 2 | Neel, θ ⟩ ⟶ θ = 0° θ = 0° 0.2 0.1 0.5 0 k = 1 ⟨ σ xj σ xj+k ⟩ t ⟨ σ zj σ zj+k ⟩ t k = 2 -0.1 0 k = 3 -0.2 -0.3 -0.5 -0.4 -0.5 -1 0 2 4 6 8 0 2 4 6 t t Similar agreement with other initial states and final H

A new method to compute the exact time evolution developed Ess Particularly effective to compute the long-time limit Applied to XXZ chain for the Neel quench: Brockmann, Wouters, Fioretto, De Nardis = � 1 + 2 7 77 h σ z 2 i sp ∆ 2 � 2 ∆ 4 + 16 ∆ 6 � 1 σ z 2 i GGE = � 1 + 2 2 ∆ 4 + 43 7 h σ z ∆ 2 � 1 σ z 8 ∆ 6 -0.56 iTEBD 0.30 GGE The difference is more evident oTBA -0.58 starting from the dimer state 0.20 -0.60 0.10 (a) -0.62 Pozsgay, Mestyan, Werner, Kormos, Zarand, Takacs ’14 0 1 2 3 0 1 t

A new method to compute the exact time evolution developed Ess Particularly effective to compute the long-time limit Applied to XXZ chain for the Neel quench: The GGE does not work?? :( Brockmann, Wouters, Fioretto, De Nardis more work to be done! = � 1 + 2 7 77 h σ z 2 i sp ∆ 2 � 2 ∆ 4 + 16 ∆ 6 � 1 σ z But what about the time evolution? 2 i GGE = � 1 + 2 2 ∆ 4 + 43 7 h σ z ∆ 2 � 1 σ z 8 ∆ 6 -0.56 iTEBD 0.30 GGE The difference is more evident oTBA -0.58 starting from the dimer state 0.20 -0.60 0.10 (a) -0.62 Pozsgay, Mestyan, Werner, Kormos, Zarand, Takacs ’14 0 1 2 3 0 1 t

1. One-point function of a primary operator with ‹ ψ 0 |O(x) | ψ 0 › ≠ 0: ‹O(t,x)› ∝ e - π x o t/2 τ o Exponential relaxa τ 0 related to the init 2. Two-point function of a primary operator with ‹ ψ 0 |O(x) | ψ 0 › ≠ 0: { e - π xor/2 τ o for t>r/2 ‹O(t,r)O(t,0)› ∝ e - π xot/ τ o for t<r/2 If ‹ ψ 0 |O(x) | ψ 0 › ≠ 0, for t<r/2 ⇒ ‹O(t,r)O(t,0)›= ‹O Connected correlations vanish for t<r/