Analytic solution of the Domain Wall initial state. Jacopo Viti. ECT & IIP (UFRN), Natal, Brazil

Analytic solution of the Domain Wall initial state Jacopo Viti ECT & IIP (UFRN), Natal, Brazil Joint work with M. Collura (Oxford Un.) and A. De Luca (Oxford Un.) based on 1707.06218

Background: Non-equilibrium dynamics in 1d quantum systems Consider a 1d quantum many body quantum system. In general we will be interested in the complicated problem ψ(t) = e iht ψ(t = 0) but H is a many-body Hamiltonian! In particular we will consider the case (quantum operators will be in bold) ψ(t = 0) = ψ l ψ r or ρ(t = 0) = ρ l ρ r The density matrices on the left and the right will be typically (Generalized)-Gibbs ensembles and we could ask, what is the value O(x, t) = Tr[O(x)ρ(t)] maybe at large times, if it is easier.

Partitioning protocol: XXZ and the DW initial state This setup with a tensor product initial state is called partitioning protocol (for instance: Bernard and Doyon, J. Phys. A (2012)) We will consider a paradigmatic example in 1d: the XXZ spin chain H = L 2 1 i= L 2 s + i s i+1 +s i s + i+1 {}}{ s x i s x i+1 + s y i sy i+1 + ( s z i sz i+1 1 ) 4 The initial state will be of the state with inhomogeneous left (L) and right (R) magnetization: s z 0 = ± 1 2 tanh(h) ρ = e2hsz L Z L z e 2hS R Z R h DW = We will discuss the gapless case 1 < < 1, the gapped being trivial. Due to spin flips, the dynamics generates a spin current

DW phenomenology I: spin and current profiles The spin profile emerging from the DW initial state was analyze first by Gobert, Kollath, Schollwoeck and Schuetz (PRE, 2005) Figure: From GKSS 2005, s z i(t) at = 0 and 1. It was also conjectured that for x 1 and t 1 s z (x, t) = F ( x t α( ) ) With α( = 1) 0.6 (best fit)

Superdiffusion (?) at = 1 and dependence on h The same problem (*) was reconsidered numerically recently by Ljubotina, Znidaric and Prosen (Nat. Comm., 2017) ρ = (1 + 2µs z ) L/2 (1 2µs z ) L/2

Superdiffusion (?) at = 1 and dependence on h The same problem (*) was reconsidered numerically recently by Ljubotina, Znidaric and Prosen (Nat. Comm., 2017) ρ = (1 + 2µs z ) L/2 (1 2µs z ) L/2 At any µ [ 1, 1], for 1 < < 1, we have ballistic transport; at = 1 superdiffusion and for > 1 pure diffusion d 1.0 0.8 0.6 α 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 μ Figure: Left: The anisotropy dependence of the exponent α. Right: The µ = tanh(h) dependence of α( = 1) (best fit).

The free case = 0 and hydrodynamics At = 0, the XXZ chain is equivalent to free fermions H = 1 2 L/2 1 i= L/2 [c i c i +1 + c i +1 c i ] -5-4 -3-2 -1 0 1 2 3 4 5 6

The free case = 0 and hydrodynamics At = 0, the XXZ chain is equivalent to free fermions H = 1 2 L/2 1 i= L/2 [c i c i +1 + c i +1 c i ] -5-4 -3-2 -1 0 1 2 3 4 5 6 π The evolution from a DW initial state has a simple hydrodynamical interpretation (same is true for finite h) p t = 0 x Particles move independently with velocity v(p) = sin(p) Only particles with v(p) > x t can contribute to the density π

The free case = 0 and hydrodynamics At = 0, the XXZ chain is equivalent to free fermions H = 1 2 L/2 1 i= L/2 [c i c i +1 + c i +1 c i ] -5-4 -3-2 -1 0 1 2 3 4 5 6 π The evolution from a DW initial state has a simple hydrodynamical interpretation (same is true for finite h) p v(p)t p + (x/t) p (x/t) x Particles move independently with velocity v(p) = sin(p) Only particles with v(p) > x t can contribute to the density π

The free case = 0 and hydrodynamics At = 0, the XXZ chain is equivalent to free fermions H = 1 2 L/2 1 i= L/2 [c i c i +1 + c i +1 c i ] -5-4 -3-2 -1 0 1 2 3 4 5 6 π π The evolution from a DW initial state has a simple hydrodynamical interpretation (same is true for finite h) p v(p)t p + (x/t) p (x/t) x Particles move independently with velocity v(p) = sin(p) Only particles with v(p) > x t can contribute to the density Proven by stat. phase (V., Stephan, Dubail and Haque; EPL 2016)

Airy kernel at the edges at = 0 Integrating, we can determine the magnetization profile for x, t large and ζ = x/t fixed (F) ρ(x, t) 1 2 + sz (x, t) = π π dp Θ(v(p) ζ) 2π = 1 π arccos(ζ) For ζ 1, the profile has a square-root singularity ρ(x, t) = 2(1 ζ )/π + o( 1 ζ )

Airy kernel at the edges at = 0 Integrating, we can determine the magnetization profile for x, t t 1/3 [1\2+ s z ζ] large and ζ = x/t fixed (F) ρ(x, t) 1 2 + sz (x, t) = π π dp Θ(v(p) ζ) 2π = 1 π arccos(ζ) For ζ 1, the profile has a square-root singularity 1.5 1 0.5 0 t = 50 t = 60 t = 70 ρ(x, t) = 2(1 ζ )/π + o( 1 ζ ) -9-6 -3 0 3 [ζ-sin(γ)]t 2/3 γ = π/2 The uniform asymptotics of the fermion density near ζ = 1 is (Airy process) ρ(x, t) ( ) 2 1/3 [ t Ai(X ) 2 X Ai(X ) 2] where X = (x t) ( ) 2 1/3 t

Separation of scales and hydrodynamics Assume in a certain macroscopic space-time region we relaxed fast to a local stationary state described by a density matrix with local conserved quantities Q α = t=t 0 dx q α (x, t 0 ) q α (x, t) Among different space-time regions at local equilibrium we could have slower variations in the mean values of the charge densities. These variations among different locally at equilibrium space-time regions should obey the continuity equation t t charge dens. charge curr. {}}{{}}{ q α (x, t) + x j α (x, t) = 0; α = 1,..., N

Generalized hydrodynamics for integrable models In a (Bethe Ansatz) integrable model such a relation exists! Consider XXZ spin chain for example.

Generalized hydrodynamics for integrable models In a (Bethe Ansatz) integrable model such a relation exists! Consider XXZ spin chain for example. The spectrum and the eigenvalues of the conserved charges are known and there are different species of bound states, their total number is F. q α (x,t) = F i=1 rapidity {}}{ dλ q α,i root density }{{} charge eigen. {}}{ ρ (x,t),i, j α (x,t) = F i=1 dλ q α,i dressed velocity Continuity eqs. boil down to the same hydro eqs. for the free fermion Wigner function (Castro-Alvaredo, Doyon, Yoshimura PRX (2016); Bertini, Collura, De Nardis, Fagotti, PRL (2016)) {}}{ v (x,t),i ρ (x,t),i t ρ (x,t),i (λ) x [v (x,t),i (λ)ρ (x,t),i (λ)] = 0, i = 1,..., F and λ R

Technical implementation in the XXZ spin chain The hydrodynamic equation can be formally solved passing to the variables fillings=occ. numb {}}{ ϑ (x,t),i (λ) = ρ (x,t),i (λ) ρ (x,t),i (λ) + ρ h ζ,i (λ), ρh (x,t),i(λ) = hole density Assuming only dependence on ζ = x t, the solution is discontinuous ϑ ζ,i (λ) = ϑ L,i (λ)θ(v ζ,i (λ) ζ) + ϑ R,i (λ)θ( v ζ,i (λ) + ζ) (1) 0. Find the fillings in the initial L/R states 1. Use (1) to obtain the fillings ϑ ζ,i at a given ζ 2.Determine the new dressed velocities v ζ,i (λ) Goto 1

Physical interpretation: relaxation along a ray t ζ + i = max λ [v ζ + i,i (λ)] ζ min ζ max = max i ζ + i ϑ L (λ) ϑ R (λ) The i-th bound state coming from the left do not contribute to the stationary state if ζ > ζ + i Inside the light cone ζ [ζ min, ζ max ] transport will be ballistic! x

Analytic solution for the DW state Remarkably we can solve analytically the hydrodynamic equations for all the initial states of the form ρ = e2hsz L Z L e 2hSz R Z R In particular we derived the exact magnetizations and current profiles for any = cos(γ) = cos πq P, MCD(Q, P) = 1 Only two bound states contribute to the determination of the magnetization and current profile. Their velocities ṽ(p) are equal and are not modified by the interactions!. ṽ(p) = ζ 0 sin(p) with p [ π P, π P ], ζ 0 = sin(γ) sin(π/p)

Remark 1: No-where continuous (fractal) behaviour Take h, summing only the contributions of the two bound states we obtain (cft with the free case (F) ) s z ζ = 1 2 1 π/p dp Θ( ṽ(p) + ζ) = 2π/P π/p ( ) 1 ζ 2π/P arcsin ζ 0 I remind: ζ 0 = sin(γ), ζ [ sin(γ), sin(γ)], ζ = x/t sin(π/p) The spin current follows from the continuity eq. [ j s z ζ = ζ 0 2π/P 1 ζ2 ζ 0 cos ( ) ] π P

Remark 1: No-where continuous (fractal) behaviour Physically we could expect intermediate relaxations in correspondence of the rational approximations of, obtained by its continuous fraction representation j s z ζ 0.4 0.3 0.2 0.1 ζ = 0 γ = π/2 0.3 0.2 0.1 ζ 0 0.1 1 10 100 t DW γ = π/φ γ = 2π/3 γ = 3π/5 γ = 5π/8 γ = π/φ γ = π/2 γ = 2π/3 γ = 3π/5 γ = 5π/8 0-1 -0.5 0 0.5 1 Figure: The central ζ = 0 current, in the DW quench with γ = π/ϕ. ϕ is the golden ration and 1/ϕ = [0; 1, 1, 1, 1... ] as continuous fraction

Remark 2: Absence of Tracy-Widom and diffusion Near the right boundary of the light cone ζ max = sin(γ) the magnetization profile as a finite derivative

Remark 2: Absence of Tracy-Widom and diffusion Near the right boundary of the light cone ζ max = sin(γ) the magnetization profile as a finite derivative This proves that the scaling near the edge is not Tracy-Widom! Take Q = 1, the magnetization follows from the density of a free fermion model with p [ γ, γ] and ṽ(p) = sin(p) Uniform asymptotics near ζ = sin(γ) is now ( ) 1 2π 1/2 2 + sz ζ F (X ) t cos(γ) π π γ γ x t sin(γ) However X = and F (X ) is given in terms of imaginary [t cos(γ)] 1/2 error functions γ π p

Remark 2: Absence of Tracy-Widom distribution and diffusion t 1/3 [1\2+ s z ζ] 1.5 1 0.5 0 t = 50 t = 60 t = 70 γ = π/2-9 -6-3 0 3 [ζ-sin(γ)]t 2/3 t 1/2 [1\2+ s z ζ] 3 2.5 2 1.5 1 0.5 0 t = 50 t = 60 t = 70 γ = π/3-3 0 3 [ζ-sin(γ)]t 1/2 t 1/2 [1\2+ s z ζ] 2.5 2 1.5 1 0.5 γ = π/4 t = 50 t = 60 t = 70 0-3 -2-1 0 1 2 3 [ζ-sin(γ)]t 1/2 t 1/2 [1\2+ s z ζ] 2 1.5 1 0.5 γ = π/5 t = 50 t = 60 t = 70 0-2 -1 0 1 2 3 [ζ-sin(γ)]t 1/2 Figure: Collapse of data points in the scaling variable X = (x t sin(γ))t 1/2 when 0 Diffusive behaviour and no-where continuous behaviour were also pointed out recently by Stephan (1707.06625) and Misguich, Mallick, Krapivsky (1708.01843)

Remark 3: The fractal Drude weight We also get an analytic expression at finite h for the current [ j s z ζ = tanh(h) ζ 0 2π/P 1 ζ2 ζ 0 cos ( ) ] π P

Remark 3: The fractal Drude weight We also get an analytic expression at finite h for the current [ j s z ζ = tanh(h) ζ 0 2π/P 1 ζ2 ζ 0 cos ( ) ] π P It is then possible to extract the infinite temperature spin Drude weight following Ilievsky, De Nardis PRL (2017) D s z β = lim δµ 0 2δµ ζmax ζ min dζ j s z ζ Where the small magnetic field gradient is δµ = 4h. One gets the exact result for β 0 [ (16/β)D s z = ζ0 2 1 sin(2π/p) ] 2π/P Proving the nowhere continuous behaviour of the Drude weight and the tightness of the lower bound obtained in Prosen and Ilievsky, PRL (2013).

Take home messages We have exactly solved the relaxation dynamics in a truly interacting 1d model (XXZ), starting from initial states where spins have uniform expectation values tanh(h) 2 on the left and tanh(h) 2 on the right. We have determined exactly the spin profile and the spin current The profiles exhibit nowhere continuous (fractal) behaviour as a function of the anisotropy Away from = 0 where the model is free, there is no more Tracy-Widom distribution at the edges At the isotropic point = 1, it seems that the right scaling variable should be x/ t, for any h. So diffusion! We proved the tightness of Prosen-Ilievsky lower bound for the infinite temperature Drude weight That s it for today, thank you!