Analysis of Planck and the Equilibrium ofantis in Tropical Physics

Transcription

1 Emergence of Fokker-Planck Dynamics wihin a Closed Finie Spin Sysem H. Niemeyer(*), D. Schmidke(*), J. Gemmer(*), K. Michielsen(**), H. de Raed(**) (*)Universiy of Osnabrück, (**) Supercompuing Cener Juelich ETQS, Sellenbosch, April 15h, 13

2 Thermalizaion in closed quanum sysems? (Non-eq.) Thermodynamics Quanum Mechanics auonomous dynamics of a few macrovariables aracive fixed poin, equilibrium ofen describable by Fokker-Planck equaions auonomous dynamics of he wavefuncion. no aracive fixed poin (Schroedinger equaion) Schroedinger is no Fokker-Planck This puzzle (parially) riggered a lo of research: quanum ypicaliy, eigensae hermalizaion hypohesis (ETH), projecion operaor mehods, open quanum sysems, decoherence, Caldeira-Lege model, ec. Quanum sysems ha show sandard Fokker-Planck relaxaion bu are no of he small sysem + large bah ype appear o be rare in he lieraure. Recen example: Aes e al., PRL 18, (1): magneizaion in an Ising model wih a ransverse field decays according o Fokker-Planck bu wih a ime-depeden FP-Operaor.

3 Model and observables spin-model L R Heisenberg-ype Hamilonian: A ladder wih anisoropic, XXZ-ype couplings which are srong along he beams and weak along he rungs. Ĥ = ij J ij (ˆσ i xˆσ j x + ˆσ i yˆσ j y +.6 ˆσ i zˆσ j z), where J ij = 1 for solid lines, J ij = κ =. for doed lines and J ij = oherwise. Toal number of spins N = 16. The z-componen of oal magneizaion S z is conserved We analyze: magneizaion difference ˆx ( ) ˆx = 1 l L ˆσ l z r R ˆσ r z eigenvalues of ˆx wihin he subspace of vanishing oal magneizaion, S z = : X = N 4, N + 1,... + N 4.

4 Naive classical descripion Assume here are raes a which muual spin-flips, i.e., simulanous, conrariwise flips of adjacen spins occur. Le hese raes be proporional o he square of he coupling consan beween he adjacen spins. Exploi local equilibrium due o ime scale separaion beween leg-dynamics (fas) and rung-dynamics (slow) Raes R (X X±1) = γκ N ( 1 X ) N coninuum limi, N,X, magneizaion difference densiy z := X/N, Kramer-Moyal expansion: p(,z) = z(( zu(z)p)+ 1 z(d(z)p)+o( 3 z) U(z) = γκ z, D(z) = γκ (1/4+4z )/N. Almos like a Brownian paricle in a parabolic poenial.

5 Exac resul vs. naive descripion iniial saes mean of X ˆρ X () = 1 Z ˆP (,)ˆP X ˆP (,) X=1 X= ˆP X : projecor ono subspace X ˆP (,) : projecor ono energy inervall a() 1 1 P X widh of X 4 x σ ().5 X= X=1 X=

6 Do we undersand hose numerical findings? We ry o! This effor involves he TCL projecion operaor mehod projecion superoperaor Pˆρ = P X ˆPX d X, P X = Tr{ˆP Xˆρ} d X = Tr{ˆP X } P = P going hrough he formalism yields: Ṗ Y = X Y RY,X()P TCL X RX,Y()P TCL Y X Y realisically compuable are. order raes: R TCL Y,X () := C Y,X ( )d ime dependence: only generaed by Ĥ, here: legs ˆV: ineracion, here: rungs C Y,X ( ) = κ d X Tr{[ˆV( ), ˆP Y ][ˆV(), ˆP X ]} Iniial correlaion funcions are proporional o naive raes: C Y,X () = δ Y,X±1 R X X±1 4γ This resul is no resriced o his model.

7 Do we undersand hose numerical findings? There are more condiions on he validiy of.order descripions han jus ime-scale separaion (Van Hove, Barsch e al.): The ineracion marix mus show feaures of a marix he elemens of which are drawn a random. This seems o hold here: random fine srucure of ransiion marix smooh coarse srucure of ransiion marix

8 Wha abou bigger sysems? - numerics: he quanum evoluion of a pure sae may be more easy o compue han he evoluion of a mixed sae - dynamical ypicaliy: adequae random pure saes may mimic he dynamics of mixed saes We use an ieraive Chebyshev scheme o implemen Schroedinger-ype propagaion Iniial saes: ψ() e (Ĥ E) τˆp X φ, wih φ random, E = shifed expecaion values of X, N = 3 variances of X, N = 3 <M_z(A)-M_z(B)> dm=7 dm=6 dm=5 dm=4 dm=3 dm= dm=1 Var(M_z(A)-M_z(B)) dm dm dm4 dm6 dm8 dm1 dm1 dm14 dm16 dm18 dm dm dm4 dm4 dm6 dm8 dm

9 How is ha comparable o wo cups of coffee hermalizing each oher? - The larges inial X() yielding Markovian decaying expecaion values ˆx() appears o scale as N. - The maximun widh δx during his decay appears o scale as N Are he final widh δx ruely independen of he iniial sae and wha does ha imply for he ETH? Since φ ˆx () φ φ ˆx φ Tr{ˆx ()ˆx }/d (ypicaliy) and for large imes Tr{ˆx ()ˆx } n n ˆx n here are ways o infer he variance σ of he disribuion of n ˆx n wihin some energy regime from pure sae evoluions. more abou his: Europhys. Le., 11, 111 (13) Thank you for your aenion! sigma/n relaive spread of n ˆx n, i.e., σ /N N