N onequilibrium Dynamics wi ti tie T ime-Dependent DMR G Salvatore R. Manmana Institute for Theoretical Physics University of Göttingen • A.C. Tiegel, S.R. Manmana, T. Pruschke, and A. Honecker, arXiv:1312.6044: frequency-space dynamics at finite temperatures • F.H.L. Essler, S. Kehrein, S.R. Manmana, and N.J. Robinson, arXiv:1311.4557 (to appear in PRB): controlled integrability breaking • J. Eisert, M. van den Worm, S.R. Manmana, and M. Kastner, PRL 111 , 260401 (2013): Lieb-Robinson-Bound & long-range interactions
Quantum Many-Body Sys tf m s in Nature and in ti e Lab Schrödinger equation: time dependent time independent i ~ ∂ ∂ t | ψ i = ˆ ˆ H | ψ i H | ψ i = E | ψ i ~ 2 ˆ ˆ X ~ X r 2 H = � i + V ( ~ x j ) x i , ~ 2 m i i h i,j i Quantum Magnetism of natural Minerals Synthetisized Materials: (Herbertsmithite, Azurite,...): “Spin-liquids”? High-Temperature Superconductors Low-dimensional systems Ultracold Gases (Optical Lattices): (e.g. TTF-TCNQ & further charge-transfer salts) Bose-Einstein condensates & Mott-Insulators
M any-Body Sys tf ms Out-Of-Equilibrium: 1 ) Linear Respons e angle-resolved photoemission scanning-tunneling (ARPES) spectroscopy photon energy source analyser ( from www.physics.rutgers.edu/ bartgroup/ ) (from Wikipedia) ☞ electronic density of states A ( k , ω ) ☞ local density of states A ( ω )
M any-Body Sys tf ms Out-Of-Equilibrium: 2 ) Highly Exci tf d Ma tf rial s F. Krausz & M. Ivanov, RMP (2009) Photo-excitation of “Light-induced Photovoltaic effects Mott insulators superconductivity” S. Wall et al., Nature Physics (2010) D. Fausti et al., Science (2011) E. Manousakis PRB (2010) Salvatore R. Manmana
M any-Body Sys tf ms Out-Of-Equilibrium: 3 ) Ul ts acold Gases & Op tj cal La tu ice s Out-of-Equilibrium Collapse and Revival “Quantum Quenches” of a Bose-Einstein-Condensate ➠ Sudden change of M. Greiner et al., Nature (2002) parameters U 0 ➟ U thermal state in 3D, not in 1D Prepared states, Expansions ➠ “Release” atoms, remove a ‘Quantum Newton Cradle’ trapping potential T. Kinoshita et al., Nature (2006) ➠ Relaxation behavior ➠ Time scales ➠ Novel (metastable) states?
E xample Quantum Simula tp rs: P olar Molecules [A.V. Gorshkov, S.R. Manmana et al., PRL & PRA (2011)] polar Molecules (e.g. KRb) in optical lattices: 2 Rotational states ⇔ two Spinstates dipolar interaction Effective Model: J ⊥ � 1 h i X X c † i S − j + S − S + i S + + J z S z i S z n i S z j + S z � � � � H = − t j, σ c j +1 , σ + h.c. j + V n i n j + W i n j + j | i − j | 3 2 j, σ i,j J -W/8 W t: nearest-neighbor hopping V: Coulomb-repulsion (long-range) W: density-spin-interaction (long-ranged) J: Heisenberg coupling (anisotropic, long-ranged) V t V ➥ generalized t-J model with dipolar long-range interactions
E xample Op tj cal La tu ices: P olar Molecules [A.V. Gorshkov, S.R. Manmana et al., PRL & PRA (2011)] 2 basic observations: H 0 = B N 2 − d 0 ~ polar molecules are rigid rotors, e.g., in electric field: E dipolar, long-ranged interactions: level scheme for a rigid rotor in a field: Idea: project dipolar operator onto two states ➠ effective S=1/2 system
P olar molecules on op tj cal la tu ices: e ff ec tj ve model s [A.V. Gorshkov, S.R. Manmana et al., PRL & PRA (2011)] level scheme for energies in microwaves a rigid rotor in a field: electric field: ➠ dressed states: (II) (I) More general: project dipolar operator onto two dressed states ➠ tunable parameters [ ⇥ m 0 | d | m 0 ⇤ � ⇥ m 1 | d | m 1 ⇤ ] 2 “Ising” = J z useful choice of coefficients: 2 ⇥ m 1 | d | m 0 ⇤ 2 = “spin flip” J ⊥ (depend on details 1 4 [ ⇥ m 0 | d | m 0 ⇤ + ⇥ m 1 | d | m 1 ⇤ ] 2 V = “density interaction” of dressed states 1 ⇥ m 0 | d | m 0 ⇤ 2 � ⇥ m 1 | d | m 1 ⇤ 2 ⇤ {|m 0 >,|m 1 >} ) ⇥ “anisotropic interaction” = W 2 (II): Arbitrary ratio between all coefficients (I): Simplest case, leads to J z = V = W = 0 ➠ Future research ➠ This talk (III): Beyond S=1/2, spatial anisotropies,topological order: S.R.M. et al., PRB (rapid comm.) 87, 081106(R) (2013); A.V. Gorshkov, K. Hazzard & A.M. Rey, arXiv:1301.5636 (2013)
E xample Ul ts acold Gases: I ons in a Tra p 9 Be + ions in a Penning trap (NIST Boulder) 171 Yb + ions (JQI/NIST Maryland) [J.W. Britton et al., Nature 484 , 489 (2012)] [K. Kim et al., Nature 465 , 590 (2010); R. Islam et al., Nature Comm. 2 ,377 (2011); NJP and more...] Realization of Ising models with transverse field on variety of lattices: Interactions ∼ 1/r α
N umerical Me ti ods for Many-Body Sys tf ms: C ha lm enge s I) Dynamical spectral functions: resolution, finite temperatures II) ‘Highly excited systems’: long times, time evolution at finite temperatures III) Recent development quantum simulators: long-range interactions Further important challenges: D>1, dissipation, infinite system size, ...
“N umerica lm y Exact Dynamics ” : E xact Diagonaliza tj o n Direct approach: No limitations: • arbitrary long times • accuracy (machine precision) • arbitrary geometry • independent on details of system or initial state Bad: ➠ Need the full spectrum...difficult ☹
“N umerica lm y Exact Dynamics ” : Itf ra tj ve Diagonaliza tj o n Lanczos procedure: H | v n i � a n | v n i � b 2 | v n +1 i n | v n − 1 i = (Krylov space method) K. Lánczos (1950) a n = h v n | H | v n i n +1 = h v n +1 | v n +1 i b 2 , , b 0 = 0 h v n | v n i h v n | v n i a 0 b 1 b 1 a 1 b 2 0 Tridiagonalization of ... b 2 a 2 T n = Hamiltonian matrix: ... ... 0 b n b n a n Projection of time evolution operator: T.J. Park and J.C. Light, J. Chem. Phys (1986) Error estimate: M. Hochbruck and C. Lubich, SIAM (1997) Larger systems possible Usually n<20 is sufficient Pro’s/Con’s similar to ‘full diagonalization’ ➠ Need to store n vectors with dimension of H ☹
“N umerica lm y Exact Dynamics ” : Ti e DMR G S.R. White, PRL (1992); U. Schollwöck, RMP (2005)/Ann. Phys. (2011); R.M. Noack & S.R. Manmana, AIP (2005) A B l Obtain ground state of finite, small lattice (e.g., using Lanczos) l Reduced density matrix of subsystem (“system block”) ➠ Schmidt decomposition (1907) dim H m Approximation: X X | ψ i = w j | α i j | β i j ⇡ w j | α i j | β i j m ⌧ dim H j =1 j =1 | α i j , | β i j : Eigenstates of reduced density matrices of A or B typically (1D) m ∼ 1000, error (discarded weight) ∼ 10 -9 X w 2 j log w 2 ➥ central quantity: entanglement entropy S = − j j The larger the entanglement, the larger m for a desired accuracy. • Problematic for D > 1 (‘area law of entanglement’) • Entanglement grows with time - inhibits (very) long times
“N umerica lm y Exact Dynamics ” : Ti e DMR G Iterative Procedure: [Webpage E. Jeckelmann]
“N umerica lm y Exact Dynamics ” : Ti e adap tj ve t-DMR G Basic idea: − Approximate time evolution operator • Suzuki-Trotter decomposition [Vidal (2003/2004); S.R. White & A. Feiguin (2004); A. Daley et al. (2004)] • Lanczos projection [P. Schmitteckert (2004); S.R. Manmana et al. (2005)] U − Adapt basis of density-matrix eigenvectors at each time step Trotter approach (n.n. interactions): Lanczos approach (arbitrary geometry)
“N umerica lm y Exact Dynamics ” : M a ts ix Product Sta tfs U. Schollwöck, Ann. Phys. (2011) Matrix product state (MPS) representation of wave functions: local complex-valued matrix ➠ underlying structure of the wave function in the DMRG Convergence: optimize M-matrices via variational principle Matrix product operator (MPO) representation of operators:
L inear Response Dynamics at T>0
L inear Response: D ynamical correla tj on fv nc tj on s ☞ time-dependent perturbation H ( t ) = H 0 − h A e i ω t A ☞ linear response: � ∞ ∞ ∞ � d � � � d t e i ω t � T B ( t ) A � 0 = � � Ψ 0 | B | n � � n | A | Ψ 0 � e i t ( ω − ( E n − E 0 )) � d t � B ( t ) � = d t � d h A � n � −∞ h A =0 −∞ −∞ � = 2 π � Ψ 0 | B | n � � n | A | Ψ 0 � δ ( ω � ( E n � E 0 )) n with H 0 | n � = E n | n � ☞ express via Green’s functions G A ( z ) = � Ψ 0 | A † ( z � H ) − 1 A | Ψ 0 � C A † ,A ( ω ) = Im G A ( ω + i η + E 0 ) ,
D ynamical proper tj es of quantum magnets: E SR on Cu-PM in magne tj c fi eld s Copper pyrimidine dinatrate: [S. Zvyagin et al., PRB(R) (2011)] (Quasi-)1D Heisenberg AFM, described by effect of staggered g-tensor + DM interaction ESR spectrum in magnetic field: DMRG results
F ini tf tf mperature me ti ods: p uri fi ca tj on wi ti ma ts ix product sta tfs ☞ Compute thermal density matrix via a pure state in an extended system: [U. Schollwöck, Annals of Physics (2011)] | Ψ T i = e − ( H P ⊗ I Q ) / (2 T ) h i ⌦ L j =1 | rung � singlet i j ) % T = e − H/T = Tr Q | Ψ T i h Ψ T | ☞ Real time evolution at finite temperature: | Ψ T i ( t ) = e − i ( H P ⊗ U Q ) t | Ψ T i ) G A ( T, t ) Fourier ) G A ( T, ω ) Problem: reach long times for large systems Ways out: linear prediction, backward time evolution in Q [T. Barthel, U. Schollwöck & S.R. White, PRB (2009); C. Karrasch, J.H. Bardarson & J.E. Moore, PRL (2012)]
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