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Dynamics of Hubbard Hamiltonians with the Multiconfigurational Time-Dependent Hartree Method for Indistinguishable Particles Axel Lode, Christoph Bruder

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Dynamics of Hubbard Hamiltonians with the Multiconfigurational Time-Dependent Hartree Method for Indistinguishable Particles Axel Lode, Christoph Bruder Dynamics of Hubbard Hamiltonians with the multiconfigurational time-dependent Hartree method for indistinguishable particles Axel Lode, Christoph Bruder To cite this version: Axel Lode, Christoph Bruder. Dynamics of Hubbard Hamiltonians with the multiconfigurational time-dependent Hartree method for indistinguishable particles. Physical Review A, American Physical Society 2016, 94, pp.013616. 10.1103/PhysRevA.94.013616. hal-02369999 HAL Id: hal-02369999 https://hal.archives-ouvertes.fr/hal-02369999 Submitted on 19 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW A 94, 013616 (2016) Dynamics of Hubbard Hamiltonians with the multiconfigurational time-dependent Hartree method for indistinguishable particles Axel U. J. Lode* and Christoph Bruder Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Received 29 April 2016; published 22 July 2016) We apply the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X) to systems of bosons or fermions in lattices described by Hubbard-type Hamiltonians with long-range or short-range interparticle interactions. The wave function is expanded in a variationally optimized time-dependent many-body basis generated by a set of effective creation operators that are related to the original particle creation operators by a time-dependent unitary transform. We use the time-dependent variational principle for the coefficients of this transform and the expansion coefficients of the wave function expressed in the time-dependent many-body basis as variational parameters to derive equations of motion. The convergence of MCTDH-X is shown by comparing its results to the exact diagonalization of one-, two-, and three-dimensional lattices filled with bosons with contact interactions. We use MCTDH-X to study the buildup of correlations in the long-time splitting dynamics of a Bose-Einstein condensate loaded into a large two-dimensional lattice subject to a barrier that is ramped up in the center. We find that the system is split into two parts with emergent time-dependent correlations that depend on the ramping time—for most barrier-raising times the system becomes twofold fragmented, but for some of the very fast ramps, the system shows revivals of coherence. DOI: 10.1103/PhysRevA.94.013616 I. INTRODUCTION For continuous systems of ultracold atoms, i.e., atoms that do not reside in an optical lattice, a theory which does not Ultracold atoms in optical lattices are a very active and wide suffer from the aforementioned problems has been formulated field of research that bridges the gap between atomic, molec- [19,20] and implemented [21–23]: the multiconfigurational ular, and optical physics and condensed matter systems. They time-dependent Hartree for indistinguishable particles. In this have been employed as quantum simulators for condensed work, we apply the same philosophy to describe ultracold matter systems that cannot be experimentally controlled to the same extent as ultracold atoms in optical lattices [1–3]. Recent atoms in optical lattices. progress in this direction includes, for instance, the realization We adopt the strategy of Refs. [19,20] and apply the time- of artificial gauge fields [4] and topological states of matter [5] dependent variational principle [24,25] to the time-dependent with cold atom systems. many-body Schrodinger¨ equation using a formally complete, To describe the dynamics of ultracold atoms in opti- time-dependent, and variationally optimized orthonormal cal lattices theoretically requires solving the equation gov- many-body basis set. We demonstrate the exactness of the ob- erning these systems, i.e., the time-dependent many-body tained theory, the multiconfigurational time-dependent Hartree Schrodinger¨ equation (TDSE). However, the dimensionality method for indistinguishable particles in lattices (MCTDH-X). of the many-body Hilbert space that hosts the solution grows In the static case, i.e., the solution of the time-independent exponentially with the number of particles and with the num- Schrodinger¨ equation, we compare MCTDH-X for a bosonic ber of sites in the considered optical lattice. Since no general Hamiltonian with exact diagonalization and find that the error analytical solution is known to date, many approximate numer- in the obtained method goes to zero roughly exponentially ical methods have been devised. Popular approaches include with the number of effective time-dependent one-particle the time-dependent density matrix renormalization group [6], basis functions employed. In the dynamical case, i.e., the matrix product states [7–9], time-evolved block decimation solution of the time-dependent Schrodinger¨ equation, we apply [10,11], dynamical mean-field theory [12–15], mean-field lat- MCTDH-X to the long-time splitting dynamics of bosons in a tice methods [16], the discrete nonlinear Schrodinger¨ equation two-dimensional lattice. We show that correlations in the split [17], and the Hartree-Fock theory for Hubbard Hamiltonians system are built up almost independently of the splitting times: [18]. These methods have problems when dealing with optical the reduced one-body density matrix acquires two eigenvalues lattices of spatial dimension larger than one (time-dependent on the order of the particle number throughout the splitting pro- density matrix renormalization group, matrix product states, cess, i.e., fragmentation emerges. Fragmentation emerges with time-evolved block dimension), when the considered lattice is a delay for short splitting times which becomes proportional not spatially homogeneous (dynamical mean-field theory), or to the splitting time for longer splitting times. Interestingly, when they oversimplify the emergent many-body correlations revivals of the uncorrelated (coherent) initial state are seen (discrete nonlinear Schrodinger¨ equation and Hartree-Fock for for very short splitting times. By quantifying the coherence of Hubbard Hamiltonians). the system with the first-order correlation function, we show that the mechanism behind the fragmentation is the loss of coherence between the left and right fractions of the bosons as *[email protected] the barrier is ramped up. 2469-9926/2016/94(1)/013616(7) 013616-1 ©2016 American Physical Society AXEL U. J. LODE AND CHRISTOPH BRUDER PHYSICAL REVIEW A 94, 013616 (2016) II. THEORY † above effective time-dependent operators aˆk(t). The resulting A. Schrodinger¨ equation and Hubbard Hamiltonians ansatz for the many-body state then reads The aim is to solve the time-dependent many-body | = |≡ | Schrodinger¨ equation, Cn(t) n Cm (t) m; t , (6) n m i∂t |=Hˆ |, (1) 1 † † | = m1 ··· mM | m; t (aˆ1(t)) (aˆM (t)) vac . (7) for the Hubbard Hamiltonian including a general long-ranged m1! ···mM ! interaction, † † † The number of coefficients C m(t) that has to be accounted for is Hˆ = bˆ bˆ − J [bˆ bˆ + bˆ bˆ ] N+M−1 M Hub j j j j k k j N for bosons and N for fermions. The number of coef- j j,k ficients is no longer directly dependent on the number of sites Ms in the treated system. Note that the multiconfigurational U † 2 2 + W(i,j)(bˆ ) (bˆj ) . (2) ansatz in Eq. (6) contains the mean-field type wave functions 2 i ij of the time-dependent Hartree-Fock for fermions and the Here and in the following, ·,· denotes neighboring sites. The so-called discrete nonlinear Schrodinger¨ equation for bosons = = full many-body state reads as special cases, for M N and M 1, respectively. The action functional [24,25] for the time-dependent many-body |= Cn(t)|n; Schrodinger¨ equation (1) can now be (re-)formulated using the {n} operator relation in Eq. (5) and reads ⎡ 1 † † |n= (bˆ )n1 ···(bˆ )nMs |vac. (3) { } { ∗ } = ⎣ | ˆ − | ··· 1 Ms S[ ukj , ukj ] dt (t) HHub i∂t (t) n1! nMs ! The sum in this ansatz runs over all possible configurations ⎛ ⎞⎤ of N indistinguishable particles in the Ms lattice sites. In the − ⎝ ∗ − ⎠⎦ exact diagonalization approach, one takes Eq. (3) as the ansatz μpq(t) upj (t)uqj (t) δpq . pq j and determines the coefficients Cn(t). However, the number of the coefficients Cn(t) in the many-body ansatz, Eq. (3), grows (8) N+Ms −1 Ms as a factorial [ N for bosons and N for fermions], i.e., exponentially, with the number of lattice sites. Here, the time-dependent Lagrange multipliers μpq(t)have To make large lattices tractable, approximations have to been introduced to enforce the orthonormality of the column be introduced to reduce the number of coefficients. In what vectors of the transform upk(t). This ensures that upk(t)is follows, we use a variational approach to obtain such an indeed unitary for all times t. To derive the equations of motion approximation: the general “MCTDH ansatz” is used to derive for upk(t), we proceed by inserting the transformed operators equations of motion. given in Eq. (5) in the above action, Eq. (8): (t)|HˆHub − i∂t |(t) B. MCTDH-X for Hubbard Hamiltonians = | ∗ † The MCTDH-X equations of motion are derived with (t) j upj (t)uqj (t)aˆp(t)aˆq (t) a time-dependent variational principle of the Schrodinger¨ j pq equation [25] for Hubbard Hamiltonians by taking into account ∗ † − J u (t)u (t)aˆ (t)aˆ (t) multiple “effective” time-dependent creation operators that act pj qk p q pq on all M lattice sites. j,k s To start, we note that one can transform the time- U ∗ ∗ † + W(i,j)u (t)u (t)u (t)u (t) independent operators bˆ in the Hubbard Hamiltonian in 2 pj rj qj sj j ij prqs Eq.
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