Many-body quantum mechanics as a symplectic dynamical system

Rowe, D.J.; Ryman, A.; Rosensteel, G.

doi:10.1103/physreva.22.2362

Cited by 106 publications

(97 citation statements)

References 22 publications

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“…Applying the TDVP for D > 1 results in time-dependent Hartree-Fock theory, and by linearizing these equations around the stationary solution, one obtains the so-called random phase approximation. 84,85 In this context, the random phase approximation can also be derived using Green's function techniques 86 or by using the equations-of-motion approach. 87 In line with what we have illustrated in the previous section, is it well known in quantum chemistry and nuclear physics that the random phase approximation can be plagued by instabilities.…”

Section: Analogies With Other Variational Methodsmentioning

confidence: 99%

Post-matrix product state methods: To tangent space and beyond

2013

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We develop in full detail the formalism of tangent states to the manifold of matrix product states, and show how they naturally appear in studying time evolution, excitations, and spectral functions. We focus on the case of systems with translation invariance in the thermodynamic limit, where momentum is a well-defined quantum number. We present some illustrative results and discuss analogous constructions for other variational classes. We also discuss generalizations and extensions beyond the tangent space, and give a general outlook towards post-matrix product methods.

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Section: Analogies With Other Variational Methodsmentioning

confidence: 99%

Post-matrix product state methods: To tangent space and beyond

2013

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“…The space X is the set of three-dimensional unit vectors v and each quantum state is associated with the probability distribution in Eq. (24). In this case, the elements of S( v) are the Hilbert space vectors |ψ such that | ψ|φ | 2 > B ≡ 1/2, |φ being the state whose bloch vector, defined by Eq.…”

Section: B the Theoremmentioning

confidence: 99%

“…In section III and here, we have considered some examples, other examples are provided in Refs. [21,22,23,24], where a N -dimensional Hilbert space is reduced to a classical phase space of 2N real variables. Our result is relevant also for quantum Monte Carlo methods.…”

Section: B the Theoremmentioning

confidence: 99%

Exponential complexity and ontological theories of quantum mechanics

Montina

2008

Phys. Rev. A

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Ontological theories of quantum mechanics describe a single system by means of well-defined classical variables and attribute the quantum uncertainties to our ignorance about the underlying reality represented by these variables. We consider the general class of ontological theories describing a quantum system by a set of variables with Markovian (either deterministic or stochastic) evolution. We provide the first proof that the number of continuous variables can not be smaller than 2N −2, N being the Hilbert space dimension. Thus, any ontological Markovian theory of quantum mechanics requires a number of variables which grows exponentially with the physical size. This result is relevant also in the framework of quantum Monte Carlo methods.

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“…Time-dependent HF theory is obtained by using an SD Ansatz in the time-dependent variational principle (TDVP). 21,22,[45][46][47][48] Time-dependent DMRG (which stays within the MPS Ansatz space) is similarly obtained by using an MPS Ansatz in the TDVP. 43,44,[49][50][51] RPA, or linear response theory for HF, is obtained by linearizing the time-dependent HF equations in the vicinity of a variational mimimum.…”

Section: Introductionmentioning

confidence: 99%

“…The Thouless theorem gives a nonredundant parameterization to generate all possible SDs from any given SD reference, by means of its OV excitation operators. [21][22][23][24] Recently, a new way to understand the qualitative structure of quantum many-body states has appeared, whereby the state is approximated by a tensor network, i.e., a contracted product of tensors where each tensor represents a local degree of freedom. These Ansätze are efficient representations of low-energy states because they capture the boundary law for the entanglement entropy.…”

Section: Introductionmentioning

confidence: 99%

Thouless theorem for matrix product states and subsequent post density matrix renormalization group methods

et al. 2013

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The similarities between Hartree-Fock (HF) theory and the density matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function Ansatz. Linearization of the time-dependent variational principle near a variational minimum allows to derive the random phase approximation (RPA). We show that the nonredundant parameterization of the matrix product state (MPS) tangent space [J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for MPS, i.e., an explicit nonredundant parameterization of the entire MPS manifold, starting from a specific MPS reference. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation (TDA), the configuration interaction (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS Ansatz with single and double excitations to improve on the ground state and to calculate low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calculations of the RPA-MPS correlation energy. With the long-range quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.

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Many-body quantum mechanics as a symplectic dynamical system

Cited by 106 publications

References 22 publications

Post-matrix product state methods: To tangent space and beyond

Post-matrix product state methods: To tangent space and beyond

Exponential complexity and ontological theories of quantum mechanics

Thouless theorem for matrix product states and subsequent post density matrix renormalization group methods

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