Ground-state properties of quantum many-body systems: entangled-plaquette states and variational Monte Carlo

Mezzacapo, Fabio; Schuch, Norbert; Cirac, J. I.

doi:10.1088/1367-2630/11/8/083026

Cited by 102 publications

(148 citation statements)

References 35 publications

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“…This recalls the correlator product state approximation (also known as an entangled plaquette state, 40,41 ), although once again the tensor is expressed in an excitation rather than occupation number picture. Naturally, we can consider many other combinations of auxiliary indices and physical indices, and the appropriateness of the particular choice depends on the problem at hand.…”

Section: A Classification Of Tensor Factorizationsmentioning

confidence: 93%

Tensor factorizations of local second-order Møller–Plesset theory

Yang

Kurashige

Manby

et al. 2011

The Journal of Chemical Physics

163

170

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Efficient electronic structure methods can be built around efficient tensor representations of the wavefunction. Here we first describe a general view of tensor factorization for the compact representation of electronic wavefunctions. Next, we use this language to construct a low-complexity representation of the doubles amplitudes in local second-order Møller-Plesset perturbation theory. We introduce two approximations-the direct orbital-specific virtual approximation and the full orbitalspecific virtual approximation. In these approximations, each occupied orbital is associated with a small set of correlating virtual orbitals. Conceptually, the representation lies between the projected atomic orbital representation in Pulay-Saebø local correlation theories and pair natural orbital correlation theories. We have tested the orbital-specific virtual approximations on a variety of systems and properties including total energies, reaction energies, and potential energy curves. Compared to the Pulay-Saebø ansatz, we find that these approximations exhibit favorable accuracy and computational times while yielding smooth potential energy curves.

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Section: A Classification Of Tensor Factorizationsmentioning

confidence: 93%

Tensor factorizations of local second-order Møller–Plesset theory

Yang

Kurashige

Manby

et al. 2011

The Journal of Chemical Physics

163

170

View full text Add to dashboard Cite

show abstract

“…38,39 ). For this system size and the same values of τ and D we now achieve the slightly lower energy per site −0.62637(2), and we can also provide the converged D = 6 result −0.62774(1).…”

Section: B Heisenberg Modelmentioning

confidence: 99%

Algorithms for finite projected entangled pair states

2014

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Projected entangled pair states (PEPS) are a promising ansatz for the study of strongly correlated quantum many-body systems in two dimensions. But due to their high computational cost, developing and improving PEPS algorithms is necessary to make the ansatz widely usable in practice. Here we analyze several algorithmic aspects of the method. On the one hand, we quantify the connection between the correlation length of the PEPS and the accuracy of its approximate contraction, and discuss how purifications can be used in the latter. On the other, we present algorithmic improvements for the update of the tensor that introduce drastic gains in the numerical conditioning and the efficiency of the algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with the Heisenberg and quantum Ising models on lattices of up to 21 × 21 sites.

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“…The approximation improves as the set of variational wavefunctions expands to include wavefunctions that have greater overlap with the true ground state |Ψ 0 , becoming exact in the limit where |Ψ 0 is included in the space. There exist various forms of variational wave-functions including the Slater-Jastrow [7], backflow [8,9], HuseElser [10] (equivalently correlator product states [11], entanglement plaquette states [12], or graph tensor network states [13]), BDG states, projected entangled pair states [14,15], etc. At finite temperature, variational techniques have been less useful, except again in onedimension where minimally entangled typical thermal states (METTS) [16,17] and finite-temperature DMRG [18] have proved valuable.…”

Section: Department Of Physics University Of Illinois At Urbana-chammentioning

confidence: 99%

Finite-temperature properties of strongly correlated systems via variational Monte Carlo

Claes

Clark

2017

Phys. Rev. B

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Variational methods are a common approach for computing properties of ground states but have not yet found analogous success in finite temperature calculations. In this work we develop a new variational finite temperature algorithm (VAFT) which combines ideas from minimally entangled typical thermal states (METTS), variational Monte Carlo (VMC) optimization and path integral Monte Carlo (PIMC). This allows us to define an implicit variational density matrix to estimate finite temperature properties in two and three dimensions. We benchmark the algorithm on the bipartite Heisenberg model and compare to exact results.Strongly correlated fermionic and frustrated spin systems span many interesting physical systems. Computing properties of these systems is difficult, particularly in dimensions greater then one where density matrix renormalization group (DMRG) Of course, it is also possible to use the variational technique at finite temperature. Most naturally one might parameterize the finite temperature many body density, optimizing again over the parameters α. However, this requires specifying a set of FTDM over which to optimize, a task that is naturally more difficult than specifying a set of wavefunctions. Additionally, this optimization needs to minimize the free energy of the system, a quantity which is harder to evaluate, requiring techniques such as coupling constant integration. For one example of this approach, see ref. [19].Producing a good variational ansatz for very high or low temperature density matrices is straightforward.At high temperature (small β), one can approximate exp[−βĤ] by either Taylor expanding as (1 − βĤ) or using the Trotter break-up. At low temperature, one can instead expand in terms of the low-energy excitations, writingρ =and E i are variational estimates for the ith eigenvector and eigenvalue respectively and k is some cutoff. Unfortunately, this can be difficult, requiring orthogonal variational estimates of a number of excited states, and as one goes to higher temperature the number of eigenstates increases exponentially. Nonetheless, this framework is often applied in the context of density functional theory (DFT) to produce finite temperature functionals [20].At any temperature, one can approximate the density matrix as a purificationρ ≈ i λ i |Ψ i Ψ i | over a set |Ψ i of variational wavefunctions. However, one cannot efficiently enumerate more than a polynomial number of |Ψ i , nor is it obvious how to optimize over these wavefunctions.In spite of these difficulties, it would be useful if we could use known variational wave-functions to compute finite temperature properties. In this paper, we describe a new variational finite temperature approach (VAFT), in which we implicitly define a variational finite temperature density matrix. Our variational FTDM is a purification of an exponentially large number of variational wavefunctions. We compute properties of this FTDM via an algorithm which stochastically samples from its diagonal without ever needing to explicitly represent...

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Ground-state properties of quantum many-body systems: entangled-plaquette states and variational Monte Carlo

Cited by 102 publications

References 35 publications

Tensor factorizations of local second-order Møller–Plesset theory

Tensor factorizations of local second-order Møller–Plesset theory

Algorithms for finite projected entangled pair states

Finite-temperature properties of strongly correlated systems via variational Monte Carlo

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