Computation of quantum phase transitions by reduced-density-matrix mechanics

Gidofalvi, Gergely; Mazziotti, David A.

doi:10.1103/physreva.74.012501

Cited by 52 publications

(73 citation statements)

References 35 publications

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“…(1) was unknown until recently, 8 an approximate class of Nrepresentability constraints has been demonstrated to be sufficient for calculating ground-state properties of the metal-to-insulator transition of hydrogen chains, 9 ground states and charge distributions of quantum dots, 10 quantum phase transitions, 11,12 dissociation channels, 13 and quantum lattice systems. [14][15][16][17][18][19][20] Variational minimization of the energy as a functional of the 2-RDM is expressible as a special convex optimization problem known as a semidefinite program.…”

Section: Introductionmentioning

confidence: 99%

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Rubin¹,

Mazziotti²

2014

Highlights in Theoretical Chemistry

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Minimizing the energy of an N -electron system as a functional of a two-electron reduced density matrix (2-RDM), constrained by necessary N -representability conditions (conditions for the 2-RDM to represent an ensemble N -electron quantum system), yields a rigorous lower bound to the groundstate energy in contrast to variational wavefunction methods. We characterize the performance of two sets of approximate constraints, (2,2)-positivity (DQG) and approximate (2,3)-positivity (DQGT) conditions, at capturing correlation in one-dimensional and quasi-one-dimensional (ladder) Hubbard models. We find that, while both the DQG and DQGT conditions capture both the weak and strong correlation limits, the more stringent DQGT conditions improve the ground-state energies, the natural occupation numbers, the pair correlation function, the effective hopping, and the connected (cumulant) part of the 2-RDM. We observe that the DQGT conditions are effective at capturing strong electron correlation effects in both one-and quasi-one-dimensional lattices for both half filling and less-than-half filling.

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Section: Introductionmentioning

confidence: 99%

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Rubin¹,

Mazziotti²

2014

Highlights in Theoretical Chemistry

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“…This third stage also comprises a ter codes based on semidefinite programming to the actual calculation of the 2-RDM subject to N-representability conditions [49][50][51]. In fact, these mathematical and computational developments made it possible to systematically assesss the effect that the progressive inclusion of tighter N -representability conditions had on the energy [52][53][54][55][56]. This has lead to the possibility of approaching the accuracy of traditional and very exact quantum chemical calculations such as those based on the Coupled-Cluster method [57,58].…”

Section: N-representability In Rdm Theorymentioning

confidence: 99%

Functional <i>N</i>-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories

Ludeña¹,

Torres²,

Costa³

2013

JMP

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The N-representability conditions on the reduced second-order reduced density matrix (2-RDM), impose restrictions not only in the context of reduced density matrix theory (RDMT), but also on functionals advanced in one-matrix theory such as natural orbital functional theory (NOFT), and on functionals depending on the one-electron density such as those of density functional theory (DFT). We review some aspects of the applications of these N-representability conditions in these theories and present some conclusions.

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“…One important idea is to study the properties of 2-RDMs is by looking at the two-and three-dimensional projections [4,5,9,10]. Since these projections are convex sets in R 2 and R 3 respectively, the hope is that the properties of the different quantum phases can be visually available.…”

Section: Introductionmentioning

confidence: 99%

“…The recent development in quantum information theory has shown that the characterization of the 2-RDMs is a hard problem even with the existence of a quantum computer [6][7][8]. Nevertheless, many practical approaches are developed to characterize the properties of 2-RDMs, and to retrieve useful information that reflects the physical properties of the system [9,10].…”

Section: Introductionmentioning

confidence: 99%

Joint product numerical range and geometry of reduced density matrices

Chen

Guo

et al. 2016

Sci. China Phys. Mech. Astron.

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The reduced density matrices of a many-body quantum system form a convex set, whose threedimensional projection Θ is convex in R 3 . The boundary ∂Θ of Θ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range Π of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of Π. We show that, a ruled surface on ∂Θ sitting in Π has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of Θ, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

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Computation of quantum phase transitions by reduced-density-matrix mechanics

Cited by 52 publications

References 35 publications

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Functional <i>N</i>-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories

Joint product numerical range and geometry of reduced density matrices

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Computation of quantum phase transitions by reduced-density-matrix mechanics

Cited by 52 publications

References 35 publications

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Functional &lt;i&gt;N&lt;/i&gt;-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories

Joint product numerical range and geometry of reduced density matrices

Contact Info

Product

Resources

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Functional <i>N</i>-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories