Solving condensed-matter ground-state problems by semidefinite relaxations.

@article{Barthel2012SolvingCG,
  title={Solving condensed-matter ground-state problems by semidefinite relaxations.},
  author={Thomas Barthel and R. H{\"u}bener},
  journal={Physical review letters},
  year={2012},
  volume={108 20},
  pages={
          200404
        }
}
We present a generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite programs (SDP). These can be solved efficiently, yielding strict lower bounds to the ground-state energy and approximations to the few-particle Green's functions. As the method is applicable for all particle statistics, it represents, in particular, a novel route for… 

Figures from this paper

A paradox in bosonic energy computations via semidefinite programming relaxations
We show that the recent hierarchy of semidefinite programming relaxations based on non-commutative polynomial optimization and reduced density matrix variational methods exhibits an interesting
Variational reduced density matrix method in the doubly-occupied configuration interaction space using four-particle N-representability conditions: Application to the XXZ model of quantum magnetism.
TLDR
The different approximations are applied to the one-dimensional XXZ model of quantum magnetism, which has a rich phase diagram with one critical phase and constitutes a stringent test for the method and shows the usefulness of the treatment to achieve a high degree of accuracy.
Bootstrapping Lattice Vacua
This paper demonstrates the application of semidefinite programming to lattice field theories, showcasing spin chains and lattice scalar field theory. Requiring expectation values of manifestly
Product-state approximations to quantum ground states Citation
The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of
Product-state approximations to quantum ground states
TLDR
A new quantum version of the de Finetti theorem is proved which does not require the usual assumption of symmetry and is described a way to analyze the application of the Lasserre/Parrilo SDP hierarchy to local quantum Hamiltonians.
Variational embedding for quantum many-body problems
TLDR
This work introduces the first quantum embedding theory that is also variational, in that it is guaranteed to provide a one-sided bound for the exact ground-state energy, and describes how the proper notion of quantum marginal should be phrased in terms of certain algebras of operators.
Product-state Approximations to Quantum Ground States
The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of
Variational determination of the two-particle reduced density matrix within the doubly occupied configuration interaction space: exploiting translational and reflection invariance
This work incorporates translational and reflection symmetry reductions to the variational determination of the two-particle reduced density matrix (2-RDM) corresponding to the ground state of
Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints
TLDR
This work derives new constraints which extend these cluster constraints to incorporate the open-system nature of a cluster on a larger lattice, at a fraction of the computational cost.
Strong Electron Correlation in Materials from Pair-Interacting Model Hamiltonians
Strong electron correlation in materials is explored within a class of model Hamiltonians that treat only pair interactions between electrons. The model is unique among typical spin Hamiltonians in
...
1
2
...