Tensor Network States and Geometry

@article{Evenbly2011TensorNS,
  title={Tensor Network States and Geometry},
  author={Glen Evenbly and Guifr{\'e} Vidal},
  journal={Journal of Statistical Physics},
  year={2011},
  volume={145},
  pages={891-918}
}
  • G. Evenbly, G. Vidal
  • Published 6 June 2011
  • Mathematics, Physics
  • Journal of Statistical Physics
Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D=1 dimensions, as well as projected entangled pair states (PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional… 
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References

SHOWING 1-10 OF 91 REFERENCES
Accurate determination of tensor network state of quantum lattice models in two dimensions.
TLDR
A novel numerical method to calculate accurately physical quantities of the ground state using the tensor network wave function in two dimensions and results for the Heisenberg model on a honeycomb lattice agree well with those obtained by the quantum Monte Carlo and other approaches.
Simulating strongly correlated quantum systems with tree tensor networks
We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the
Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
This work explores the use of a tree tensor network ansatz to simulate the ground state of a local Hamiltonian on a two-dimensional lattice. By exploiting the entropic area law, the tree tensor
Finitely correlated states on quantum spin chains
We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a
Fermionic multiscale entanglement renormalization ansatz
In a recent contribution [P. Corboz, G. Evenbly, F. Verstraete, and G. Vidal, arXiv:0904.4151 (unpublished)] entanglement renormalization was generalized to fermionic lattice systems in two spatial
Tensor-product representations for string-net condensed states
We show that general string-net condensed states have a natural representation in terms of tensor product states (TPSs). These TPSs are built from local tensors. They can describe both states with
Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states
We explain how to implement, in the context of projected entangled-pair states (PEPSs), the general procedure of fermionization of a tensor network introduced in P. Corboz and G. Vidal, Phys. Rev. B
Criticality, the area law, and the computational power of projected entangled pair states.
TLDR
It is proved that coherent versions of thermal states of any local 2D classical spin model correspond to PEPS, which are in turn ground states of local2D quantum Hamiltonians, and this correspondence maps thermal onto quantum fluctuations.
Explicit tensor network representation for the ground states of string-net models
We provide a simple expression for the ground states of arbitrary string-net models in the form of local tensor networks. These tensor networks encode the data of the fusion category underlying a
Ground-state phase diagram of the two-dimensional t-J model
The ground-state phase diagram of the two-dimensional t-J model is investigated in the context of the tensor network algorithm in terms of the graded Projected Entangled-Pair State representation of
...
1
2
3
4
5
...