Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [Phys. Rev. A 82, 050301 (2010)] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group
G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, associated, e.g., with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1)-symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multiscale entanglement renormalization Ansatz.
The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to select a specific symmetry sector (e.g., with a given particle number or spin), to ensure the exact preservation of total charge, and to significantly reduce computational costs. Compared to the case of a generic tensor network, the practical implementation of symmetries in the MPS is simplified by the fact that tensors only have three indices (they are trivalent, just as the Clebsch-Gordan coefficients of the symmetry group) and are organized as a one-dimensional array of tensors, without closed loops. Instead, a more complex tensor network, one where tensors have a larger number of indices and/or a more elaborate network structure, requires a more general treatment. In two recent papers, namely, (i) [Singh, Pfeifer, and Vidal, Phys. Rev. A 82, 050301 (2010)] and (ii) [Singh, Pfeifer, and Vidal, Phys. Rev. B 83, 115125 (2011)], we described how to incorporate a global internal symmetry into a generic tensor network algorithm based on decomposing and manipulating tensors that are invariant under the symmetry. In (i) we considered a generic symmetry group
G that is compact, completely reducible, and multiplicity free, acting as a global internal symmetry. Then, in (ii) we described the implementation of Abelian group symmetries in much more detail, considering a U(1) symmetry (e.g., conservation of global particle number) as a concrete example. In this paper, we describe the implementation of non-Abelian group symmetries in great detail. For concreteness, we consider an SU(2) symmetry (e.g., conservation of global quantum spin). Our formalism can be readily extended to more exotic symmetries associated with conservation of total fermionic or anyonic charge. As a practical demonstration, we describe the SU(2)-invariant version of the multiscale entanglement renormalization ansatz and apply it to study the low-energy spectrum of a quantum spin chain with a global SU(2) symmetry.
In this thesis I develop a formalism whereby a tensor network may be understood in terms of a unitary braided tensor category, and represented in a particularly efficient manner corresponding to the exploitation of this mathematical structure. This approach permits the exploitation of the global Abelian and non-Abelian internal group symmetries of a lattice model, both to facilitate the study of particular symmetry sectors and to provide a considerable reduction in computational cost. It also enables the study of models with non-trivial exchange statistics (fermions, Abelian and non-Abelian anyons) using tensor network algorithms which scale polynomially in the system size.
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group
G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group
G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance in the context of tensor network algorithms as well, thus setting the stage for cross-fertilization between these two areas of research.
Due to the unfavorable scaling of tensor-network methods with the refinement parameter
M, new approaches are necessary to improve the efficiency of numerical simulations based on such states, in particular for gapless, strongly entangled systems. In one-dimensional density matrix renormalization group methods, the use of Abelian symmetries has led to large computational gain. In higher-dimensional tensor networks, this is associated with significant technical efforts and additional approximations. We explain a formalism to implement such symmetries in two-dimensional tensor-network states and present benchmark results that confirm the validity of these approximations in the context of projected entangled-pair state algorithms.
The simulation and numerical study of large, strongly correlated quantum systems containing Fermions or using real-time evolution in finite dimensions is still an essentially unsolved problem, primarily due to the exponential growth of the Hilbert state space with system size and the occurrence of the so-called sign problem in Monte Carlo studies. In this area, the use of tensor-network methods, for one-dimensional systems chief among them the density matrix renormalisation group (DMRG) and matrix-product states (MPS), has grown in importance in recent years. This thesis first recapitulates the use of non-abelian symmetries such as SU(2)-Spin in arbitrary tensor networks with an extensive review of the published literature including detailed algorithms and implementation hints. Implementing such symmetries can lead to a considerably more efficient representation of states in the tensor network. This part is intended to be suitable as an implementation-oriented introduction to tensor networks in general and the implementation of non-abelian symmetries in particular. Second, it introduces a series of technical improvements for the MPS methods. These improvements include a faster convergence scheme for MPS-DMRG, a systematic approach to the construction of matrix-product operators and an improved Krylov time evolution method as well as the combination of several well-known techniques into a single tensor network toolkit, SYTEN. The effectiveness of these improvements is demonstrated in numerical examples. Third, the toolkit is applied to the study of two models of current research interest: A one-dimensional spin chain in a staggered external magnetic field is studied and confinement of the elementary spinon excitations, as predicted by analytical arguments, found numerically using real-time evolution and evaluation of the dynamical structure factor. Additionally, the Hubbard model in two dimensions is studied extensively at various system sizes, geometries, interaction strengths U and filling factors n using up to 30'000 SU(2)-Spin-symmetric states equivalent to approx. 100'000 states in other MPS-DMRG implementations. Hints of a possible phase coexistence in the region 0.85 < n < 0.95 are found at intermediate interaction strengths U = 4 and U = 6 as well as a consistently striped ground state in the region n ≈ 0.875.