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arXiv:1008.4774 · Physical Review B

Tensor network states and algorithms in the presence of a global U(1) symmetry

Sukhwinder Singh, Robert N. C. Pfeifer, Guifré Vidal

Published Mar 15, 2011• 287 citations
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Abstract

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 $\mathcal{G}$, 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.

DOI
10.1103/physrevb.83.115125

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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\mathcal{G}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.
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\mathcal{G}G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G\mathcal{G}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.
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\mathcal{G}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 we extend the formalism of tensor network algorithms to incorporate global internal symmetries. We describe how to both numerically protect the symmetry and exploit it for computational gain in tensor network simulations. Our formalism is independent of the details of a specific tensor network decomposition since the symmetry constraints are imposed at the level of individual tensors. Moreover, the formalism can be applied to a wide spectrum of physical symmetries described by any discrete or continuous group that is compact and reducible. We describe in detail the implementation of the conservation of total particle number (U(1) symmetry) and of total angular momentum (SU(2) symmetry). Our formalism can also be readily generalized to incorporate more exotic symmetries such as conservation of total charge in anyonic systems.
Tensor networks offer a variational formalism to efficiently represent wave functions of extended quantum many-body systems on a lattice. In a tensor network N\mathcal{N}N, the dimension $\ensuremath{\chi}$ of the bond indices that connect its tensors controls the number of variational parameters and associated computational costs. In the absence of any symmetry, the minimal bond dimension ${\ensuremath{\chi}}^{\mathrm{min}}$ required to represent a given many-body wave function $|\ensuremath{\Psi}\ensuremath{\rangle}$ leads to the most compact, computationally efficient tensor network description of $|\ensuremath{\Psi}\ensuremath{\rangle}$. In the presence of a global, on-site symmetry, one can use a tensor network Nsym{\mathcal{N}}_{\mathrm{sym}}Nsym​ made of symmetric tensors. Symmetric tensors allow one to exactly preserve the symmetry and to target specific quantum numbers, while their sparse structure leads to a compact description and lowers computational costs. In this paper we explore the trade-off between using a tensor network N\mathcal{N}N with minimal bond dimension ${\ensuremath{\chi}}^{\mathrm{min}}$ and a tensor network Nsym{\mathcal{N}}_{\mathrm{sym}}Nsym​ made of symmetric tensors, where the minimal bond dimension ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}$ might be larger than ${\ensuremath{\chi}}^{\mathrm{min}}$. We present two technical results. First, we show that in a tree tensor network, which is the most general tensor network without loops, the minimal bond dimension can always be achieved with symmetric tensors, so that ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}={\ensuremath{\chi}}^{\mathrm{min}}$. Second, we provide explicit examples of tensor networks with loops where replacing tensors with symmetric ones necessarily increases the bond dimension, so that ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}>{\ensuremath{\chi}}^{\mathrm{min}}$. We further argue, however, that in some situations there are important conceptual reasons to prefer a tensor network representation with symmetric tensors (and possibly larger bond dimension) over one with minimal bond dimension.
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.
Exploiting symmetries in tensor network algorithms plays a key role for reducing the computational and memory costs. Here we explain how to incorporate the Hermitian symmetry in double-layer tensor networks, which naturally arise in methods based on projected entangled-pair states. For real-valued tensors the Hermitian symmetry defines a Z2{\mathbb{Z}}_{2}Z2​ symmetry on the combined bra and ket auxiliary level of the tensors. By implementing this symmetry, a speedup of the computation time by up to a factor 4 can be achieved, while expectation values of observables and reduced density matrices remain Hermitian by construction. Benchmark results based on the corner transfer matrix renormalization group and higher-order tensor renormalization group are presented. We also discuss how to implement the Hermitian symmetry in the complex case, where a similar speedup can be achieved.