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arXiv:1004.2563

Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states

Zheng‐Cheng Gu, Frank Verstraete, Xiao-Gang Wen

Published Apr 15, 2010• 36 citations
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Abstract

The projective construction (the slave-particle approach) has played an very important role in understanding strongly correlated systems, such as the emergence of fermions, anyons, and gauge theory in quantum spin liquids and quantum Hall states. Recently, fermionic Projected Entangled Pair States (fPEPS) have been introduced to effciently represent many-body fermionic states. In this paper, we show that the strongly correlated bosonic/fermionic states obtained both from the projective construction and the fPEPS approach can be represented systematically as Grassmann tensor product states. This construction can also be applied to all other tensor network states approaches. The Grassmann tensor product states allow us to encode many-body bosonic/fermionic states effciently with a polynomial number of parameters. We also generalize the tensor-entanglement renormalization group (TERG) method for complex tensor networks to Grassmann tensor networks. This allows us to approximate the norm and average local operators of Grassmann tensor product states in polynomial time, and hence leads to a variational approach for describing strongly correlated bosonic/fermionic systems in higher dimensions.

DOI
10.48550/arxiv.1004.2563

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The projective construction (the slave-particle approach) has played an very important role in understanding strongly correlated systems, such as the emergence of fermions, anyons, and gauge theory in quantum spin liquids and quantum Hall states. Recently, fermionic Projected Entangled Pair States (fPEPS) have been introduced to effciently represent many-body fermionic states. In this paper, we show that the strongly correlated bosonic/fermionic states obtained both from the projective construction and the fPEPS approach can be represented systematically as Grassmann tensor product states. This construction can also be applied to all other tensor network states approaches. The Grassmann tensor product states allow us to encode many-body bosonic/fermionic states effciently with a polynomial number of parameters. We also generalize the tensor-entanglement renormalization group (TERG) method for complex tensor networks to Grassmann tensor networks. This allows us to approximate the norm and average local operators of Grassmann tensor product states in polynomial time, and hence leads to a variational approach for describing strongly correlated bosonic/fermionic systems in higher dimensions.
Tensor network states, and in particular projected entangled pair states, play an important role in the description of strongly correlated quantum lattice systems. They do not only serve as variational states in numerical simulation methods, but also provide a framework for classifying phases of quantum matter and capture notions of topological order in a stringent and rigorous language. The rapid development in this field for spin models and bosonic systems has not yet been mirrored by an analogous development for fermionic models. In this work, we introduce a tensor network formalism capable of capturing notions of topological order for quantum systems with fermionic components. At the heart of the formalism are axioms of fermionic matrix-product operator injectivity, stable under concatenation. Building upon that, we formulate a Grassmann number tensor network ansatz for the ground state of fermionic twisted quantum double models. A specific focus is put on the paradigmatic example of the fermionic toric code. This work shows that the program of describing topologically ordered systems using tensor networks carries over to fermionic models.
Recently, the Grassmann-tensor-entanglement renormalization group (GTERG) algorithm has been proposed as a generic variational approach to study strongly correlated boson/fermion systems [Gu et al., arXiv:1004.2563]. However, the weakness of such a simple variational approach is that generic Grassmann tensor product states (GTPS) with large inner dimension DDD will contain a large number of variational parameters which are hard to be determined through usual minimization procedures. In this paper, we first introduce a standard form of GTPS which significantly simplifies the representations. Then we describe a simple imaginary-time-evolution algorithm to efficiently update the GTPS based on the fermion coherent state representation and show that all the algorithms developed for usual tensor product states (TPS) can be implemented for GTPS in a similar way. Finally, we study the environment effect for the GTERG approach and propose a simple method to further improve its accuracy. We demonstrate our algorithms by studying some simple two-dimensional free and interacting fermion systems on honeycomb lattice, including both off-critical and critical cases.
The success of tensor network approaches in simulating strongly correlated quantum systems crucially depends on whether the many-body states that are relevant for the problem can be encoded in a local tensor network. Despite numerous efforts, strongly correlated projective states, including fractional quantum Hall states, have not yet found a local tensor network representation. Here we show that one can encode the calculation of averages of local operators in a Grassmann tensor network which is local. Our construction is explicit and allows the use of physically motivated trial wave functions as starting points in tensor network variational calculations.
We present a 2D bosonization duality using the language of tensor networks. Specifically, we construct a tensor network operator (TNO) that implements an exact 2D bosonization duality. The primary benefit of the TNO is that it allows for bosonization at the level of quantum states. Thus we use the TNO to provide an explicit algorithm for bosonizing fermionic projected entangled pair states (fPEPs). A key step in the algorithm is to account for a choice of spin structure, encoded in a set of bonds of the bosonized fPEPS. This enables our tensor network approach to bosonization to be applied to systems on arbitrary triangulations of orientable 2D manifolds.
Abstract The strongly correlated fermions play a vital role in modern physics. For a given fermionic Hamiltonian system, the most widely used approach to exploring the underlying physics is to study the wave function that incorporates Fermi–Dirac statistics, which can be obtained variationally by energy minimization or by imaginary-time evolution. In this work, we develop an accurate tensor network method for one-dimensional interacting fermionic models based on the coherent-state path-integral representation of the fermionic partition function. Employing the coherent-state representation, the partition function is effectively represented as a (1+1)-dimensional anisotropic Grassmann-valued tensor network, and the Grassmann version of the corner transfer-matrix renormalization group algorithm is developed to contract the tensor network and evaluate physical quantities. We validate our method on the one-dimensional fermionic Hubbard model with a magnetic field, where the essential features of the phase diagram in the ( μ , B ) plane are quantitatively captured. Our work offers a promising approach to interacting fermionic models within the framework of tensor networks.
By combining the Grassmann algebra with multi-scale entanglement renormalization ansatz (MERA), we introduce a new unbiased and effective numerical method for simulating 2D strongly correlated electronic systems. The new GMERA method inherits all the advantages of MERA, which constructs the variational wave function based on complicated tensor network. Besides it can deal with fermionic properties of the system due to Grassmann algebra through local tensor contractions. This general method can treat different tensor network structures in a universal way. We show several benchmark calculations of the GMERA method, including the free fermion model, tight binding model, as well as the t-J model with hole doping.