Grassmann Tensor Networks

• Grassmann tensor networks are a class of states that represent many-body quantum wavefunctions using local tensors dressed with Grassmann variables.
• The networks preserve fermionic anticommutation exactly, enabling unified treatment of fermions and bosons without auxiliary sign adjustments.
• Scalable renormalization methods like Grassmann TERG enable polynomial-time contractions and efficient simulations of complex quantum systems.

Grassmann tensor networks are a class of tensor network states and associated algorithms that systematically encode strongly correlated quantum states, including both fermionic and bosonic systems, by employing Grassmann algebra to represent and contract many-body wavefunctions. In this approach, the tensors that form the network are “dressed” with Grassmann variables, allowing the exact preservation of fermionic anticommutation relations and providing a unifying formalism for both particle statistics. Key elements such as the projective (slave-particle) construction, fermionic projected entangled pair states (fPEPS), and traditional tensor product states (TPS) are incorporated within the Grassmann tensor network (“GTN”) framework, leading to scalable and efficient descriptions of complex strongly correlated phases (Gu et al., 2010).

1. Construction of Grassmann Tensor Network States

A Grassmann tensor network is constructed by translating a many-body wavefunction into a network of local tensors whose components are elements of the Grassmann algebra. For a system of spinless fermions, the wavefunction

$$|\Psi_f\rangle = \exp\left(\sum_{\langle ij\rangle}u_{ij}c_i^\dagger c_j^\dagger\right)|0\rangle$$

is encoded as a tensor trace over Grassmann-valued site tensors $T_i$ and bond tensors $G_{ij}$: $$\Psi(\{m_i\}) = \int \prod_{i}T^{m_i}_{i; a_K a_L \ldots} \prod_{\langle ij\rangle} G_{ij; a_I a_J}$$ The integration is over all Grassmann link variables. The tensors $T$ and $G$ are formed by algebraically mimicking the structure of the original fermionic operators (creation/annihilation) with Grassmann derivatives and multiplications, so the anticommutation relations are captured exactly. For example, on site $i$, in a free fermion “bond” form,

$$T^m_i = \begin{cases} 1 & m=0 \ \sum_{I\in i} d\theta_I & m=1 \end{cases}$$

with $G_{ij} = 1 + u_{ij} \theta_J \theta_I$ so that each link (bond) carries its own local Grassmann variables. Multi-species Grassmann variables can be attached to each link for more complex systems. This formalism represents arbitrary projective constructions and fPEPS states as GTNs, with a number of parameters that scales polynomially in system size.

2. Renormalization and the Grassmann TERG

Contracting a Grassmann tensor network exactly is exponentially hard, so practical computations employ the Grassmann generalization of the tensor entanglement renormalization group (TERG) algorithm. The coarse-graining process iteratively combines and splits network tensors, exactly keeping track of all Grassmann sign structures and anticommutation factors.

• Basic move: Merge two site tensors $T_i$ and $T_j$ with a bond tensor $G_{ij}$ to form a new rank-4 tensor:

$$T_{p_1p_2p_3p_4} = \sum_{p_5p_6} \int_{ij} T_{i;p_5p_1p_2} T_{j;p_6p_3p_4} G_{ij;p_5p_6}$$

Integration here explicitly follows the Grassmann algebra, preserving the correct sign ordering.

• Splitting and truncation: The resulting higher-rank tensor is split (via Grassmann SVD or its generalization) into rank-3 tensors plus a new bond tensor, analogous to the traditional SVD-truncation in bosonic TERG, but constructed so as to respect the Grassmann parity and contracted variable order.
• Successive RG steps: Repeat coarse-graining until the network contracts to a single effective tensor, whose trace gives the norm or expectation value. At each step the truncation dimension $D_{\text{cut}}$ controls the computational cost/accuracy, and all steps proceed in polynomial time as long as $D_{\text{cut}}$ is polynomial in system size.

3. Norms, Observables, and Local Operators

The expectation value of a local operator, or the norm, is expressed as a tensor network integral: $$\langle\Psi|\Psi\rangle = \int \prod_i \mathcal{T}_i \prod_{\langle ij\rangle}\mathcal{G}_{ij}$$ with

$$\mathcal{T}_i = \sum_m \bar{T}^m_{i; \bar{a}_K \bar{a}_L \ldots} T^m_{i; a_K a_L \ldots}, \qquad \mathcal{G}_{ij} = \bar{G}_{ij; \bar{a}_I \bar{a}_J} G_{ij; a_I a_J}$$

where the bar denotes complex conjugation and proper reordering of integration measures. Local operators are included (“as impurities”) by modifying the site tensors at the affected locations. The locality of the tensors—each carrying only polynomially many Grassmann variables and indices—guarantees that operator insertions can be handled with only a local change in the network structure.

4. Advantages and Scalability

Grassmann tensor networks have several critical advantages in describing strongly correlated quantum systems:

• Efficient encoding (polynomial complexity): Strongly correlated fermionic and bosonic states (including those arising from projective/“slave particle” constructions and fPEPS) are represented with a polynomial number of parameters and local tensors, sidestepping the exponential growth of conventional wavefunction representations.
• Exact fermionic statistics: All sign structures arising from fermion exchange or parity are exactly realized by the algebraic properties of the Grassmann tensors. In contrast to “bosonic” tensor networks, there is no need for Jordan–Wigner transformations or auxiliary sign-keeping gadgets.
• Unified formalism for bosons/fermions: By introducing a “parity sign” function $s(m_i)$ (assigning $+1$ for bosons and $-1$ for fermions) and using Grassmann-valued site/link tensors, the approach naturally treats both quantum statistics within a single mathematical structure.
• Scalable RG-based contraction: The Grassmann TERG method enables the approximate contraction of the network in polynomial time, even in two dimensions, making practical variational simulations feasible for large lattices and complex models.

5. Mathematical Structure and Generalization

A Grassmann tensor product state (GTPS) is generally defined as

$$\Psi(\{m_i\}) = \sum_{\{a_I\}} \int \prod_i T_{i; a_K a_L \ldots}^{m_i} \prod_{\langle ij\rangle} G_{ij; a_I a_J}$$

with the norm

$$\langle\Psi|\Psi\rangle = \sum_{\{p_I\}} \int \prod_i \mathcal{T}_{i; p_K p_L \ldots} \prod_{\langle ij\rangle} \mathcal{G}_{ij; p_I p_J}$$

which can be renormalized and contracted using the Grassmann TERG moves and truncations previously described.

In this framework, projective states, fPEPS, and more general tensor network constructions are all included as special cases. The only requirement is that the physical wavefunction coefficients can be encoded within site and link tensors using a number of parameters and Grassmann variables that scale polynomially with system size.

6. Implications and Applications

The Grassmann tensor network formalism provides a variational class well suited for describing two-dimensional quantum spin liquids, fractional quantum Hall states, and other topologically ordered or strongly correlated systems with emergent fermion/boson statistics. It offers a route to scalable numerical simulations that preserve correct quantum statistics, enable systematic RG-based truncation, and allow the variational optimization of wavefunctions in higher dimensions. By generalizing tensor network methods to the Grassmann algebra, the approach unifies several previously distinct methods (projective constructions, fPEPS, fermionic TPS) and opens new directions for the efficient paper of quantum matter with nontrivial entanglement and exchange properties.

In summary, Grassmann tensor networks encode many-body quantum states as local networks of tensors built from Grassmann variables, with exact treatment of fermionic anticommutation relations and scalable contraction/renormalization strategies. This framework underpins practical, efficient, and statistically faithful simulations of complex quantum systems, both bosonic and fermionic, and stands as a central tool in modern strongly correlated quantum matter research (Gu et al., 2010).