The purpose of this section is to introduce a generic way to implement both on-site symmetries[104, 171, 123, 124, 8, 149, 125] and lattice space group symmetries[104] on PEPS. We firstly discuss the finite size symmetric quantum state that can be represented by a single PEPS; i.e., such a state would form a one-dimensional repre- sentation of the symmetry group. Then we define the symmetric PEPS on an infinite lattice, which is the main object to be (partially) classified in the current study.
On-site unitary symmetries
The action of a global on-site unitary symmetry 𝑆 on a finite size PEPS wavefunction is defined as 𝑆|𝜓⟩ = | ̃︀𝜓⟩ =∑︁ {𝑘s} tTr(︀(𝑇1)𝑘1. . . (𝑇𝑁s)𝑘𝑁s𝐵 1. . . 𝐵𝑁b )︀ 𝑈𝑆⊗ 𝑈𝑆. . . |𝑘1𝑘2. . . 𝑘𝑁s⟩, (3.5)
𝑈𝑆 is the representation of 𝑆 on Hilbert space of physical leg. These local actions of an on-site symmetry give a new TN, with site tensors ̃︀𝑇s and bond tensors ̃︀𝐵
bdefined as, ̃︀ 𝑇s = 𝑆 ∘ 𝑇s =∑︁ 𝑙 (𝑈𝑆)𝑘𝑙(𝑇s)𝑙 ̃︀ 𝐵b = 𝑆 ∘ 𝐵b= 𝐵b (3.6)
We focus on those PEPS that are invariant under the global symmetry up to an overall U(1) phase factor. Following the discussion in the previous section, we consider the PEPS |𝜓⟩ that differs from the transformed PEPS | ̃︀𝜓⟩ only by gauge transformations together with overall phase factors, as shown in Fig.(3-2):
𝑇s = Θ𝑆𝑊𝑆𝑆 ∘ 𝑇s
𝐵b = 𝑊𝑆𝑆 ∘ 𝐵b (3.7)
Here, gauge transformation 𝑊𝑆 and phase factor Θ𝑆 associated with symmetry 𝑆 is defined as Θ𝑆∘ 𝑇s = ei𝜃𝑆(s)(𝑇s)𝑘𝛼𝛽𝛾𝛿 𝑊𝑆∘ 𝑇s = [𝑊𝑆(s, 1)]𝛼𝛼′[𝑊𝑆(s, 2)]𝛽𝛽′. . . (𝑇s)𝑘 𝛼′𝛽′... 𝑊𝑆 ∘ 𝐵b = [𝑊𝑆(b, 1)]𝛼𝛼′[𝑊𝑆(b, 2)]𝛽𝛽′(𝐵b)𝛼′𝛽′. (3.8)
According to the definition of a gauge transformation, if site virtual leg (s, 𝑖) and bond leg (b, 𝑗) are connected, then 𝑊𝑆(s, 𝑖) = [𝑊𝑆(b, 𝑗)−1]t. Further, we always choose 𝑊𝑆 such that only site tensors transform with extra U(1) phase factors. Note that so far we do not require matrices on the leg (s, 𝑖) 𝑊𝑆(s, 𝑖) to form a representation of the on-site symmetry group when 𝑆 is tuned. We will come back to this shortly.
Time reversal symmetry
The representation of the global time reversal symmetry 𝒯 on a many-body wave- function is 𝑈𝒯 ⊗ 𝑈𝒯 . . . 𝐾, where 𝐾 denotes the complex conjugation and 𝑈𝒯 is a unitary matrix acting on local physical Hilbert space. Its action on PEPS is defined as 𝒯 |𝜓⟩ =∑︁ {𝑘s} tTr(︀(𝑇1)𝑘1. . . (𝑇𝑁s)𝑘𝑁s𝐵 1. . . 𝐵𝑁b )︀* 𝑈𝒯 ⊗ 𝑈𝒯 . . . |𝑘1𝑘2. . . 𝑘𝑁s⟩, (3.9)
Namely, the local actions on a single site or a bond tensor read
̃︀ 𝑇s = 𝒯 ∘ 𝑇s =∑︁ 𝑙 (𝑈𝒯)𝑘𝑙(𝑇s)*𝑙 ̃︀ 𝐵b = 𝒯 ∘ 𝐵b = 𝐵b* (3.10)
We consider the PEPS that is symmetric under 𝒯 . Similar to the previous discussion, we consider a PEPS satisfying:
𝑇s = Θ𝒯𝑊𝒯𝒯 ∘ 𝑇s
𝐵b = 𝑊𝒯𝒯 ∘ 𝐵b
(3.11)
where 𝑊𝒯 belongs to the gauge transformation group of the PEPS.
Lattice symmetry
The definition of a lattice space group symmetry 𝑅 on PEPS is
̃︀ 𝑇s= 𝑅 ∘ (𝑇s)𝑘≡∑︁ 𝛼𝛽... (𝑇𝑅−1(s))𝑘𝑅−1(𝛼𝛽... ) ̃︀ 𝐵b = 𝑅 ∘ 𝐵b≡ ∑︁ 𝛼𝛽 (𝐵𝑅−1(b))𝑅−1(𝛼𝛽) (3.12)
The action of 𝑅 on site and bond tensor follows the natural definition of lattice symmetries. For instance, for a square lattice, after a translation along the right direction by one lattice spacing, the transformed site tensor at a given position equals the original site tensor on the left neighboring site. Note that the symmetry 𝑅 not only acts on site and bond indices; it may also act nontrivially on virtual legs. For example, the 90∘rotation of a site tensor on the square lattice permute the four virtual legs. Again, we consider those PEPS symmetric under 𝑅 satisfying the following conditions:
𝑇s = Θ𝑅𝑊𝑅𝑅 ∘ 𝑇s
𝐵b = 𝑊𝑅𝑅 ∘ 𝐵b
(3.13)
where 𝑊𝑅 belongs to the gauge transformation group of the PEPS.
Symmetric PEPS on infinite lattices
Space groups of lattices are usually defined for infinite lattices. This is because for a finite size sample, the lattice symmetry group is a finite group whose group structure is non-generic. In this chapter, we will focus on PEPS on infinite lattices satisfying Eq.(3.7,3.11,3.13) under symmetry transformations. And we define such PEPS as symmetric PEPS on infinite lattices, or simply as symmetric PEPS. They form the main object to be (partially) classified in the current investigation.
A natural question that arises at this point is: are symmetric PEPS defined above general enough to capture ground states of quantum phases? Let us limit our discus- sion within those quantum phases whose entanglement entropies do not violate the boundary law so that in principle they may be represented as PEPS.
Basically, we expect that the symmetric PEPS on infinite lattices defined above are capable to capture all non-symmetry-breaking liquid phases. After putting on fi- nite lattices and performing a scaling with respect to both the bond dimension 𝐷 and lattice sizes, we expect the symmetric PEPS are also capable to capture the neighbor-
ing ordered phases of the liquid phases. Here by “neighboring” (or “in the vicinity below), we mean that the symmetry breaking in these phases is only sharply defined in the thermodynamic limit (namely, in the long-range physics). Note that we do not have a proof supporting the statement above. Nevertheless we are not aware of any counterexamples, so at least it is a reasonable conjecture.1.
Sometimes one is forced to use more than one PEPS to represent ground state quantum wavefunctions. For instance, in a quantum spin system with 𝑆𝑈 (2) spin rotation symmetry, this happens for the ferromagnetic phase, whose ground states form a large spin representation. However, such ferromagnetic phases are not in the vicinity of any non-symmetry-breaking liquid phases.