$\mathbb{Z}_N$ symmetry breaking in Projected Entangled Pair State models

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Let τ N v = l ​ i ​ m N h ⁡ N h ​ σ N h , N v 2 \tau_{N_{v}}=\mathop{lim}\displaylimits_{N_{h}}N_{h}\sigma^{2}_{N_{h},N_{v}} , and let S := l ​ i ​ m N v ⁡ τ N v S:=\mathop{lim}\displaylimits_{N_{v}}\tau_{N_{v}} . Then, for any ϵ > 0 \epsilon>0 , ∃ N v 0 ​ ∀ N v ≥ N v 0 : | ​ 1 N v ​ τ N v − S ​ | < ϵ 2 ​ and \displaystyle\exists N_{v}^{0}\ \forall N_{v}\geq N_{v}^{0}:\ {\hbox{$\left|\vbox to1.5pt{}\right.$}}\genfrac{}{}{}{1}{1}{N_v}\tau_{N_{v}}-S{\hbox{$\left|\vbox to1.5pt{}\right.$}}<\frac{\epsilon}{2}\hbox{\quad and} ∃ N h 0 ​ ( N v ) ​ ∀ N h ≥ N h 0 ​ ( N v ) : 1 N v ​ | ​ N h ​ σ N h , N v 2 − τ N v ​ | < ϵ 2 . \displaystyle\exists N_{h}^{0}(N_{v})\ \forall N_{h}\geq N_{h}^{0}(N_{v}):\frac{1}{N_{v}}{\hbox{$\left|\vbox to1.5pt{}\right.$}}N_{h}\sigma_{N_{h},N_{v}}^{2}-\tau_{N_{v}}{\hbox{$\left|\vbox to1.5pt{}\right.$}}<\frac{\epsilon}{2}\ .\hskip-20.00003pt Thus, | N h N v ​ σ N h , N v 2 − S | ≤ ϵ , \left|\frac{N_{h}}{N_{v}}\sigma_{N_{h},N_{v}}^{2}-S\right|\leq\epsilon\ , and finally σ 2 = l ​ i ​ m N h , N v → ∞ ⁡ σ N h , N v 2 ≤ N h N v ​ σ N h , N v 2 = S , \sigma^{2}=\mathop{lim}\displaylimits_{N_{h},N_{v}\to\infty}\sigma_{N_{h},N_{v}}^{2}\leq\frac{N_{h}}{N_{v}}\sigma_{N_{h},N_{v}}^{2}=S\ , as long as we couple the limits such that both N h ≥ N h 0 ​ ( N v ) N_{h}\geq N_{h}^{0}(N_{v}) and N h ≥ N v N_{h}\geq N_{v} . (If S = ∞ S=\infty , the inequality ( 13

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When left and right fixed points are described by different MPS M L M_{L} and M R M_{R} one should consider the mixed transfer matrix 𝔼 O = ∑ i M L i ⊗ B O ⊗ \mathaccentV ​ b ​ a ​ r ​ 016 ​ M R i \mathbb{E}_{O}=\sum\displaylimits_{i}M_{L}^{i}\otimes B_{O}\otimes\mathaccentV{bar}016{M_{R}^{i}}

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Strictly speaking, due to finite size effects, these are not all fixed points

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