Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry
Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry
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DOI：
10.1103/physrevb.94.205124
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发表时间：
2016-08
影响因子：
3.7
通讯作者：
M. Mambrini;R. Orús;D. Poilblanc
中科院分区：
文献类型：
--
作者：
M. Mambrini;R. Orús;D. Poilblanc
We elaborate a simple classification scheme of all rank-5 SU(2) spin rotational symmetric tensors according to (i) the onsite physical spin $S$, (ii) the local Hilbert space ${V}^{\ensuremath{\bigotimes}4}$ of the four virtual (composite) spins attached to each site, and (iii) the irreducible representations of the ${C}_{4v}$ point group of the square lattice. We apply our scheme to draw a complete list of all SU(2)-symmetric translationally and rotationally invariant projected entangled pair states (PEPS) with bond dimension $D\ensuremath{\le}6$. All known SU(2)-symmetric PEPS on the square lattice are recovered and simple generalizations are provided in some cases. More generally, to each of our symmetry class can be associated a $(\mathcal{D}\ensuremath{-}1)\text{-dimensional}$ manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of $\mathcal{D}\text{-independent}$ tensors of a given bond dimension $D$. In addition, generic (low-dimensional) families of PEPS explicitly breaking either (i) particular point-group lattice symmetries (lattice nematics) or (ii) time-reversal symmetry (chiral spin liquids) or (iii) SU(2) spin rotation symmetry down to $\text{U}(1)$ (spin nematics or N\'eel antiferromagnets) can also be constructed. We apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum. In particular, we show how the symmetrization of a double-layer PEPS leads to a chiral topological state with a gapless edge described by a $\mathrm{SU}{(2)}_{2}$ Wess-Zumino-Witten model.
We elaborate a simple classification scheme of all rank-5 SU(2) spin rotational symmetric tensors according to (i) the onsite physical spin $S$, (ii) the local Hilbert space ${V}^{\ensuremath{\bigotimes}4}$ of the four virtual (composite) spins attached to each site, and (iii) the irreducible representations of the ${C}_{4v}$ point group of the square lattice. We apply our scheme to draw a complete list of all SU(2)-symmetric translationally and rotationally invariant projected entangled pair states (PEPS) with bond dimension $D\ensuremath{\le}6$. All known SU(2)-symmetric PEPS on the square lattice are recovered and simple generalizations are provided in some cases. More generally, to each of our symmetry class can be associated a $(\mathcal{D}\ensuremath{-}1)\text{-dimensional}$ manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of $\mathcal{D}\text{-independent}$ tensors of a given bond dimension $D$. In addition, generic (low-dimensional) families of PEPS explicitly breaking either (i) particular point-group lattice symmetries (lattice nematics) or (ii) time-reversal symmetry (chiral spin liquids) or (iii) SU(2) spin rotation symmetry down to $\text{U}(1)$ (spin nematics or N\'eel antiferromagnets) can also be constructed. We apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum. In particular, we show how the symmetrization of a double-layer PEPS leads to a chiral topological state with a gapless edge described by a $\mathrm{SU}{(2)}_{2}$ Wess-Zumino-Witten model.