Table 1 Summary for all 7 PGs with non-vanishing PET.

From: Piezoelectricity and topological quantum phase transitions in two-dimensional spin-orbit coupled crystals with time-reversal symmetry

PGs

p1

p1m1, c1m1, p1g1

p3

p3m1, p31m

Scenario

(i)

(ii)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(i)

(ii)

(iii)

(iv)

Codim-1 GC

 Ã— 

(a)

 Ã— 

 Ã— 

(b and c)

(d and e)

(f)

(g)

(h)

(i and j)

 Ã— 

(k–m)

(n and o)

Topo. Inv.

N/A

Z2

N/A

N/A

Z2

VCN

Z2

Z2

Z2

Z2

N/A

Z2

VCN

PET Jump

N/A

✓

N/A

N/A

✓

✓

✓

✓

✓

✓

N/A

✓

✓

  1. The scenarios are classified by the symmetries that leave the GC momenta invariant, as shown in the Methods. Codim-1 GC means the GC cases with 1 fine-tuning parameter or codimension 1. If at least one GC case between gapped states with 1 fine-tuning parameter exists in the corresponding scenario, the subfigures in Fig. 1 that illustrate the GC momenta are referred to; otherwise, we fill in a  Ã— . Topo. Inv. labels the topological invariant changed by the codimensoin-1 GC, Z2 means the Z2 index, and VCN means the corresponding case changes the valley CN when the valley is well-defined.
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