1. Entropy-driven first-order phase transition in quantum compass model with Ly>3 Tian Liang

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3. DUALITY TRANSFORMATION Compass model Plaquette model

4. The plaquette model quantum fluctuation h = 0: classical model of Ising spins one-dimensional nearest-neighbor Ising model Symmetry: system energy remains the same under spin flip for each row and each column Ground state degeneracy: Given arbitrary values of the spins on one row and one column, there is a unique ground of the system compatible with these values. h > K xy : quantum fluctuations leads to proliferation of defects and loss of long-ranged order.

5. QUANTUM-CLASSICAL MAPPING Path integral representation Self-duality

11. Numerical Results Two chain problem: Ly=2

12. Finite size scaling 2D Ising model γ= 1.75 β =0.125 ν = 1

13. Numerical Results Two chain problem: Ly=3

14. Finite size scaling Four-state Potts model α = 0.69 γ = 1.17 β =0.085 ν = 0.66

15. the nature of the disorder transition B. Doucot et al., PRB 71, 024505 (2005) S. Wenzel and W. Janke, cond-mat/0804.2972v1

16. Numerical Results Two chain problem: Ly=4

17. Numerical Results Two chain problem: Ly=Lx (Quantum 2D system )

18. First-order phase transition for Ly>3

19. q-state Potts model and coloring entropy continuous mosaic Coloring entropy (q>4) Line tension, vanishes at t = 0

20. q-state Potts & XY model and coloring entropy vs. Four color theorem The current work

21. The end