Entropy-driven first-order phase transition in quantum compass

model with Ly>3

Tian Liang

Orbital compass model with directional coupling

K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP 37, 725 (1973).

Y. Tokuraand N. Nagaosa, Science 288, 462 (2000)D. I. Khomskiiand M. V. Mostovoy, J. Phys. A 36, 9197 (2003)J. van den Brink, New J. Phys. 6, 201 (2004)J.B. Kogut, RMP 51, 659 (1979)Z. Nussinovand E. Fradkin, PRB 71, 195120 (2005)J. E. Moore and D.-H. Lee, PRB 69, 104511 (2004)A. Yu Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003)L.B. Ioffeet al., Nature 415, 503 (2002)B. Douçotet al., PRB 71, 024505 (2005)A. Micheli, G.K. Brennenand P. Zoller, Nature Physics 2, 341 (2006)C.K. Xuand M. P. A. Fisher, PRB 75, 104428 (2007)

`

x x y y z zi i x i i y i i z

i

H J s s s s s s

DUALITY TRANSFORMATION

Compass model Plaquette model

ˆ ˆx x y y

x i i x y i i yi i

H J J ˆ ˆ ˆ ˆz z z z x

xy i i x i x y i y ii i

H K h

ˆ ˆ, x x x y z zi i i x i i i x

, x y xyJ h J K

The plaquette modelˆ ˆ ˆ ˆ

z z z z xxy i i x i x y i y i

i i

H K h

12 x yL L

xL

yL

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 > Kxy: quantum fluctuations leads to proliferation of defects and loss of long-ranged order.

QUANTUM-CLASSICAL MAPPING

Tr( )HZ e

,

,

c i k

i k

H S

S

e

ˆ ˆ ˆ ˆ, , , , , , 1, ,

i k i x k i x y k i y k z i k i ki k i k

K S S S S J S S

Path integral representation

Self-duality

Numerical ResultsTwo chain problem: Ly=2

Finite size scaling

2D Ising model

γ= 1.75

β =0.125 ν = 1

Numerical ResultsTwo chain problem: Ly=3

Finite size scaling

Four-state Potts model

α = 0.69γ = 1.17β =0.085 ν = 0.66

the nature of the disorder transition

2, : Ising

3, : 4 state Potts

3, : ?

...

Full 2+1 dimensional: 1st order

y x

y x

y x

y x

L L

L L q

L L

L L

B. Doucot et al., PRB 71, 024505 (2005)

S. Wenzel and W. Janke, cond-mat/0804.2972v1

ˆ ˆ ˆ ˆ, , , , , , , 1, ,

c i k i k i x k i x y k i y k z i k i ki k i k

H S K S S S S J S S

Numerical ResultsTwo chain problem: Ly=4

Numerical ResultsTwo chain problem: Ly=Lx (Quantum 2D system )

First-order phase transitionfor Ly>3

q-state Potts model and coloring entropy

1 2 3 4E E E E continuous

mosaic

mosaic continuous

2( )

F F F

Nl t Ts q

l

Coloring entropy (q>4)

Line tension, vanishes at t = 0

q-state Potts & XY model and coloring entropyvs.

Four color theorem

The current work

The end