1. Phase Transitions in Quantum Triangular Ising antiferromagnets
Ying Jiang
Inst. Theor. Phys., Univ. Fribourg, Switzerland
Y.J. & Thorsten Emig, PRL 94, (2005)
Y.J. & Thorsten Emig, PRB 73, (2006)

2. Introduction Non-frustrated Ising system: LiHoF4 2006-6-17
[Ronnow et al, Science 308, 389 (2005); Bitko et al, PRL 77, 940 (1996)]
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3. ? Triangular Ising Antiferromagnets (TIAF)
Classical antiferromagnetic Ising system ?
Geometrical frustration
Highly degenerated ground states: exactly one frustrated bond per triangle
Macroscopic degeneracy
Continuous symmetry of the system
For triangular Ising antiferromagnets
Extensive entropy density
[Wannier, Hautappel (1950)]
T = 0
Spin correlation: algebraic decay
[Stephenson (1970)]
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4. ? Triangular Ising Antiferromagnets (TIAF) Quantum system
Transverse field: intends to flip spins
Zero exchange field flippable spins
T = 0
Quantum fluctuation order from disorder ?
Quantum critical point expected
G/J
T/J ?
QLRO
QCP
Order ?
disorder
T ≠ 0
Competition between thermal and
quantum fluctuations
Phase diagram ?
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5. Spin--string mapping in classical 2D TIAF
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6. From spin configuration to dimer covering
Properties of classical TIAF ground states
Hardcore dimer covering on dual lattice
Height profile
on sites of lattice
:dimer crossed
:no dimer crossed
single spin flip:
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7. From dimer covering to fluctuating lines+
Dimer covering
XOR
Reference pattern
Fluctuating lines
non-zero entropy density
fluctuation
reference covering
directed
geometrical frustration
non-crossing
frustrated Ising spin configuration fluctuating strings
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8. Free energy functional of strings
displacement field
Global offset of flat strings
average string distance
Lock-in potential
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9. The lock-in potential Equivalent flat states: shifts by a/2
2D self-avoiding non-crossing strings = 1D free fermions
stiffness
irrelevant
quantum fluctuations increase the string stiffness
relevant
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10. Spin—spin correlations
stiffness
Vortex pair
system unstable with defects
T=0 no defect quasi-long range ordered phase
T≠0 unbound defects disordered phase
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11. Phase diagram of quantum TIAF
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12. From 2D quantum system to classical 3D system
mapping to 3D classical system (Suzuki-Trotter theorem)
correspondence becomes exact
size in imaginary time direction
T=0: real 3D system
T≠0: finite size 3D system
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13. Mapping to stacked string layers
Spin-string mapping
spin-height relation
3D XY model clock term
Topological defects
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14. Universality class of quantum phase transition
Decoupling of layers?
No!
[Korshunov, (1990)]
p-fold clock term is irrelevant at
transition point for 3D if
Hs = 3D XY Hamiltonian
+ 6-fold clock term
[Aharony, Birgeneau, Brock and Litster, (1986)]
QCP: 3d XY Universality
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15. } Quantum critical point
Decoupling of “spin waves” + topological defects (Villain mapping)
Villain coupling }
Dimensional crossover approach for layered XY models
[Ambegaokar, Halperin,Nelson and Siggia, 1980]
[Schneider and Schmidt, 1992]
 ~ 2/3 (3D XY)
Quantum phase transition point
Simulation: c/J ~ 1.65
Renormalization effects of clock term increases
[Isakov & Moessner, 2003]
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16. Phase boundaries Finite size scaling approach Phase boundaries at
[Ambegaokar, Halperin, Nelson & Siggia (1980); Schneider and Schmidt, 1992]
Phase boundaries at
Relevance of the 6-clock term
[José, Kadanoff, Kirkpatrick and Nelson (1977)]
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17. Phase diagram of quantum TIAF
Log-rough strings with bound defects
Strings locked-in by clock term
[Monte Carlo Simulations, Isakov & Moessner, 2003]
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18. Summary Transverse field TIAF system stacked 2D string lattice
Strongly anisotropic 3D XY model with 6-clock term
obtained in a microscopic way
Quantum critical point D XY universality
Reentrance of the phase diagram due to the frustration and the
competition between the thermal and quantum fluctuations
Phase diagram in excellent agreement with the recent simulations
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