Post on 08-Jan-2016
description
Phase Transitions in Quantum Triangular Ising antiferromagnets
Ying Jiang
Inst. Theor. Phys., Univ. Fribourg, Switzerland
Y.J. & Thorsten Emig, PRL 94, 110604 (2005)Y.J. & Thorsten Emig, PRB 73, 104452 (2006)
2006-6-17 @ Les Houches -2-
Introduction
Non-frustrated Ising system: LiHoF4
[Ronnow et al, Science 308, 389 (2005); Bitko et al, PRL 77, 940 (1996)]
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Triangular Ising Antiferromagnets (TIAF)
Geometrical frustration
Highly degenerated ground states: exactly one frustrated bond per triangle
T = 0Extensive entropy density
Spin correlation: algebraic decay
[Wannier, Hautappel (1950)]
?
Classical antiferromagnetic Ising system
[Stephenson (1970)]
Macroscopic degeneracy Continuous symmetry of the system
For triangular Ising antiferromagnets
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Triangular Ising Antiferromagnets (TIAF)
Quantum system
Quantum fluctuation order from disorder ?
/J
T/J
QLRO QCPOrder ?
dis
ord
erPhase diagram ?
Transverse field: intends to flip spins
Zero exchange field flippable spins
T = 0
Quantum critical point expected
T ≠ 0
Competition between thermal and quantum fluctuations
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Spin--string mapping in classical 2D TIAF
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From spin configuration to dimer covering
Hardcore dimer covering on dual lattice
Properties of classical TIAF ground states
Height profile
on sites of lattice
:dimer crossed
:no dimer crossed
single spin flip:
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From dimer covering to fluctuating lines
+
Dimer covering Reference pattern Fluctuating lines
++++
+++++
+++-
-- -
- -
-
--
--
---
---
+ ++++
++
++
+- -
--
---
--
-
+++
+++ +
+++ -
-
-
---
---
- -+ ++
++++
+-
---
--
---
-
+ +++
+++
++-
- --
--
--- -+ ++ ++ --- -
non-zero entropy density fluctuation
reference covering directed
geometrical frustration non-crossing
frustrated Ising spin configuration fluctuating strings
XOR
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Free energy functional of strings
Lock-in potential
average string distance
displacement field
Global offset of flat strings
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The lock-in potential
Lock-in potential
irrelevant
quantum fluctuations increase the string stiffness
relevant
Equivalent flat states: shifts by a/2
2D self-avoiding non-crossing strings = 1D free fermions
stiffness
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Spin—spin correlations
++++
+++++
+++-
-- -
- -
-
--
--
---
---
+ ++++
++
++
+- -
--
---
--
-
+++
+++ +
+++ -
-
-
---
---
- -+ ++
++++
+-
---
--
---
-
+ +++
+++
++-
- --
--
--- -+ ++ ++ --- -
Spin-spin correlation
Vortex pair
stiffness
system unstable with defects
T=0 no defect quasi-long range ordered phaseT≠0 unbound defects disordered phase
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Phase diagram of quantum TIAF
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From 2D quantum system to classical 3D system
mapping to 3D classical system (Suzuki-Trotter theorem)
2D Quantum system
correspondence becomes exact
size in imaginary time direction
T=0: real 3D system T≠0: finite size 3D system
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Mapping to stacked string layers
spin-height relation
Spin-string mapping
3D XY model + 6-clock term
Topological defects
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Universality class of quantum phase transition
Decoupling of layers?
No!
QCP: 3d XY Universality
[Korshunov, (1990)]
p-fold clock term is irrelevant attransition point for 3D if
[Aharony, Birgeneau, Brock and Litster, (1986)]
Hs = 3D XY Hamiltonian + 6-fold clock term
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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 ~ [Isakov & Moessner, 2003]Renormalization effects of clock term increases
}
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Phase boundaries
Finite size scaling approach[Ambegaokar, Halperin, Nelson & Siggia (1980); Schneider and Schmidt, 1992]
Relevance of the 6-clock term
[José, Kadanoff, Kirkpatrick and Nelson (1977)]
Phase boundaries at
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(C)
(PM)
(O)
Log-rough strings with bound defects
Strings locked-in by clock term
Phase diagram of quantum TIAF
[Monte Carlo Simulations, Isakov & Moessner, 2003]
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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 3D XY universality
Reentrance of the phase diagram due to the frustration and thecompetition between the thermal and quantum fluctuations
Phase diagram in excellent agreement with the recent simulations