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)

Introduction Non-frustrated Ising system: LiHoF4 2006-6-17 [Ronnow et al, Science 308, 389 (2005); Bitko et al, PRL 77, 940 (1996)] 2006-6-17 @ Les Houches

? 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)] 2006-6-17 @ Les Houches

? 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 ? 2006-6-17 @ Les Houches

Spin--string mapping in classical 2D TIAF 2006-6-17 @ Les Houches

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: 2006-6-17 @ Les Houches

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 2006-6-17 @ Les Houches

Free energy functional of strings displacement field Global offset of flat strings average string distance Lock-in potential 2006-6-17 @ Les Houches

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 2006-6-17 @ Les Houches

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 2006-6-17 @ Les Houches

Phase diagram of quantum TIAF 2006-6-17 @ Les Houches

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 2006-6-17 @ Les Houches

Mapping to stacked string layers Spin-string mapping spin-height relation 3D XY model + 6-clock term Topological defects 2006-6-17 @ Les Houches

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 2006-6-17 @ Les Houches

} 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] 2006-6-17 @ Les Houches

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)] 2006-6-17 @ Les Houches

Phase diagram of quantum TIAF Log-rough strings with bound defects Strings locked-in by clock term [Monte Carlo Simulations, Isakov & Moessner, 2003] 2006-6-17 @ Les Houches

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 the competition between the thermal and quantum fluctuations Phase diagram in excellent agreement with the recent simulations 2006-6-17 @ Les Houches