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by: Hailey Halvorson
Hailey Halvorson
GPA 3.8


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Class Notes
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This 28 page Class Notes was uploaded by Hailey Halvorson on Thursday October 22, 2015. The Class Notes belongs to PHYS 25 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/227147/phys-25-university-of-california-santa-barbara in Physics 2 at University of California Santa Barbara.

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Date Created: 10/22/15
Order by disorder and spiral spin liquid in frustrated diamond lattice antiferromagnets a Simon Trebst Microsoft Research Q University of California santa Barbara httpstationqucsb du Outline Motivation Frustrated magnetism degeneracies amp emergent phenomena Experimental observations Theory of frustrated diamond lattice AFMs Groundstates stability and orderbydisorder Numerical simulations The spiral spin liquid regime Comparison to experiments Frustrated magnetism amp degeneracies Frustration not all interactions satis able simultaneously Microscopic interactions can produce many nearly degenerate lowenergy states Weak residual effects can give rise to remarkable emergent phenomena 2D electron systems cuprates partial Landaulevel occupation several competing orders exotic quantum Hall liquids highTc superconductivity Experimental signatures of frustration longrange high temperature order Spm llquld paramagnet 0 Tc CW T frustration parameter spin liquid CurieWeiss law 90w 1 f system uctuates X N W TC amongst lowenergy CW cc n con gurations but hlghly frusuated no longrange order f gt 5 10 Key challenges Low temperature ordering mechanisms Characterizing spin liquid correlations Diamond lattice antiferromagnets Materials V Fritsch et al PRL 92 116401 2004 N Tristan et 51 PRB 72 174404 2005 T Suzuki et al 2007 Many materials take on the normal spinel structure AB2X4 Focus Spinels with magnetic Asites only I I s52 s2 CoRh204 C0304 MnS62S4 Fesczs4 orbital egeneracy I l I I 4 I l I I I I I 1 5 10 20 900 f cwTc MnA1204 C0A1204 S32 4amp4 Very limited theoretical understanding Frustration in the diamond lattice Naive Hamiltonian H lt11 antiferromagnetic classical spins 332 552 diamond lattice two FCC lattices coupled Via J1 bipartite lattice no frustration Frustration in the diamond lattice 2nd neighbor a j exchange H J1 2 Si S 7 12739 J1 3 J2 similar exchange path W L Roth J Phys 25 507 1964 J 2 generates strong frustration Stability of the N el ground state HJ1J2 j ltijgt ltltijgtgt 0 18 J2J1 J2 J1 lt 18 m Ne el state is favored Stability of the N el ground state HJ1 jJ2 j ltijgt ltltijgtgt r 0 18 JgJl 722 Hg 17 1 720 39 7 e 08 08 18 g g s 067 706 16 m E 14 8 04 704 on 7 I a 12 c 027 Neel 2 702 0 10 quotE r o y A 1 1 1 1 1 1 1 1 1 1 1 1 u 1 1 U 025 0 075 1 116 18 316 14 coupling J2 J1 5 temperature T Spiral ground states HJl 5353J2 E i U ltltijgtgt highly degenerate Ne el coplanar spirals O 18 J2 J1 coplanar spiral Direction amp pitch of spirals is characterized by a wavevector residing on a surface in momentum space JgJl 04 J2J1 085 LowT physics Longrange order At nite temperature stability nontrivial due to massive spiral degeneracy Expand in small uctuations around spiral ground states 6QS4 y r rquot x1 lt arbitrary groundstate order y Entropy stabilizes longrange order at nite temperature by lifting the degeneracy in the free energy along the spiral surface A N T2 3 thermally induced splitting CU N T13 unconventional thermodynamic behavior The system undergoes a thermal orderbydisorder transition Orderbydisorder selection At low temperatures which state does entropy select Free energy on spiral surface F Q E T S Q Entropy favors states with highest density of nearby lowenergy states Ordering temperature 05 05 N 512 NH O 4 Tc rapidly diminishes N 1728 O 4 o 39 in Neel phase H N 4096 39 s a g Order by disorder 3 with sharply reduced TC 3 02 02 no i 39 S 33 l E 01 39 01 o 39 39 39u 4 0 39 I 39 I I 39 I I I 39 I 39 I 39 0 0 01 02 3 04 05 06 07 08 09 1 spiral surface forms coupling 12 11 Firstorder transitions TJ1 C ordering temperature 05 05 04 l04 03 2 1 0 1 2 03 J2J1 085i farquot i 39 02 n a 02 ii 001 0 01 I2J1 02 01 Neel ifquot quot a quot coplanar spiral 0 I I I I 39 I I 39 I 39 I 39 I 39 0 0 01 02 03 04 05 06 07 08 09 1 coupling J2 J1 Parallel tempering K Hukushima and Y Nemoto J Phys Soc Jpn 65 1604 1996 Simulate multiple replicas of the system at various temperatures W M O O O O O O 0 T1 T2 TN swap temperature gt replicas 19E7L7 T7 gt Ei17 n1 mi 17 XpA AE Single replica performs random walk in temperature space How do we choose the temperature points J Ensemble optimization S Trebst DA Huse M Troyer PRE 70 046701 2004 HG Katzgraber S Trebst DA Huse M Troyer J STAT P03018 2006 102 10392 Feedback algorithm 103 K V quot103 Q gt904 104 Measure local diffusivity DT E 105 105 U of current 1n temperature space Q 6 76 quot510 JQJ1 02 10 Optimal choice of temperatures 10 10 003 004 005 006 007 008 009 01 3 1 temperature T I Ilbv f t N 39quot y M 0OOOOO0OOOOOOOOOOOOOOOOOOOOOOOOO den51ty of g nuunnmnn D D D D D D D D D D 0 QB XXXXXXXMIXXXXXXXXXXXIX X Iterate feedback of dlffuswlty longrange Spiral Spin liquid 4 order I I I I 004 005 006 007 008 009 01 temperature T Ensemble optimization frustrated magnets S Trebst et al Phys Rev E 2004 dense liquids S Trebst et al J Chem Phys 2005 small proteins S Trebst et al J Chem Phys 2006 Z The 3H Zgn n n20 quantum systems S Wessel et al JSTAT 2007 Spin liquid physics high temperature paramagnet longrange order spin liquid TC CW Orderbydisorder selects loW temperature coplanar spiral state Broad spin liquid regime emerges due to loW TC How can we characterize the spin liquid regime Is there an experimental probe to observe the spin liquid J Magnetic correlations in spin liquid numerical structure factor analytic free energy IQJ1 085 MnSC2S4 Spin structure factor directly images spiral surface Free energy corrections Visible for TC lt T lt 13TC longrange order splral sp1n llquld sp1n llquid 0 TC 13TC 3TC T Spin liquid correlations 15 JZJl 0 85 I 1 9 Monte Carlo data T 024 J1 MnSC2S4 33 4 g 0 5 l i39 T E 0 39 H w gwmt that hquot Iquot 1 VrV VA I I I I I I I I I I I I I I I I I I I no T 043 J a 1 fr I 1 Eigss 39 E surface nu peak s1gnature 1 025 structure factor for at quot uvdun o one FCC sublattlce 0 I I W 06 o T 070 J1 04 Wigwam I AA 1 37 02 M39h S Z 5 0 I I I I I I I u IquotII Ilrnnruu K 47r1 0 0 0 0395 1 1395 2 q 2IC0s2 if cos2 if cos2 qf sin2 if sin2 sin2 q4 Z surface function Spin liquid correlations 15 b units Spherical model Monte Carlo data r2024a a2 Sj 1 gt E Sj 339 number ofspins describes spin liquids in kagorn pyrochlore antiferrornagnets predicts data collapse for structure factor data i U lo Spin liquid correlations 15 J2J1 085 H 1 Monte Carlo data T 024 J1 MHSCZS4 analytical fit Q 5 O 5 45 39 I I I I I I I I I I I I I I I I I I T 043 J1 quantitative agreement structure factor for one FCC sublattice Nontrivial eperimental test 395 I neutron scattering on single crystals surface function Multistage ordering Realistic a 3 n o f Hamiltonian H J1 8 SJ J 2 2 8 SJ 17 Z 6H J3 Z degeneracy breaking ltltltijgtgtgt perturbations Entropic corrections vanish as T gt 0 free energy F E TS Energetic corrections from SE inevitably dominate at lowest T longrange order energetically entropically favored favored O T02 T01 T spiral spin liquid MnSczs4 A Krimmel er al PRB 73 014413 2006 M Mi lksch er al 2007 Spiral order s iral order 7 2743 4 3 4 0 op Spm hqmd high temperature d1ffus1ve scattering diffusive scattering paramagnet 0 19K 23 cw a T experiment Theoretical implications observes 110 order 0 J1 is ferromagnetic favored by J3 cw m 221K gives t J1 m 12K J2 m 10K TC 24K 9 mpy favors lowT order determined energetically 100 order not entropically MnSC2S4 A Krimmel et al PRB 73 014413 2006 M Muksch et a 2007 spiral order Spiral order 67 27r34 34 0 Spin liqmd high temperature diffusive scattering diffusive scattering paramagnet 6 0 19K 23K CW v 600 25 25 0g 16 K c 17 K E g 39 7o 7 LR K 2 2 ltgt 19K quotif Q 400 20K 15 15 lt 7 4 2 311 2 1 1 i in Q l 200 E0305 05 Q Hume 39 5 2 AAAA i 0 h39 39 39 39 1 0 0 036 04 05 06 a 07 08 05 075 1 125 15 Q A Q Zna Intensity shifts from If I to spiral surface as T washes out J3 Consistent with spiral spin liquid COA1204 N Tristan et 51 PRB 72 174404 2005 A Krimmel et 51 Physica B 378380 583 2006 T Suzuki et a 2007 Experimental observations Theoretical implications 0 strong 39ustration sample dependent powder neutron data frustration no sharp transition observed yet suggeSt J2 J1 18 05 7 7 05 04 MnSCzS4 504 amp E 03 l 03 g i a a E 027 39 COA1204 502 7 i quotLBJ 017 i 7701 O 7 i N el coplanar spiral 7 0 l l l l l l l 0 i i i i i i i i i 0 01 02 03 04 05 06 07 08 09 1 coupling J2 J1 COA1204 N Tristan et 51 PRB 72 174404 2005 A Krimmel et 51 Physica B 378380 583 2006 T Suzuki et al 2007 Experimental observations Theoretical implications 0 strong 39ustration sample dependent powder neutron data frustration no sharp transition observed yet suggeSt J2 J1 18 experiment theory 200 E o o g M 67 f6 8 100 i D i E i ii riiii i Wquot l i 5 4 4 0 2 2 W 902 12 100 91 O 2 G l l x l 7 0 Q 7 1 Q2 2mg Summary Many spinels constitute frustrated diamond lattice AFMs MnSC2S4 CoAle4 etc Simple J 1J 2 model captures essential physics Continuous spiral groundstate degeneracy Manifestation of orderbydisorder physics Spin correlations in spiral spin liquid regime reveal ordering surface and entropic effects Theoretical predictions consistent With experiments D Bergman J Alicea E Gull S Trebst and L Balents Nature Physics 3 487 2007


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