Geometrical Frustration in Ising Magnets

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KAM

Collaborators:

M. Khatun (Ball State) J. Barry (UF) �� AF

AF AF AF AF

AF AF HAPPY ! FRUSTRATED !!

Geometry determines if lowest energy state for each bond can be satisfied. Leads to large ground state degeneracy Leads to disordered ground states (e.g. spin-liquids) �� Figures from Wikipedia

Triangles in 2D Tetrahedrons in 3D

Can be present in different types of lattices in 2D or 3D �� Figures from Wikipedia

AF FM

Spins along easy axes Frustrated spins Zero moment with AF interactions with FM interactions

Frustration can be for ferro or antiferromagnetic interactions �� Giauque and Stout (1936)

Residual entropy Estimated from expt 185.3 J/(mol.K) Compare with theory Ideal gas 188.7 J/(mol.K)

Missing entropy: 3.4 J/(mol.K)

Pauling: configurational entropy 푁푘 ln 3/2 = 3.4 J/(mol.K) !! 퐵 Ramirez (1994) site disordered

Ramirez (1994)

site ordered ��

triangular kagomé

푧 푧 퐻=−퐽 2 5 σ 푖σ 푗 :;,=> Both have disordered ground states with finite residual entropy 푆 0 = 0.3231 Triangular Ising Antiferromagnet (TIA) 푁푘퐵 = 0.5018 Kagomé Ising Antiferromagnet (KIA)

Is residual entropy a good measure of the degree of frustration? ��

Usable Power: Ramirez: 1994

Θ퐶푊 Empirical measure of frustration: 푓=− 푇푁 Ramirez (1994) ��

Add field parallel to Ising spin axis: −ℎ 5 σ푧 ⇒ χ 푖 ⎪⎪ ;

perpendicular to Ising spin axis: −ℎ 5 σ푥 ⇒ χ 푖 ⊥ ; ↑ ��

χ χ Both ⎪⎪ and ⊥ can be expressed in terms of spin-correlations.

χ ⎪⎪ requires knowledge of pair correlations at all distances. χ ⊥ requires knowledge of finite number of local pair and multi-spin correlations �� • Will consider two exactly solvable models

(1) Kagome Ising Antiferromagnet (KIA), with strong geometrical frustration. (2) A triplet interaction model (KIA-‐T) with no frustration.

• For KIA: Obtain χ ⊥ exactly for all temperature and χ accurately at high temperatures. ⎪⎪

• For KIA-‐T: Obtain χ and χ as well as entropy ⊥ ⎪⎪ exactly at all temperatures Propose a quantitative measure of the degree of frustration ��

푧 푧 퐻=−퐽 2 5 σ 푖σ 푗 :;,=> ��

Barry and Khatun: 1987

JKLM [O∑V Q T] 2 WSXY RS ] 휒 =푚 훽푁 V ; 푄= 훽퐽; 훽≡ ⊥ O ∑ Q T ^ ` WSXY RS _

m: magnetic moment q: number of nearest neighbors

Sum is over all nearest neighbors ��χ⊥ ��

휒 + − ⊥a0 =푞 푄 [퐴 +2퐴0(푥 +푥 +푥 ) +퐴 푥 ] 휒2 2 01 05 06 1234 푚2 1 tanh4푄 tanh2푄 0 퐴 ±≡ 2 ± 4 2 + 3 휒2 ≡ 푞|퐽2| 8 4푄2 2푄2 1 tanh4푄 퐴0 ≡ 2 −1 8 4푄2

푧 푧 푥푖푗 ≡ σ 푖σ 푗 ≡ 푖푗

푧 푧 푧 푧 푥1234 ≡ σ 1σ 2σ 3σ 4 ≡ 1234

Need pair correlations <01>, <05>, <06> and quartet correlation <1234> �� Barry and Khatun (1997);; KAM, Khatun and Barry (unpublished)

= k BT/4J2 Note: pair correlations at distances farther apart decay faster with temperature �� KAM, Khatun and Barry (2017, unpublished)

= kBT/4J2 Note: The zero temperature limit corresponds to a 1/T divergence with an amplitude smaller than 1 (~ 0.4458) ��

퐾퐼퐴 2 푧 푧 Need pair correlations at all 휒 푇=푚 훽 5 σ0 σ푖 0 ∥ distances. Only six are known ; 휒 퐾퐼퐴 ∥ a 0 = 푞훽퐽 [1 + 4 01 + 4 05 + 2 15 + 4 06 휒2 2 + 8 16 + 4 26 + …..]

Strategy: Compare results with four-, five- and six-pairs How much do the results change? ��

1/|4Q2|= kBT/4J2

th Adding 6 pair changes result by 0.006% at 1/|4Q 2| ~ 10 and by 0.0002% at 1/|4Q 2| ~ 30. Thus high-T limit is essentially exact.

Extrapolation gives Θ퐶푊= 4J 2/kB ;; agrees with high -T expansion ��

• Perpendicular susceptibility diverges as c/T, as T goes to zero, with c <1. Follows 1/T at large T.

Compare free spins: c=1

• Parallel susceptibility could not be calculated exactly at low-T limit. Follows Curie-Weiss law

1/(T-Θ 퐶푊) at large T with Θ퐶푊= 4J2/kB <0. Agrees with high temperature expansion results.

Compare free spins: Θ 퐶푊= 0. Is any of these a good measure of geometrical frustration?

Compare with another exactly solvable model on kagome lattice that has no geometrical frustration ��

푧 푧 푧 퐻푡푟푖푝푙푒푡 =−퐽 3 5 σ 푖σ 푗σ 푘 :;=^>

2D kagome All spins are `satisfied’;; flipping any one would + increase energy + -‐

No frustration ��

푧 푧 푧 퐻푡푟푖푝푙푒푡 =−퐽 3 5 σ 푖σ 푗σ 푘 :;=^> 2D kagome

푧 푧 푧 푧 퐻푞푢푎푟푡푒푡 =−퐽 4 5 σ 푖σ 푗σ 푘 σ 푙 :;=^x> Barry and Wu (1989) 3D pyrochlore �� Both triplet and quartet models belong to a class of multi-spin (cluster) interactions of `only one kind’ whose partition functions Z are exactly calculable.

Diagrammatic expansion of Z in powers of tanh Q in the thermodynamic limit has contribution only when the path length is zero (no finite length path consisting exclusively of basic interaction clusters can totally close upon itself).

Barry and Wu (1989) used graph theory to show that in the quartet interaction model, only two terms survive in Z due to the corner-sharing nature of the 3D pyrochlore lattice corresponding to two complementary graphs.

Because 2D kagome has corner sharing triangles, it is possible to adapt the same techniques. ��

O ∑| { { { 푍푁 =5푒 z ; = ^ { = cosh푄 2푁/3 5€ 1 +휎 휎 휎 tanh 푄 3 푖 푗 푘 3 ‚{ 2푁/32푁 2푁/3 = cosh푄 3 [1 + (tanh푄 3) ]

Differs from the quartet model only by the exponent 2N/3

The magnetic Helmholtz free energy per spin is

1 2/3 훽푓 푄3 = lim ln푍 푁 푄3 = ln [2(cosh푄3) …→‡푁 �� tanh [푄 휎 휎 +휎 휎 ] 휒 =푚 2훽 3 1 2 3 4 ⊥ 푄3(휎1휎2 +휎 3휎4)

tanh [ 푄3 휎1휎2 +휎 3휎4 ] + − =퐵 + 퐵 휎1휎2휎3휎4 푄3(휎1휎2 +휎 3휎4) 1 tanh 2푄 퐵± = 3 ± 1 2 2푄3 Thermal average of the “bow-tie” unit

1 O ∑ { { { 휎 휎 휎 휎 = 5 휎 휎 휎 휎 푒 z | ; = ^ 1 2 3 4 1 2 3 4 푍푁 { 1 2푁/3 = cosh푄 3 5 휎1휎2휎3휎4 € 1 +휎 푖휎푗휎푘 tanh푄 3 푍푁 { ‚ Graph theory: only two terms in the sum survive! ��

2 휎1휎2휎3휎4 = tanh Q3 휒 퐾퐼퐴푇 푞 ⊥ a0 = 푄 1−tanh 2 푄 + tanh |푄 | 휒3 2 3 3 3 푚2 0 휒3 ≡ 푞퐽3 �� All pair correlations of KIA -T vanish, giving

휒 퐾퐼퐴푇 4퐽 ∥ 3 0 = 휒3 푘퐵푇 ��χ ��χ �� ⎪⎪ ⊥

= k BT/4|J3|

Contrast with KIA: Absence of 1/T divergence of χ⊥ at T=0 and zero intercept of (χ )-‐1 implying Θ = 0. ⎪⎪ 퐶푊 �� 휕 푆 = 푘퐵 1−훽 ln푍 푁 휕훽 푁 In the thermodynamic limit N→ 0, 푆퐾퐼퐴푇 2 = ln2 + [ln (cosh 푄3)−푄 3 tanh 3 푄 ] 푁푘퐵 3

KIA-T Finite residual entropy S 0 /NkB = (ln 2)/3 = 0.2310 ….

KIA Compare: S 0 /NkB = 0.5018

TIA S0 /NkB = 0.3231 Example of large ground state degeneracy but zero frustration ��

• Perpendicular susceptibility remains finite as T goes to zero. Follows 1/T at large T.

Compare KIA: 휒 diverges as T goes to zero ⊥ • Parallel susceptibility follows Curie-Weiss law 1/T with Θ퐶푊=0.

Compare KIA: Θ 퐶푊 < 0.

• Residual entropy finite: 0.2310 Nk B (has no frustration)

Compare KIA: 0.5018 NkB (has large frustration) ��

• KIA is highly frustrated, but KIA-T has no geometrical frustration. This implies the following:

• Residual entropy can not be a measure of

geometrical frustration. Compare finite S 0 for both KIA and KIA-T.

• Curie-Weiss temperature Θ퐶푊 is finite for frustrated KIA but zero for unfrustrated KIAT

• Perpendicular susceptibility diverges for KIA but remains finite for KIA-T.

Compare with other available exact results �� Wannier (1950);; Houtappel (1950);; Kano and Naya (1953);; Sykes and Zucker (1961);; Stephenson (1964);; Barry and Khatun (1987) •TIA is the Triangular Ising Antiferromagnet and KIF is the Kagome Ising Ferromagnet.

•TIA highly frustrated, KIF is not frustrated.

•TIA remains disordered down to zero temperature, KIF orders at a finite Tc.

• Residual entropy of TIA is 0.3231. It is zero for KIF.

•Θ 퐶푊 is positive for KIF and negative for TIA. • Perpendicular susceptibility diverges for TIA but remains finite for KIF. ��η(T) 푘 푇 η 푇 ≡ 퐵 휒 (푇) η ≡ η(푇 → 0) 푚2 ⊥ 0

Consider four different Ising models:

(1) KIA: strong geometric frustration, disordered ground state, finite residual entropy

(2) KIA-T: no geometrical frustration disordered ground state, finite residual entropy

(3) KIF: no geometrical frustration ordered ground state, zero residual entropy

(4) TIA: strong geometrical frustration disordered ground state, finite residual entropy ��

η0

KIA: 0.4458 TIA: 0.2900 KIA-T: 0 KIF: 0

Proposal: η0 is a measure of degree of frustration ��

• Considered two exactly solvable models, KIA and KIA-T

• Obtained exact χ ⊥(T) and accurate ΘCW for KIA

• Obtained exact χ (T), χ (T) and entropy S(T) for KIA -T ⊥ ⎪⎪ • Comparing the results with known results for KIF and

TIA leads to the proposal that η 0 is a good quantitative measure of geometrical frustration.

• Different zero temperature limits of χ (T) and χ (T) ⊥ ⎪⎪ imply different types of degenerate states. �� •A finite residual entropy implies large ground state degeneracy for both KIA and KIA-T, but the nature of low-energy excitations must be very different since one has large geometrical frustration but the other has none.

• This difference shows up in the zero temperature limit of the perpendicular susceptibility.

• In a more general case with both pair and triplet interactions simultaneously present , the interplay of disorder with and without frustration might lead to more interesting possibilities.

• This is not just a mathematical curiosity. Presence of weak triplet interaction in addition to a dominant pair interaction is actually quite ubiquitous in nature. ��