Order by Disorder

1. The 93rdStatistical Mechanics Conference RUTGERS UNIVERSITY Order by Disorder • A new method for establishing phase transitions in certain systems with continuous, n–component spins. • Applications to transition metal oxides

2. Cast of characters: M. Biskup (UCLA Math) And, the physicists: Relevant papers Z. Nussinov (Th. Div. Los Alamos) M. Biskup, L. Chayes, Z. Nussinov and J. van den Brink, Orbital order in classical models of transition-metal compounds, Europhys. Lett. 67 (2004), no. 6, 990–996 M. Biskup, L. Chayes and S.A. Kivelson, Order by disorder, without order, in a two-dimensional spin system with O(2) symmetry, Ann. Henri Poincaré 5 (2004), no. 6, 1181–1205. J. van den Brink (Lorentz ITP, Leiden) M. Biskup, L. Chayes and Z. Nussinov, Orbital ordering in transition-metal compounds: I. The 120-degree model, Commun. Math. Phys. 255 (2005), no. 2, 253–292 S.A. Kivelson UCLA Physics … more to come … (Plenty.)

3. What can this system do? MW: Certainly no magnetization Ground states: Now add “weak” NN ferromagnetic coupling. General considerations: Ground states still ¿ Disorder? (Exactly zero message from nearest neighbors.) Systems of interest: Continuous spins O(2), O(3), … – huge degeneracy in the ground state. Example: 2D NNN Antiferromagnet. (Say XY spins.)

4. = direction of bond Transition Metal Compounds • Levels in 3d shell split by crystal field. eg d–orbitals t2g • Single itinerant electron @ each site with multiple orbital degrees of freedom. Super–exchange approximation(and neglect of strain–field induced interactions among orbitals): [Kugal–Khomskii Hamiltonian] 120º–model (eg–compounds) V2O3, LiVO2, LaVO3, … orbital compass–model (t2g–compounds) LaTiO3, …

5. • Orbital only approximation: Neglect spin degrees of freedom. • Large S limit (for pseudo–spin operators): Go classical. Classical 120º Hamiltonian: unit vectors spaced @ 120º. Classical orbital compass Hamiltonian: – usual Heisenberg spins. For simplicity, today focus on 2D version of orbital compass.

6. write L ´ L torus May as well take The 2D Orbital Compass Model + constant. Attractive couplings (ferromagnetic). $ other ground–states but these play no rôle and will not be discussed. Couples in x–direction with x–component. Couples in y–direction with y–component. Clear: Any constant spin–field is a ground state. * O(2) symmetry restored * (a) Not clear what are the “states”. Can’t even begin to talk about contours: (b) No apparent “stiffness”. ¿Hints from SW–theory?

7. Disaster: Very IR–divergent. Indeed, spherical version of this model has no phase transition. But infrared bounds only give upper bounds on the scattering function. And MW–theorem (strictly speaking) does not apply. –– The O(2) symmetry is in the ground states, not the Hamiltonian itself. –– Theorem. For the d = 2 orbital compass model, for all b sufficiently large,$ (at least) two limiting translation invariant Gibbs states. One has close to one. The other has close to one.

8. Really clarified matters; put things on a firm foundation in a general context. Key ideas: In the physics literature since the early 80’s J. Villain, R. Bidaux, J. P. Carton and R. Conte, Order as an Effect of Disorder, J. Phys. (Paris) 41 (1980), no.11, 1263–1272. E. F. Shender, Antiferromagnetic Garnets with Fluctuationally Interacting Sublattices, Sov. Phys. JETP 56 (1982) 178–184 . C. L. Henley, Ordering Due to Disorder in a Frustrated Vector Antiferromagnet, Phys. Rev. Lett. 62 (1989) 2056–2059. Plus infinitely many papers (mostly quantum) in which specific calculations done. Our contribution to physics general theory: Modest. All of this works even in d = 2. (But TMO models of some topical interest.) At b < ∞, weighting of various ground states must take into account more than just energetics: • Fluctuations of spins will contribute to overall statistical weight. These (spin–fluctuation) degrees of freedom will themselves organize into spin–wave like modes. • Can be calculated (or estimated).

9. Key ideas: 1) At b < ∞, weighting of various ground states must take into account more than just energetics: Gaussian like SW–free energetics will tell us which of the ground states are actually preferred @ finite temperature • Fluctuations of spins contribute to overall statistical weight. 2) These (spin–fluctuation) degrees of freedom will themselves organize into spin–wave like modes. • Can be calculated (or estimated). Remarks (1) Not as drastic an approximation as it sounds; • Infrared divergence virtually non–existent at the level of free–energetics. (2) In math–phys, plenty of “selection due to finite–temperature excitations”. • Excitation spectrum always with (huge) gap. But: • Finite (or countable) number of ground states.

10. Will square this. Neglect quadratic (and beyond). Let’s do calculation. Write: [q = fixed “ground state”] Look @ H, neglect terms of higher order than quadratic in j’s Well, got: so Get and similarly

11. Fact that we are talking about “q” means that we do not integrate over the k = 0 mode. Approximate Hamiltonian is therefore: Go to transform variables: So, after some manipulations, Now, total weight can easily be calculated: Take logs:

12. A free energy; > –or– Want as small as possible. Scales with b. Pause to refresh: No difficulty doing these integrals; some infrared “action” but no big deal (logarithmic). We are only interested in a free energy. Now log is a (strictly) concave function: Do kx, ky integrals on RHS, these come out the same. We learn Use strict concavity,

13. D(b): – and hence – Calculation indicates: There are “states” at which will dominate any other “state”. Outline of a proof. (1) Define a fluctuation scale. Look @ situation where each deviation variable jr has: Fact: means that quadratic approximation is “good”. Fact: means that the effective Gaussian variables are allowed to get large.

14. Proposition: With the constraints (globally) enforced, the spin wave formula for the q – dependent limiting free energy is asymptotic as Not important; for technical reasons, this is notD(b). (a) Each neighboring pair of fluctuation variables satisfies satisfies (b) Each spin variable Not hard to see: (2) Define a running length scale, B – and another, interrelated spin–scale, e. Definition. A block LB of scale B is defined to be good if (You can add .) Clear: There are two types of good blocks.

15. (ii) Energetics good, but q not particularly near 0 or . Can use method of chessboard estimates. [FSS] [FLIS(1)] [FLIS(2)] [LF] [DLS] [FLIS(3)] A] Gives estimates on probability of bad blocks in terms of constrained partition function where all blocks are bad blocks. Type (i) indeed suppressed exponentially. Type (ii) has probability bounded by where N is the block size. This goes to zero as Note: Type (ii) is independent of . Also: Two types of bad blocks. Energetic disaster. (Should be suppressed exponentially with rate ~ bD2 ) More important, more interesting:

16. Reiterate: Two distinct types of goodness. A single box cannot exhibit both types of goodness. Thus, regions of the distinctive types of goodness must be separated by closed contours. Bad blocks form contour element which separate regions of the two distinct types of goodness. [FSS] [FLIS(1)] [FLIS(2)] [LF] [DLS] [FLIS(3)] B] Gives estimates on probability of contour bad blocks by the product of the previously mentioned probability estimates. • Standard Peierls argument, implies existence of two distinct states. Remarks: Same sort of thing true for antiferromagnet, 120º–model & 3D orbital compass model (sort of). Interesting feature: Limiting behavior of model as T goes to zero is not the same as the behavior of the model @ T = 0. In particular, non–trivial stiffness at T = 0 (and presumably  as well)..

17. The End