The Coulomb phase in frustrated systems

Transcription

1 The Coulomb phase in frustrated systems Christopher L. Henley, [Support: U.S. National Science Foundation] Waterloo/Toronto, March

2 Outline 1. Examples: lattices/models/materials w/ Coulomb phase 2. Power-law correlations and diffraction 3. charge defects [= emergent magnetic monopoles] 4. application: dynamics 5. application: transitions to ordered phases 6. application: quenched disorder Disclaimer: journalism, not my research. Many others (e.g. John Chalker) have done more on this. 2

3 Highly frustrated magnets lattice systems with highly constrained and highly degenerate ground states Theorist s difficulty: navigation among these states. How label/enumerate states so as to evaluate partition sum? If a particular state is selected from the degenerate ensemble, due to fluctuations or to perturbation terms in the Hamiltonian, how to find that needle in a haystack? One answer: coarse-graining (Converts lattice to continuum model) Retain local averages of certain variable as degrees of freedom. Should be Conserved quantities. But (often) spin isn t conserved (spins flip). 3

4 Constraints and fluxes The answer: another sort of conservation: constraints on degrees of freedom around each lattice point. Mapping: local variables (weighted) arrows along bonds. Constraint signed sum of incoming arrows = 0 (in any allowed configuration.) arrows = lattice fluxes (analog: magnetic/electric flux) Vector field P(r) = coarse graining of lattice flux (i.e. averaged over some smallish volume) constraint divergence condition P = 0 (in absence of charges) 4

5 Also (to be shown!) coarse-grained free energy is d d r K 2 P 2 just like (magnetic/electric) field energy. called COULOMB PHASE (another condition: liquid like no long range order) [Preview: what we can do with this spin correlations have spatial dependence of a dipole-dipole interaction,] 5

6 1. Examples Lattices Start from parent lattice B which is bipartite (= even+odd sites, a.k.a. nodes Bond midpoints of parent lattice = medial lattice L. (Degrees of freedom and fluxes on these sites.) 6

7 Table 1: Parent lattices and medial lattices. d parent latt. Brav. latt. medial latt. 2 square square checkerboard honeycomb triangular. kagomé 3 simple cubic s.c. octahedral diamond f.c.c. pyrochlore Laves graph b.c.c. half-garnet 7

8 Realizations of pyrochlore lattice: B sublattice (octahedral sites) of spinel oxides; two interpenetrating lattices in pyrochlore crystal structure; majority (small atom sites) in Laves phase (alloys). Half-garnet lattice: corner-sharing equilateral triangles. (= hyperkagomé for nearest neighbors, but higher in symmetry.) 8

9 Degrees of freedom ice models Arrow on each edge. ice rule constraint: half of arrows point in, half point out [Bernal 1933, Pauling 1935!] Arrow is the flux. dimer models A dimer covers two nodes of the parent lattice, and every node is covered exactly once. Flux: weight Z B 1 on occupied edge, 1 on unoccupied edge. 9

10 Mappings of the ice model on the diamond lattice Change to 3 in a row... (a). (b). (c). (d). O H A B abstract ice model. Water ice Compound of species A and B Ising ground state. 10

11 Structural/electronic realizations Water ice The arrow represents which O the H is covalently bonded to, in the hydrogen bond. Lattice-gas orders Mutually repelling particles on the medial lattice. Constrain net filling to rational values, then (all) ground states satisfy the constraint. Pyrochlore lattice with 50% ice model (diamond lattice) with 25% dimer covering ( ) What particles? Ni and Cr in pyrochlore CsNiCrF 6 valence states in spinels: Fe +2 and Fe +3, in magnetite Fe 3 O 4, or V +3 and V +4 in heavy-fermion metal LiV 2 O 4. 11

12 Spin realizations in antiferromagnets Nearest-neighbor pyrochlore AFM: H = J ij S i S j = 1 2 tetra. α L α 2 + const. L α i α S i = 0 (tetrahedron sum in ground state) Classically, any state with L α = 0 is ground state. This is a divergence constraint! (Either for discrete Ising spins, or continuous vector spins). Highly degenerate cooperative paramagnet state. 12

13 Spin ice pyrochlore magnets Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 ferromagnetic int. + local 111 spin anisotropies represent by Ising spins (local axis even-to-odd) effective Hamiltonian is antiferromagnetic Ising AFM ground states: 2 spins in, 2 spins out maps to ice model. dipolar spin ice Actual interactions are (mostly) dipolar All microstates satisfying ice rules have (nearly) the same energy [Melko & Gingras 2004, Isakov et al 2005, Castelnovo et al 2008] 13

14 Just replace each dipole by a dumbbell pair of opposite effective charges ±q having the same dipole moment µ. Separate them by bond length d of parent lattice, then iff dipoles satisfy the ice rule all effective charges cancel! (a). (b). (showing 2D version for pedagogical purposes) 14

15 2. Coarse-graining and correlations Now ready to use coarse-grained polarization P. Want an effective free energy to predict correlations. 15

16 Local rearrangements? Can I flip the flux on one bond touching site α? No, I must flip the flux on another bond, the opposite way, to preserve the constraint. The string of flips goes on and on. Must flip an entire closed loop! Loop never changes the total P. (But if string spans the system, flipping it does changes total P.) 16

17 Entropy as function of polarization No. microstates exp(s(p) volume). Recall: rearrangements require loops large P hard to find reverse arrows few loops low entropy density (vanishes at saturated polarization P max ) P = 0 many loops entropy maximum entropy density S(P) 0 P max P 17

18 So s(p) s K P 2 total free energy (all entropy) for (slowly) fluctuating P: ) F tot ({P(r)} = const + d d r 1 T 2 K P(r) 2. Divergence constraint becomes P(r) = 0 Like field energy of mag../electric field w/o monopoles/charges. In dipolar spin ice, the polarization is proportional to the real magnetization M. The total free energy has the same form, but now it s all energy actual magnetic energy. 18

19 Obtaining correlations Fourier transform P(r): F tot T = q 1 2 K P(q) 2 Naive equipartition (components a, b = x, y, z): But divergence constraint is P µ (q)p ν ( q) = 1 K δ µν q P(q) = 0. So we must project out the components along q/ q. Result is P µ (q)p ν ( q) = 1 ( δ µν q ) µq ν K q 2 19

20 Diffraction consequences Observables P struc. factor has same singularities. No divergence, but limit depends on direction of approach! q y pseudo dipolar diffraction singularity q x When coefficients relating observable to P(r) alternate in space at wavevector Q, singularity gets displaced to Q. 20

21 Reciprocal space indeed shows these pinch points : [T. Fennell et al, Science 2009] 21

22 Real space correlations From F.T. of P µ (q)p ν ( q). Recall that ( ) 1 q 2 = F.T. V Coul (R) 1 4π R Hence, the singular term q µ q µ q 2 = F.T. ( µ ν V Coul (R) which is the form of a dipole-dipole coupling 1 ( KR 3 δ µν 3 ˆR ) µ ˆRν Surprise: in this liquid-like state, correlations don t fall off exponentially, but by power-law as in a critical state. ) 22

23 Application: magnetizations near a defect spin [Arnab Sen, Kedar Damle, R. Moessner, unpublished. See also CLH paper for first HFM conference, Waterloo 2000] Replacing some atoms with non-magnetic: site dilution [= bond dilution on the parent lattice ] (half) orphan spins appear on tetrahedra/triangles where just one edge (spin) is left after dilution: no way to satisfy flux constraint there. Defect spin responds to ext. field like free spins. Perturbation of the surrounding spins decays as a power law. This explains NMR observations of the 2-layer kagomé system, SrCr 9p Ga 12 9p O

24 3. Charge defects e.g. monopoles What if we allow (dilute) places that violate the flux constraint? (e.g. thermal excitation costing E Q out of ground state ensemble). Label the defect by (pseudo) charge Q = net outward flux. So Gauss s Law holds can be detected nonlocally and pseudo charge conserved in time: a topological defect In Coulomb phase, effective defect-defect potential is Coulomb: F int (r 1, r 2 ) T = KQ 1Q 2 4π r 1 r 2 Recall lattice gases that map to dimer covering on parent lattice: remove one particle creates a defect/antidefect pair. Thus each defect carries particle number 1/2! v. elementary example of fractionalization [Moessner & Sondhi PRL 2001] 24

25 Thermal population of defects Analogous to intrinsic semiconductor with a gap 2E Q. Defects having density n Q exp( E Q /T ) imply Debye screening of the effective Coulomb interactions, i.e. V coul gets multiplied by e κ r 1 r 2 where κ = nq KQ 2. T F.T.: in structure factor, this means replacing q 2 q 2 + κ 2 in the denominators, softening the singularity. 25

26 Magnetic monopoles in dipolar spin ice Look again at the construction dipole (pair ±q). At defect node this puts a net magnetic charge Q. (a). (b). Total energy for the microstates with a given set of defects is almost independent of configuration of intervening dipoles, and given by V Coul (R): they are emergent magnetic monopoles [Castelnovo, Moessner, & Sondhi, Nature 2008] 26

27 In what sense a monopole? Not microscopic monopoles! so there must be a backflow of the actual magnetic flux. (Notice that here the emergent Coulomb interaction is energetic and seen in any microstate; whereas, in the entropic Coulomb phase, it is entropic and seen only if you sum over microstates, conditional on the defect positions.) 27

28 So is it any different from the effective monopoles at the ends of any bar magnet? (Yes.) (i) these monopoles move freely in response to forces; (ii) their magnetic charge is quantized (depending on material properties); (iii) numerous cute experiments 28

29 Experiments about monopoles? not observations of single monopoles, but thermodynamic/transport measurements. Magnetization dynamics (see below)), whereby the time derivative of magnetization maps to a monopole current How thermal monopoles (also induced by magnetic fields) cut off pseudodipolar correlations (see above) Best: measure monopole charge by Wien effect that is, magnetic field reduces the ionization energy, increases thermal monopole density, reduces autocorrelation of the actual spins, increases autocorrelation time of precessing implanted muons (muon spin resonance) 29

30 Muon spin relaxation in field Expt by [Bramwell et al, Nature 461, 956 (2009)] First: muon spin relaxation 1. Motional narrowing effect: a steady ext. field causes bigger precession, thus faster noise means slower relaxation 30

31 2. (left) In a given field, the barrier to dissociate depends on B (barrier at length scales bigger than lattice const!) 3. (right) Relaxation rate λ at a particular B (here 2 mt) has activated T dependence infer barrier, hence Q. Result: it gives same Q at each T < 0.3K, and satisfies µ = Q(d/2) as expected. 31

32 4. Dynamics The Coulomb phase idea also illuminates the dynamics, as (i) the P(r) field has the slow degrees of freedom [sometimes spin is an independent and conserved field] (ii) we know the constraints on what can change. Here only discuss (pseudo charge) defects tend to dominate the dynamics 32

33 Implementing loop updates First recall: if we re limited to microstates obeying the constraint, we must flip an entire loop at once. Not likely! 33

34 Instead, first nucleate a defect pair. One spin flip one step of defect. Walk the defects around the loop and finally re-annihilate loop update accomplished. (For simulations, or in real systems.) 34

35 Instead, first nucleate a defect pair. One spin flip one step of defect. Walk the defects around the loop and finally re-annihilate loop update accomplished. (For simulations, or in real systems.) 35

36 Instead, first nucleate a defect pair. One spin flip one step of defect. Walk the defects around the loop and finally re-annihilate loop update accomplished. (For simulations, or in real systems.) 36

37 Instead, first nucleate a defect pair. One spin flip one step of defect. Walk the defects around the loop and finally re-annihilate loop update accomplished. (For simulations, or in real systems.) 37

38 Instead, first nucleate a defect pair. One spin flip one step of defect. Walk the defects around the loop and finally re-annihilate loop update accomplished. (For simulations, or in real systems.) 38

39 Defect current and total magnetization Defects matter for a second reason in cases, e.g. spin ice or water ice, where the model polarization is a physical polarization (magnetization or electric polarization) that couples to external fields: they control the relaxation of the polarization. Consider spin ice with thermal monopole density n(t ); we are back to the semiconductor analogy. 39

40 Whenever a monopole (defect) moves by r, it changes the system s total polarization by P tot = Q r. Corollary 1: dp tot dt = J mono. Corollary 1: external B-field applies force on each defect F B = QB. Hence J mono = n(t )µ drift QB where µ drift = drift mobility. Magnetization relaxation (where M tot = µp tot ) dm tot dt = n(t )µ drift QB. See [Jaubert et al PRL]: this explains T dependence of magnetic relaxation rates observed years ago by [Snyder et al (Schiffer)]. 40

41 5. Transitions out of Coul. phases Various Hamiltonians lead to a crystal of dimers, arrows, etc. (Still has a polarization, but not fluctuating hence not Coulomb phase.) (Chalker and coworkers). Can map configurations a set of world lines which always run in y direction... (a). (b). (c). 41

42 ... Reinterpret as imaginary time direction in a path integral; thus, a quantum gas of particles in d 1. Transition controlled by bias as to direction of dimers. Near limit of saturated flux, the world-lines are dilute. a continuum description (Bose condensate in d = 2) is good. Kasteleyn transition : one side has usu. critical fluctuations. On other side ((saturated side), no fluctuations at all! 42

43 6. Disorder Dilute defects interact mediated by Coulomb phase [Chalker +] E.g. in dynamics of spin ice, following the semiconductor analogy, defects could serve as traps for pseudocharge defects. With vector spins: strong modulation of bond strength local pseudospins interact w/ dipole-dipole int. [Chalker & Andreanov or Saunders]. Long known: randomly sited dipoles have [spin-glass] transition. (Coulomb phase picture helps explain why spin glass temperature T SG disorder strength.) 43

44 Depleted antiferromagnets I turn to constrained disorder, special disorder ensembles which satisfy a flux constraint. This makes them more tractable than plain uncorrelated randomness. Hyperkagomé lattice = pyrochlore lattice with 1/4 of magnetic sites removed in a highly regular pattern. lattice of corner-sharing triangles. Found in spinel Na 4 Ir 3 O 8. This depletion pattern corresponds to a dimer covering of the diamond lattice (whose edge midpoints are pyrochlore sites). Fig. from [Lawler, Kee, Jim & Vishwanath, PRL 2008] 44

45 Let s specify the depletion by a random dimer covering. Parent lattice is still 3-coordinated everywhere (actual lattice still consists of corner-sharing triangles) still has Coulomb phase. The quenched disorder itself also is specified by a Coulomb phase! The actual polarization of the spin configuration has an offset quenched polarization. Hence actual correlations = superposition of 2 pseudo-dipolar terms: (quenched term) + (fluctuating term). [CLH, J. Phys. Conf. Series 2008] 45

46 Loop-disordered ensemble Another example of constrained disorder It was proposed by [Banks et al] that in CsNiCrF 6, A (=Cr) and B (=Ni) spins populate the pyrochlore such that every tetrahedron has two of each. This is (recall) lattice gas realization of ice model. Exchange constants: J AA, J AB, or J BB, depending which species at the ends of the bond. In one tetrahedron, for plausible couplings, the ground state has A spins point along ±ˆn, B spins point along ±ˆn. Whole lattice breaks up into alternating loops of A or B spins; each loop chooses a staggered direction indep. of the others. 46

47 r s r s r s p q p q p q n o n o n o l m l m l m t u t u t u j k j k j k v w v w v w b c b c b c d e d e d e ~ ~ ~ \ \ x y x y x y z { z { z { f g f g f g ] ] } } } h i h i h i Q S O M Y [ U W _ a ` a The pink arrows form an alternating loop which describes the chains of identical A or B species. (in the 3D pyrochlore analog), the spin directions alternate along each loop as shown. Disorder-averaged spin correlation = loop connectedness correlation function C ij. (probability, averaged over the ensemble, that sites i and j are on same loop.) Numerically [S. T. Banks] C ij 1/r ij : (we don t know why!) The same kind of loops are formed by electrons in the wavefunctions envisaged for LiV 2 O 4 [Fulde,Pollmann...] 47

48 Conclusion The polarization field is a comprehensive framework not yet exhausted to map out long-range correlations, charge-like defects, dynamics, and disorder effects. Calculational tools: (1) Continuum theories (form of correlations) (2) The large-n limit of n-component spin models, to evaluate correlations in particular lattices. (SKIPPED IN THIS TALK) (3) World lines used to model some transitions into ordered phases 48

49 Comparison of two kinds of loops (a) (b) 49