Magnetic Monopoles in Spin Ice

Transcription

1 Magnetic Monopoles in Spin Ice Tim Herfurth Institut für Theoretische Physik Goethe-Universität Frankfurt July 6th, 2011

2 Outline 1 Spin Ice Water Ice What is Spin Ice? 2 Magnetic Monopoles Origin of Magnetic Monopoles Real Monopoles? 3 Experiment

3 Water Ice and Ice Rule Cells of water ice have shape of tetraheda (see Figure) For each oxygen atom, two of the neighboring hydrogen atoms are near (covalent bonds) and two are further away (neighboring molecules) ice rule number of congurations conforming to this rule grows exponentially with the system size residual entropy (i.e. intrinsic randomness) for T 0 Figure: arrangement of hydrogen atoms (black circles) about oxygen atoms (open circles) in ice

4 Structure of Spin Ice examples: Dy 2 Ti 2 O 7, Ho 2 Ti 2 O 7, Ho 2 Sn 2 O 7 magnetic rare earth-atoms reside on a polychlore lattice (gure) spins on the corners behave like classical Ising spins spins represent magnetic moments that point along the local Ising axes that connect the centers Figure: (C. Castelnovo, R. Moessner, S.L. Sondhi)

5 Why Ice? Spin Congurations nearest-neighbor interaction is ferromagnetic, J < 0. ferromagnetic interaction is frustrated minimal-energy-states are highly degenerate always two-in-two-out (Ice Rule) ice rule implies strong correlations Figure: (L. Balents et al.)

6 Why Ice? Analogy to Water Ice for low temperatures system uctuates almost entirely within this manifold of states number of congurations that fullll ice rule is exponentially large residual entropy residual entropy was rstly described by Pauling considering water ice (therefore spin ice)

7 What's it all about? In 1931 Paul Dirac showed that magnetic monopoles would explain the quantization of electric charges. But no elementary particles with a net magnetic charge have ever been observed. What kind of monopoles we are talking about here? emergent particles How do they occur?

8 Interaction in Spin Ice Materials Ising spins on corners carry magnetic moment µ i = µs i e i, where S i = ±1, e i Ising axes direction and µ 10µ B. Hamiltonian of Interaction H = J [ S i S j + Da 3 ei e j 3 r ij 3 3(e ] i r ij )(e j r ij ) r ij 5 S i S j i,j }{{}}{{} long-range dipolar interaction nearest-neighbor ex. a 3.54A D = µ 0 µ 2 /(4πa 3 ) pyrochlore nearest-neighbor distance coupling constant of dipolar interaction

9 Going to Magnetic Charges concept: replace magnetic moments by magnetic charges living on the bonds ends (depicted by red and blue dumbbells) dipol interaction magnetic Coulomb interaction two ways of assigning charges reproduce the two possible orientations of the original dipole

10 Going to Magnetic Charges now the dipole interaction energy is given by the magnetic Coulomb law V (r αβ ) = { µ0 4π Q αq β r αβ α β 1 2 ν 0Q 2 α α = β Q i : total magnetic charge at site i with charges xed at ±µa d, where a d = 3/2a. self-energy ν 0 /2 gives n-n-interaction

11 Excitations all Q i = 0: minimal energy according to ice rule but: even when k B T J violations of ice rule occur inverting a single dipole/dumbbell generates local net dipole moment 2µ two adjacent sites with net magnetic charge Q α = ±q m = ±2/a d monopole-antimonopole pair (what you can see as red/blue sphere)

12 Excitations this means a tetraheda with three spins pointing in and one out (or vice versa) requires energy of 2J relative to ground state this gives sink/source of magnetic ux center of tetraheda can be seen as a local magnetic monopole magnetic charge is weak (1/14, 000 of electric interaction at same distance) but still measurable

13 Separation of the Monopoles real monopoles are non-local objects. How do they seperate? monpoles can move by ipping a chain of adjacent dumbbells each ip moves a monopole to the next site between monopole/antimonople a Dirac String of ipped dipoles is created

14 Spin Ice Magnetic Monopoles Separation and Energy n-n-interaction: ip does not cost energy energy cost: magnetic Coulomb interaction ( only 1 r, see plot below ) nite energy necessary to seperate monopoles to in nity! quasi-independent monopoles Experiment

15 Spin Ice as Particularity condition for occurrence of monopoles: energy for creating strings of dipoles remains bounded in vaccum this needs creating additional dipoles in conventional magnetic materials a string is accompanied by costly domain walls particular ground state gives unusual properties of spin ice: two dipole strings enter and exit each lattice site

16 Real Monopoles? monopoles are sources and sinks of magnetic eld H B is also monopolar but compensating ux is created by 'Dirac String' of ipped dipoles magnetic monopoles are emergent particles a monopole can only occur accompanied by an antimonopole monopoles are not entirely independent

17 Spin Ice Magnetic Monopoles Experiment Jonathan Morris, Alan Tennant et al. showed existence of monopoles by applying an external magnetic eld they could increase the density of Dirac strings scattering of neutrons made magnetic monopoles visible K) Dy2 Ti2 O7 Experiment