SU(N) magnets: from a theoretical abstraction to reality

Transcription

1 1 SU(N) magnets: from a theoretical abstraction to reality Victor Gurarie University of Colorado, Boulder collaboration with M. Hermele, A.M. Rey Aspen, May 2009

2 In this talk 2 SU(N) spin models are more controllable theoretically than their SU(2) counterparts and have richer phase diagrams [Marston, Affleck, 89; Read, Sachdev, 89; many others afterwards] SU(N) spins can be naturally realized by the alkaline earth ultracold atoms [A.M. Gorshkov, A.M. Rey, et al, 09] SU(N) spin models, of the kind easiest to realize, can form chiral spin liquids, states of matter having fractional and possibly nonabelian statistics [M. Hermele, A.M. Rey, VG, 09]

3 2D Heisenberg antiferromagnets 3 Nearest neighbors A collection of spin-1/2 on a square lattice in the presence of the antiferromagnetic interactions at T=0 forms a Néel state with a long range antiferromagnetic order.

4 Valence bond states 4 Suppose some bonds are made stronger than others.

5 Valence bond states 4 Suppose some bonds are made stronger than others.

6 Valence bond states 4 Suppose some bonds are made stronger than others. Spins on these bonds will form singlets and break the antiferromagnetic order

7 Spontaneous valence bond states 5 In the 80s, inspired by some ideas to understand high Tc superconductors, people looked for the valence bond states in the absence of strong bonds. They found them along the following routes: Adding frustration (additional interactions) Switching from the SU(2) spin to larger groups, such as SU(N) spin.

8 From SU(2) to SU(N) spin 6 Spin-up wave function Spin-down wave function fundamental (spin-1/2) representation of the SU(2) group symmetric combination of two spin-1/2s produces spin 1 symmetric combination of nc spin-1/2s produces spin nc /2 Antisymmetric combination of two spin-1/2s produces spin 0, or a scalar.

9 From SU(2) to SU(N) spin 7 fundamental representation of the SU(N) group

10 From SU(2) to SU(N) spin 7 fundamental representation of the SU(N) group Antisymmetric combination of two fundamentals produces an antisymmetric tensor with N(N-1)/2 components.

11 From SU(2) to SU(N) spin 7 fundamental representation of the SU(N) group Antisymmetric combination of two fundamentals produces an antisymmetric tensor with N(N-1)/2 components. Antisymmetric combination of m fundamentals produces an antisymmetric tensor with N(N-1)...(N-m+1)/m! components.

12 From SU(2) to SU(N) spin 7 fundamental representation of the SU(N) group Antisymmetric combination of two fundamentals produces an antisymmetric tensor with N(N-1)/2 components. Antisymmetric combination of m fundamentals produces an antisymmetric tensor with N(N-1)...(N-m+1)/m! components. If m=n-1, this N-dim representation is called antifundamental.

13 From SU(2) to SU(N) spin 7 fundamental representation of the SU(N) group Antisymmetric combination of two fundamentals produces an antisymmetric tensor with N(N-1)/2 components. Antisymmetric combination of m fundamentals produces an antisymmetric tensor with N(N-1)...(N-m+1)/m! components. If m=n-1, this N-dim representation is called antifundamental. If m=n, this is a scalar. Antisymmetric combination of the fundamental and antifundamental representations = scalar. More generally, [m]+[n-m] = scalar.

14 From SU(2) to SU(N) spin 7 fundamental representation of the SU(N) group Antisymmetric combination of two fundamentals produces an antisymmetric tensor with N(N-1)/2 components. Antisymmetric combination of m fundamentals produces an antisymmetric tensor with N(N-1)...(N-m+1)/m! components. If m=n-1, this N-dim representation is called antifundamental. If m=n, this is a scalar. Antisymmetric combination of the fundamental and antifundamental representations = scalar. More generally, [m]+[n-m] = scalar. Symmetric representations with nc components are also possible, just like for SU(2).

15 Interesting antiferromagnets with the 8 SU(N) spins The general interest is to see if valence bond states can exist for N>2 with nearest neighbor coupling only. Place [m] and [N-m] on even and odd bonds m singlets Read & Sachdev, 1989 Marston, Affleck, 1988 Many others afterwards N-m The ground state is Néel if N<4, VBS if N>4

16 Alkaline earth atoms 9 two electrons in the outer shell Ground state Excited state Both of these states have J=0, so the nuclear spin is decoupled from the electronic spin. This (sometimes large) nuclear spin can play the role of the SU(N) spin. For example, 87 Sr: I=9/2, N=10. A.V. Gorshkov, M. Hermele, VG, C. Xu, P.S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, A.M. Rey. arxiv:

17 From SU(N) Hubbard to an antiferromagnet 10 Fermionic alkaline earth atoms in a deep lattice form a Mott insulator nuclear spin One atom per site + virtual hops = antiferromagnetism One atom per site: } } fundamental m=1 representation on SU(N) spins each site.

18 Identical representations on every site 11 Theoretically it s easiest to study large N, so usually we consider m=n/k per site at large N. k=2: dimers k=4: plaquettes N=4, k=4: Pokrovsky, Uimin, 1971; Li, Ma, Shi, Zhang, 1998; M. van den Bossche, F-C Zhang, F. Mila, 2000; F. Wang, A. Vishwanath, 2009; M. Hermele, 2009 New. k 5: chiral spin liquid. M. Hermele, A.M. Rey, VG, 09

19 Chiral spin liquid Proposed as a state of matter by X.-G. Wen, F. Wilczek, A. Zee, Decouple the interactions by the hopping amplitude In a k=5 chiral spin liquid, t fluctuates about a saddle point where it corresponds to a constant magnetic field, 1/5 of a unit flux per plaquette (large N stabilizes the saddle) Celebrated Chern-Simons theory Fermions acquire fractional statistics with the angle θ:

20 Phase diagram Experiment Most likely CSL k=5: strong numerical evidence for CSL VBS 1 2 # of atoms per site

21 Non-Abelian chiral spin liquid 14 Place two species of fermionic atoms on each site, in the states 1 S0 and 3 P0. a, b = 1 S0, 3 P0 labels species This is non-abelian Chern-Simons SU(2)N theory!

22 Non-Abelian chiral spin liquid 14 Place two species of fermionic atoms on each site, in the states 1 S0 and 3 P0. a, b = 1 S0, 3 P0 labels species This is non-abelian Chern-Simons SU(2)N theory! Topological quantum computing?

23 Conclusions 15 SU(N) magnets are a useful theoretical tool due to the existence of the large N techniques SU(N) magnets can have phases going beyond the phases of the SU(2) magnets Nuclear spin of the alkaline earth atoms a perfect realization of the SU(N) spin A version of the SU(N) magnets particularly well suited to realization by the alkaline earth atoms forms chiral spin liquids, a state of matter with fractionalized excitations Possibility of the topological quantum computing with the SU(N) spin magnets??

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