Kun Yang National High Magnetic Field Lab and Florida State University In Collaboration with

1. Kun Yang National High Magnetic Field Lab and Florida State University In Collaboration with Yafis Barlas, Hsin-Hua Lai (NHMFL) Yue Yu (ITP, Beijing) Hui Zhai (Tsinghua, Beijing) Novel Quantum Criticality and Emergent Particles in Trapped Cold Atom Systems ---Realizing and Probing Supersymmetry

2. Quantum Control in Cold Atom Systems • Atoms/molecules with both Fermi and Bose statistics and their mixtures are available. • Optical lattice potential can be controlled through frequency and intensity of light; band structure/effective mass controllable. • “Magnetic field” controlled through rotation of optical lattice/trap (Coriolis’ force “ ≈ ” Lorentz force), or synthetic gauge field. • Interaction strength controlled via (real) magnetic field tuned Feshbach resonance.

3. Outline • Realizing supersymmetry in Bose-Fermi mixtures and its breaking via quantum control, and emergent Goldstino mode and its detection. (Y. Yu and KY, PRL 08; PRL 10; H.-H. Lai and KY, unpublished) • New universality class of superfluid-insulator transition (SIT) in Bose-Fermi mixtures, and “high-Tc” p-wave fermion pairing near SIT critical points. (KY, PRB 08) • Quantum Hall transitions near Feshbach resonances in rotating fermion gases, and emergent quantum particles obeying exotic statistics at the quantum phase transitions. (KY and H. Zhai, PRL 08; Y. Barlas and KY, PRL 11)

4. Supersymmetry in Bose-Fermi Mixtures • Bose-Fermi mixtures (like mixture of 6Li and 7Li) have been realized experimentally. • Through quantum control, one can tune parameters such that the bosons and fermions have same dispersion and interaction, to realize supersymmetry. • Supersymmetry always broken in a non-relativistic system, either spontaneously or explicitly, resulting in a fermionic Goldstone mode called Goldstino. • Goldstino detectable experimentally! Thus supersymmetry and its breaking can be studied in cold atom labs! (in addition to billion $ accelerators)

5. Model and Generators of Supersymmetry Grand canonical ensemble Hamiltonian: Generators of supersymmetry: Generators of supersymmetry are fermionic operators!

6. Supersymmetric Bose-Fermi Mixture If then Hamiltonian supersymmetric: If we further have grand Hamiltonian supersymmetric: Consequences of supersymmetry and its breaking related to those of ordinary symmetries, but with important differences.

7. Spontaneous Breaking of Supersymmetry Ground state of supersymmetric grand Hamiltonian has zero or at most one fermion!If more than one fermions: “Ground State”: Bosonic trial state: As a result, Ground state of two-component bosons spin-fully polarized (Eisenberg and Lieb 02; KY and You-Quan Li 02); thus same as single component boson.

8. Gapless Fermionic Goldstone Mode or Goldstino Zero momentum, zero energy state with one fermion and one fewer boson: Promoting supersymmetry generator to finite momentum: Goldstino mode at finite momentum: Quadratic dispersion identical to spin wave (which is a bosonic goldstone mode) of corresponding two-component boson system (KY and You-Quan Li 02); only difference in statistics.

9. Explicit Breaking of Supersymmetry and Gapped Goldstino Mode In order to have finite density of fermions, need to (minimally) break supersymmetry by a finite chemical potential difference: Effect similar to an magnetic field in a ferromagnet. Supersymmetry explicitly broken for grand Hamiltonian: Consequences: (Gapped, hole-like Goldstino). (Otherwise energy lower than ground state).

10. Experimental Detection of Goldstino Hole-like Goldstino is an excited state with one more boson and one fewer fermion; hard to couple to. However in the presence of Bose condensate, boson number NOT fixed in the first place; can couple to it by creating single fermion hole: Fermion hole state has finite weight on Goldstino; will show up as sharp peak in spectral function of fermion Green’s function!

11. Fermion Spectral Function at q=0 With Supersymmetry Without Supersymmetry

12. Noninteracting Boson-Fermion mixture Fermion Green’s function: Boson Green’s function: Goldstino Green’s function:

13. Perturbation studies: first order p p q Bose-Fermi contact repulsion p p p q Interaction vertex: p + + Simply shift the simple pole of the Goldstino Green’s function

14. p p p q p q Goldstino Self-energy Since we need two pairs of Boson-Fermion lines to define two external Goldstino Green’s function in the Dyson series, the next irreducible diagram that can give a finite self energy to the Goldstino mode comes at fourth order p p q p Dyson equation for the Goldstino Green’s function: p q = + p q +

15. Relaxation of Goldstino Cancel = + + Complex Goldstino self-energy : Spectral function has the Lorentzian form around Relaxation rate of the Goldstino mode

16. p p p p q Turn on Bose-Bose interaction p p p q p + Interaction vertex: Perturbation: first order + + + Simply shift the simple pole of the Goldstino Green’s function

17. More diagrams Cancel = + + + + + Expectations: • Complex Goldstino self-energy : • Relaxation rate vanishes when UBF = UBB

18. For realization of Wess-Zumino model with (emergent) space time SUSY, see Y. Yu and KY, PRL 10 For detection of Goldstino without boson BEC, see Shi, Sun and Yu PRA 10.

19. Outline • Realizing supersymmetry in Bose-Fermi mixtures via quantum control, and emergent Goldstino mode and its detection. (Y. Yu and KY, PRL 08; PRL 10) • New universality of superfluid-insulator transition in Bose-Fermi mixtures, and “high-Tc” p-wave pairing. (KY, PRB 08) • A quantum Hall transition near a Feshbach resonance in a rotating fermion gas, and emergent quantum particles obeying semionic statistics at the quantum phase transition. (KY and H. Zhai, PRL 08)

20. Observation of Superfluid-Mott Insulator Transition (SIT) on an Optical Lattice M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch 02

21. SIT Well Understood Using Bose Hubbard Model (M. P. A. Fisher, P. Weichman, G. Grinstein and D. S. Fisher 89) • Two universality classes for SIT: • Away from tips of Mott-Insulator lobes, either particles or holes condense; dilute Bose gas transition with z=2. • At the tips, particles and holes condense simultaneously; emergent particle-hole and Lorentz symmetry, z=1.

22. Boson SIT in Bose-Fermi Mixtures • Motivations • Experimentally, Bose-Fermi mixtures (like mixture of 6Li and 7Li) have been realized, including on optical lattices. • Theoretically, boson SIT best understood quantum phase transition (QPT). But historically QPT started with magnetic transitions in metals (Hertz 76), which turn out to be notoriously difficult due to presence of fermions. • The present case hybrid between the two; turns out to be better behaved than Hertz theory, and hopefully will shed light on QPT involving fermions in general.

23. Tracing out Fermions Generate effective boson interaction terms represented by Feynman diagrams: Compare with Hertz Theory: Origin of difference lie in the boson-fermion vertex; quartic (density-density coupling) in present case, cubic in Hertz case.

24. Analysis of Leading Singularity No singularity in quadratic terms; leading singularity a quartic term: Tree level RG equation: z=2 case: λ4 irrelevant; dilute Bose gas QCP stable. z=1 case: λ4 irrelevant at d=2 Wilson-Fisher fixed point; marginal at d=3; detailed loop analysis similar to Sachdev and Morinari (02) in a different context finds (3+1) dimensional XY QCP unstable, and no fixed point at weak coupling.

25. Possible New Phase Diagrams in 3D First Order

26. Quantum Critical Fluctuation Mediated Fermion p-wave Pairing

27. Outline • Brief introduction of quantum control in cold atom systems, especially controlling interaction strength using Feshbach resonace. • Realizing supersymmetry in Bose-Fermi mixtures via quantum control, and emergent Goldstino mode and its detection. (Y. Yu and KY, PRL 08; PRL 10) • New universality of superfluid-insulator transition in Bose-Fermi mixtures. (KY, PRB 08) • Quantum Hall transitions near Feshbach resonances in rotating fermion gases, and emergent quantum particles obeying exotic statistics at the quantum phase transitions. (KY and H. Zhai, PRL 08; Y. Barlas and KY, PRL 11)

28. Same quantized Hall conductance (e*: atom “charge”): Same quasiparticle charge: e*=2e*/2 Quantum Hall States in a Rotating Trap • Integer quantum Hall (IQH) states: fully-filled Landau levels, filling factor ν = m; fermions only. • Fractional quantum Hall (FQH) states: Laughlin type of states with ν = 1/m; even m for boson and odd m for fermion. • Starting with fermionic atom IQH state at ν= 2; turning on strong pairing interaction  bosonic molecule FQH state at ν = ½! (Haldane+Rezayi 04)

29. Spin QH state. Quasiparticles are spin-1/2 fermions. Two edge modes. There must be a quantum phase transition from the IQH state to FQH state! • Spin insulator. • Quasiparticles are spinless semions. • One edge mode.

30. Chern-Simons Theory of Phases and Phase Transition Conserved charges: Constraint from CS term: Mean-field approximation:

31. Mean-field Description of Phases Within mean-field treatment of statistical flux attached to particles, Quantum Hall Effect = Bose condensation of Chern-Simons bosons! (Zhang, Hanson and Kivelson 89) Atomic IQH phase: two condensates and both U(1) symmetries broken: Molecular FQH phase: one condensates and only one U(1) symmetry broken:

32. Mean-field Description of Phase Transition Bogliubov transformation: Integrate out massive field : Effective theory is that of 2+1D XY transition with Lorentz invariance! But coupling to CS field must be included.

33. Full Theory of QH Phase Transition Perform a similar transformation on the CS gauge field: Integrate out massive field : Theory describes condensation of emergent particles with zero charge, spin-1/2, and semion statistics!

34. Comments on the Effective Field Theory • Properties of emergent particle different from both the fundamental particles and quasiparticles of both phases; similar to what happens at “deconfined quantum critical points” (Senthil, Vishwanath, Balents, Sachdev, Fisher 04) • Properties of emergent particle expected from difference between two phases. • Same theory studied earlier in context of QH-Insulator transition on a lattice; nature of transition controversial: Large-N limit suggests 2nd order transition with θ’-dependent critical behavior (Wen and Wu 92, Chen, Fisher and Wu 93); while Pryadko and Zhang (94) found fluctuation-driven 1st order transition in certain parameter range. • Order of transition resolvable in cold atom system by measuring spin-gap using RF absorption. Not (yet) realizable in electronic systems.

35. Same quantized Hall conductance (e*: atom “charge”): Different quasiparticle charge: e* ≠ 2e*/4=e*/2; charge fractionalization! Different statics as well. Same number of edge modes; similar edge physics! A Simpler (but More Interesting) Case Starting with spinlessfermionic atom IQH state at ν= 1; turning on strong p-wave pairing interaction  bosonic molecule FQH state at ν = ¼ !

36. CSGL Theory of Phases and Phase Transition • Differences from previous case: • Only one Chern-Simons Gauge Field (coupled to charge). • Only one condensate in both phases! • Gauge field has an Anderson-Higgs mass (overwhelming Chern-Simons term), and plays no-role at the transition! • Reduces to an Ising model:

37. Theory identical to that of transition between atomic and molecular BECs without rotation (Radzihovsky+Park +Weichman; Romans+Duine+Sachdev+Stoof, PRLs 04); Ising transition! • Stable 2nd order transition; unlike transition between atomic and molecular BECs unstable due to collapse of system near Feshbach resonance. • Critical properties known accurately: z=1, υ≈0.63; neutral gap vanishing with exponent υz near critical point. • However: Looks like a conventional (Landau-type) transition; topological nature of phases involved and transition not explicit. • Change of topological properties not manifested, like change of torus degeneracy, charge fractionalization etc.

38. (Dual) Chern-Simons/Z2 Gauge Theory Conserved atom 3-current. σ =± 1: Z2 gauge field. φ: phase of molecular (half) vortex field. • Phases and Phase Transition: • Small β: Z2 vortices (or visons) condensed; molecular vortices confined; only atomic vortices present, which represent original charge-1, fermionic quasiparticles. IQH phase. • Large β: Z2 vortices (or visons) gapped; molecular vortices de-confined = Laughlin quasiparticle; fractionalization! FQH phase. • At transition: vison gap closes; Ising criticality.

39. Non-trivial checks: • Torus degeneracy changes from 1 to 4 due to the phase transition in Z2 sector. Topological nature of phases and phase transition explicitly manifested. • No additional edge modes! (See Y. Barlas and KY, PRL 11 for more details; simulation underway by Haldane and Rezayi) Story similar to, but simpler than Senthil-Fisher theory for spin-charge separation in cuprates.

40. Summary/Conclusions Trapped cold atom systems offer opportunities to study strongly correlated systems different from those encountered in electronic systems. We discussed examples of new universality classes of quantum phase transitions, especially exoticemergentparticles like semions and Goldstinos originally proposed in condensed matter and high energy physics respectively, but have not yet found concrete realization in these fields. Quantum manipulation allows us to generate such emergent particles, and study their properties in cold atom labs.