Supersymmetric Bose-Fermi Mixture

Novel Quantum Criticality and Emergent
Particles in Trapped Cold Atom Systems
Kun Yang
National High Magnetic Field Lab
and Florida State University
In Collaboration with
Yafis Barlas (NHMFL → UC Riverside)
Yue Yu (ITP, Beijing)
Hui Zhai (Tsinghua, Beijing)
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.
• 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)
• New universality class of superfluid-insulator
transition (SIT) in Bose-Fermi mixtures, and “highTc” 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)
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)
Model and Generators of Supersymmetry
Generators of
Generators of supersymmetry are fermionic operators!
Supersymmetric Bose-Fermi Mixture
then Hamiltonian
If we further have
grand Hamiltonian
Consequences of supersymmetry and its
breaking related to those of ordinary
symmetries, but with important differences.
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.
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 twocomponent boson system (KY and You-Quan Li 02);
only difference in statistics.
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.
explicitly broken for
grand Hamiltonian:
(Gapped, hole-like Goldstino).
(Otherwise energy lower than ground state).
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!
Fermion Spectral Function at q=0
Without Supersymmetry
With Supersymmetry
For detection of Goldstino without boson
BEC, see Shi, Sun and Yu PRA 10.
For realization of Wess-Zumino
model with (emergent) space time
SUSY, see Y. Yu and KY, PRL 10
• 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)
Observation of Superfluid-Mott Insulator
Transition (SIT) on an Optical Lattice
M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch 02
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.
Boson SIT in Bose-Fermi Mixtures
• 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.
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.
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.
Possible New Phase Diagrams in 3D
Quantum Critical Fluctuation Mediated
Fermion p-wave Pairing
• Brief introduction of quantum control in cold atom systems,
especially controlling interaction strength using Feshbach
• 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 BoseFermi 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)
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)
Same quantized Hall conductance (e*: atom “charge”):
Same quasiparticle charge: e*=2e*/2
There must be a quantum phase transition
from the IQH state to FQH state!
• Spin insulator.
• Quasiparticles are
spinless semions.
• One edge mode.
• Spin QH state.
• Quasiparticles are
spin-1/2 fermions.
• Two edge modes.
Chern-Simons Theory of Phases and Phase Transition
Conserved charges:
Constraint from CS term:
Mean-field approximation:
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:
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.
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!
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.
A Simpler (but More Interesting) Case
Starting with spinless fermionic atom IQH state at
ν = 1; turning on strong p-wave pairing interaction
 bosonic molecule FQH state at ν = ¼ !
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!
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:
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.
(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 deconfined = Laughlin quasiparticle; fractionalization! FQH phase.
•At transition: vison gap closes; Ising criticality.
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.
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 exotic emergent
particles 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.

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