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Fermi surface change across quantum phase transitions
Phys. Rev. B 72, 024534 (2005)Phys. Rev. B 73 174504 (2006)
cond-mat/0609106
Hans-Peter Büchler (Innsbruck) Predrag Nikolic (Harvard)
Stephen Powell (Yale+KITP) Subir Sachdev (Harvard)
Kun Yang (Florida State)
Talk online at http://sachdev.physics.harvard.edu
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Consider a system of bosons and fermions at non-zero density, and N particle-number (U(1)) conservation laws.
Then, for each conservation law there is a “Luttinger” theorem constraining the momentum space volume enclosed by the locus of gapless single particle excitations, unless:
• there is a broken translational symmetry, and there are an integer number of particles per unit cell for every conservation law;
• there is a broken U(1) symmetry due to a boson condensate – then the associated conservation law is excluded;
• the ground state has “topological order” and fractionalized excitations.
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OutlineOutline
A. Bose-Fermi mixtures Depleting the Bose-Einstein condensate in trapped
ultracold atoms
B. Fermi-Fermi mixturesNormal states with no superconductivity
C. The Kondo Lattice The heavy Fermi liquid (FL) and the
fractionalized Fermi liquid (FL*)
D. Deconfined criticality Changes in Fermi surface topology
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OutlineOutline
A. Bose-Fermi mixtures Depleting the Bose-Einstein condensate in trapped
ultracold atoms
B. Fermi-Fermi mixturesNormal states with no superconductivity
C. The Kondo Lattice The heavy Fermi liquid (FL) and the
fractionalized Fermi liquid (FL*)
D. Deconfined criticality Changes in Fermi surface topology
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Mixture of bosons b and fermions f
(e.g. 7Li+6Li, 23Na+6Li, 87Rb+40K)
Tune to the vicinity of a Feshbach resonance associated with a molecular state
† †
† †
Conservation laws:
b
f
b b N
f f N
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Phases
k
²k
f Ãb
º
Detuning º
1 FS + BEC2 FS + BEC2 FS, no BEC
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Phase diagram
0b
0b 0b
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Phase diagram
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2 FS, no BEC phase
“molecular” Fermi surface
“atomic” Fermi surface
Volume = bN Volume = f bN N
2 Luttinger theorems; volume within both Fermi surfaces is conserved
0b
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Phase diagram
0b
0b 0b
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2 FS + BEC phase
“molecular” Fermi surface
“atomic” Fermi surface
Total volume = fN
1 Luttinger theorem; only total volume within Fermi surfaces is conserved
0b
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Phase diagram
0b
0b 0b
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1 FS + BEC phase
“atomic” Fermi surface
Total volume = fN
1 Luttinger theorem; only total volume within Fermi surfaces is conserved
0b
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OutlineOutline
A. Bose-Fermi mixtures Depleting the Bose-Einstein condensate in trapped
ultracold atoms
B. Fermi-Fermi mixturesNormal states with no superconductivity
C. The Kondo Lattice The heavy Fermi liquid (FL) and the
fractionalized Fermi liquid (FL*)
D. Deconfined criticality Changes in Fermi surface topology
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Tune to the vicinity of a Feshbach resonance associated with a Cooper pair
† †
† †
Conservation laws:
f f N
f f N
Mixture of fermions and f f
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chemical potential; "magnetic" field; detuningh
D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 96, 060401 (2006); M. Y. Veillette, D. E. Sheehy, and L. Radzihovsky, cond-mat/0610798.
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chemical potential; "magnetic" field; detuningh
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chemical potential; "magnetic" field; detuningh
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2 FS, normal state
minority Fermi surface
majority Fermi surface
Volume = N Volume = N
2 Luttinger theorems; volume within both Fermi surfaces is conserved
0
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1 FS, normal state
minority Fermi surface
majority Fermi surface
0N Volume = N
2 Luttinger theorems; volume within both Fermi surfaces is conserved
0
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Superfluid
minority Fermi surface majority Fermi surface
Volume Volume N N
1 Luttinger theorem; difference volume within both Fermi surfaces is conserved
0
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Magnetized Superfluid
minority Fermi surface majority Fermi surface
Volume Volume N N
1 Luttinger theorem; difference volume within both Fermi surfaces is conserved
0
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Sarma (breached pair) Superfluid
minority Fermi surface majority Fermi surface
Volume Volume N N
1 Luttinger theorem; difference volume within both Fermi surfaces is conserved
0
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Any state with a density imbalance must have at least one Fermi surface
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OutlineOutline
A. Bose-Fermi mixtures Depleting the Bose-Einstein condensate in trapped
ultracold atoms
B. Fermi-Fermi mixturesNormal states with no superconductivity
C. The Kondo Lattice The heavy Fermi liquid (FL) and the
fractionalized Fermi liquid (FL*)
D. Deconfined criticality Changes in Fermi surface topology
T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003).
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The Kondo lattice
+
Local moments fConduction electrons c
† †'K ij i j K i i fi fi fj
i j i ij
H t c c J c c S J S S
Number of f electrons per unit cell = nf = 1Number of c electrons per unit cell = nc
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†
Define a bosonic field which measures the
hybridization between the two bands:
Analogy with Bose-Fermi mixture problem: is the analog of the "molecule"
i i i
i
b c f
c
† †
† †
(Global)
(Lo
Main di
Conservation laws:
fference: second conse
1
1
rvation law
is so there is a U(1) gauge field.
al) c
cf f c c n
f
loc
b
l
f b
a
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DecoupledFL
If the f band is dispersionless in the decoupled case, the ground state is always in the 1 FS FL phase.
1Fk cV n
0b 0b
1 FS + BEC Heavy Fermi liquid (FL) Higgs phase
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2 FS + BEC Heavy Fermi liquid (FL) Higgs phase
A bare f dispersion (from the RKKY couplings) allows a 2 FS FL phase.
FL0b
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2 FS, no BEC Fractionalized Fermi liquid (FL*)
Deconfined phase
FL*
0b
The f band “Fermi surface” realizes a spin liquid (because of the local constraint)
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+
Local moments fConduction electrons c
† †' ,ij i j K i i fi H fi fj
i j i i j
H t c c J c c S J i j S S
Another perspective on the FL* phase
Determine the ground state of the quantum antiferromagnet defined by JH, and then couple to conduction electrons by JK
Choose JH so that ground state of antiferromagnet is a Z2 or U(1) spin liquid
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+
Local moments fConduction electrons c
Influence of conduction electrons
Perturbation theory in JK is regular, and so this state will be stable for finite JK.
So volume of Fermi surface is determined by(nc+nf -1)= nc(mod 2), and does not equal the Luttinger value.
At JK= 0 the conduction electrons form a Fermi surface on their own with volume determined by nc.
The (U(1) or Z2) FL* state
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OutlineOutline
A. Bose-Fermi mixtures Depleting the Bose-Einstein condensate in trapped
ultracold atoms
B. Fermi-Fermi mixturesNormal states with no superconductivity
C. The Kondo Lattice The heavy Fermi liquid (FL) and the
fractionalized Fermi liquid (FL*)
D. Deconfined criticality Changes in Fermi surface topology
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, cond-mat/0702119.
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s
Phase diagram of S=1/2 square lattice antiferromagnet
*
Neel order
~ 0
(Higgs)
z z
VBSVBS order 0,
1/ 2 spinons confined,
1 triplon excitations
S z
S
or
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Area 4 Area 8