Topology & Interaction:
QCPs in Helical Fermi Liquids
Cenke Xu
Harvard University
Quantum Critical Points in Helical Fermi Liquids
Outline:
1, Introduction, helical fermi liquids, Hertz-Millis
theory
2, QPT with time-reversal symmetry breaking
3, QPT with continuous symmetry breaking
4, other systems, general formalism
Quantum Critical Points in Helical Fermi Liquids
1 QCP of helical fermi liquids
1,
Example 1: edge state of 3d topological band insulator
Topologically protected edge
states, bulk gapped, all the
low energy
gy pphysics
y
at edge.
g Hsieh et.al.
Hi h
l
0904.1260
Chen et.al.
0904.1829
Quantum Critical Points in Helical Fermi Liquids
E
Example
l 2:
2 Rashba
R hb model
d l
Electric
El
i field
fi ld induced
i d d spini
orbit coupling in 2dEG,
Electric field becomes
magnetic field under
Lorentz transformation:
moving electrons see
Zeeman splitting.
Quantum Critical Points in Helical Fermi Liquids
Example 3, Iron-pnictides:
Fee
As
As
Orbital helical liquid.
dyz
dxz
dxz
Raghu, 2008
dyz
Orbital order important to lattice
structure distortion observed
p
y Orbital order
experimentally.
transition, similar to our analysis.
Quantum Critical Points in Helical Fermi Liquids
Symmetry:
B k
Broken
Quantum Critical Points in Helical Fermi Liquids
Hertz-Millis theory of QCP in fermi liquid
Example: FM transition in fermi liquid
Wilson-Fisher
FM order parameter can decay into particle-hole pairs
Couples most strongly
to the intersection,
where the particle-hole
excitation is softest.
softest
Quantum Critical Points in Helical Fermi Liquids
Hertz-Millis theory of QCP in fermi liquid
Example: FM transition in fermi liquid
Wilson-Fisher
FM order parameter can decay into particle-hole pairs,
i the
i.e.
th imaginary
i
i
partt off the
th self-energy.
lf
Quantum Critical Points in Helical Fermi Liquids
Conclusion:
l i
FM transition
i i in
i ordinary
di
f
fermi
i liquid
li id has
h z
= 3. Perturbations of g does not lead to divergence, i.e. it
is a mean field transition.
transition
The self-energy dominates the fermi liquid behavior,
behavior at
the quantum critical point there is no well-defined QP.
Same analysis
l i for
f all
ll other
h orders
d at zero momentum.
Quantum Critical Points in Helical Fermi Liquids
H li l fermi
Helical
f
i liquid:
li id spin
i orbit
bi coupling,
li topological
l i l band.
b d
Phase transition: interaction
The goal of this talk: two effects together
TBI w
with 5d transition
s o metal?
e ? Pesin,
es , Balents,
e s, 20099
Strategy: Assume the existence of the phase transition,
transition
discuss the nature of the universality class.
Quantum Critical Points in Helical Fermi Liquids
1a, quantum critical point with time-reversal and
1a
reflection symmetry breaking.
Using
i edge
d states as example
l off helical
h li l FL
The Dirac mass
Breaks T, Px, Py and opens a
gap in the spectrum, but keeps
the fermi surface gapless.
gapless
Quantum Critical Points in Helical Fermi Liquids
Take chemical potential to zero first.
first
3d Ising
relevant
Higgs-Yukawa model, can use 1/N expansion to study its
universality class.
N = 1? expansion hard to converge. 4-d
expansion may lead to useful results
Dirac mass gap drives the system into a quantum Hall
states with integer Hall conductance N/2.
N/2
Quantum Critical Points in Helical Fermi Liquids
Take chemical potential to nonzero.
nonzero
Uniform and static renormalization
Tune chemical potential
can drive
d i the
h transition
ii
Quantum Critical Points in Helical Fermi Liquids
Following the standard procedure,
the most singular correction to
self-energy
lf
off order
d parameter
t is
i
the imaginary part
Matrix element suppression:
Real part of self-energy is nonsingular,
so the Lagrangian still has z = 1 (in
contrast to z = 3 in usual case):
Quantum Critical Points in Helical Fermi Liquids
Self-energy behaves (almost) the same as the fermi
liquid,
q , the qquasiparticle
p
is well-defined at the
quantum critical point. (in contrast to non fermi
liquid behavior in usual case)
Quantum Critical Points in Helical Fermi Liquids
To show the suppression explicitly
explicitly, take isolated
patches on the fermi surface:
suppression
Quantum Critical Points in Helical Fermi Liquids
d + z = 3,
3 is this transition a Wilson-Fisher 3d Ising
transition?
3d Isingg universalityy class,, ggives order pparameter a
positive anomalous dimension:
Ag
g2 iss irrelevant,
e ev , [Ag
[ g2] = - ηη<00
Still have to check higher order loop
diagrams: Cenke Xu, arXiv:0908.2147
Quantum Critical Points in Helical Fermi Liquids
Bottom line: Order parameter and fermions interact very
weakly in the infrared limit (very different from HM).
Other systems: graphene with finite chemical
potential gamma matrices operate on two sublattices.
potential,
sublattices
Quantum Hall
3d Ising universality
Quanutm Spin
Q
S i Hall
H ll
3d O(3) universality
Quantum Critical Points in Helical Fermi Liquids
In real system,
system when chemical potential is large,
large we
need to consider higher order terms, which may align
spin
p alongg z direction.
Chen et.al.
Chen
et al
0904.1829
Fu
0908.1418
Will lead to over-damp of the order
parameter, i.e. z = 3.
Zhang et.al.
0901.2762
Quantum Critical Points in Helical Fermi Liquids
1b, quantum critical point with rotation symmetry
1b
breaking
3d XY
Order
O
d parameter
t couples
l to
t the
th Dirac
Di current,
t breaks
b k
lattice rotation, reflection symmetries.
Quantum Critical Points in Helical Fermi Liquids
1b, quantum critical point with rotation symmetry
1b
breaking, finite chemical potential
A static and uniform XY order moves the entire fermi
surface, without developing any extra spin polarization.
i.e. the
h static
i andd uniform
if
susceptibility
ibili vanishes.
ih
Quantum Critical Points in Helical Fermi Liquids
The system
y
does not have XY spin
p rotation symmetry,
y
y,
only has spin-space combined rotation symmetry,
therefore transverse mode and longitudinal mode can
b h
behave
diff
differently.
l
Longitudinal
L it di l mode
d
interacts strongly with
the fermi surface.
surface
Transverse
mode
T
d
interacts weakly with
the fermi surface.
surface
Quantum Critical Points in Helical Fermi Liquids
Explicit calculation:
A critical point with both z = 3 and z = 2 quantum
critical modes.
modes At low temperature,
temperature z = 3 modes
dominate the thermal dynamics.
Quantum Critical Points in Helical Fermi Liquids
Both z = 3 and z = 2 quantum critical points are above
or equal to the critical dimension d + z = 4, the
transition is expected to be mean field like.
like
Fermion self-energy also
dominated by longitudinal mode
non-fermi liquid at the quantum critical regime.
Another difference betweeen T and L modes:
T-mode mixes with charge
g fluctuation,, Bernevig
g 2004,, Raghu
g 2009,,
Quantum Critical Points in Helical Fermi Liquids
In the XY ordered phase,
phase Goldstone mode:
Directional
i i l dependent
d
d dynamical
d
i l exponent.
Quantum Critical Points in Helical Fermi Liquids
For general points on the fermi surface:
For special point:
Quantum Critical Points in Helical Fermi Liquids
M t generall form:
Most
f
pp-wave: Rashba model,, edge
g states of TBI
d-wave: orbitals in Iron-pnictides
Decompose XY order into Btransverse and B-longitudinal:
3d,, two transverse modes and one longitudinal
g
mode.
Cenke Xu, arXiv: 0909.2647
Quantum Critical Points in Helical Fermi Liquids
Summary:
Ordinary FM transition:
z = 3, mean field, non fermi liquid,
T breaking Ising transition in HFL:
T-breaking
z = 1, 3d Ising, fermi liquid
Rotation-breaking XY transition in HFL:
z = 2 and z = 3,
3 mean field
field, non fermi liquid
Direction dependent Goldstone mode.
Cenke Xu, arXiv:0908.2147, arXiv: 0909.2647
Deconfinement in SC
SC-FM-SC
FM SC Josephson Lattice
2 Fractionalization
2,
F ti li ti off SC-FM-SC
SC FM SC Josephson
J
h
lattice
l tti
SC
FM
Domain wall of the 1d Dirac mass gap localizes a Majorana
fermion zero mode, Liang Fu and C. L. Kane, PRL, 2009
j
1
j+x
2
The
h Majorana
j
zero modes
d
enable single charge tunneling.
Liangg Fu,, 2009
Deconfinement in SC
SC-FM-SC
FM SC Josephson Lattice
2a, 1d as warm up
Constraint
Constraint
}
}
Deconfinement in SC
SC-FM-SC
FM SC Josephson Lattice
Mapped to an ordinary 1d charge
charge-11 rotor model,
model by
tuning t2/U there is a KT transition between SF and MI
at integer filling, all the scaling dimensions are known.
Deconfinement in SC
SC-FM-SC
FM SC Josephson Lattice
2b 2d and
2b,
d deconfined
d
fi d phase
h
off Z2 gauge field
fi ld
2d Z2 gauge field has nontrivial gauge dynamics!
Deconfinement in SC
SC-FM-SC
FM SC Josephson Lattice
Consider a phase with large U, MI phase:
Deconfinement in SC
SC-FM-SC
FM SC Josephson Lattice
Ordinary
O
d
y Z2 ggauge
uge field:
e d:
By tuning h/K, there is a transition between confined /
deconfined phase transition.
In this SC-FM-SC lattice, h term is very nonlocal product
of Majorana fermions,
fermions therefore forbidden,
forbidden the Z2 gauge
field is always deconfining.
Whatt doest
Wh
d t deconfine
d
fi mean??
1, An electron will fractionalize into a charge-1 boson and
topological defect.
2, 3d XY* SF-MI phase transition.
Deconfinement in SC
SC-FM-SC
FM SC Josephson Lattice
2 extensions
2c,
t i
11, More general intra-SC
intra SC
tunneling, toric code model
with three magnetic field:
2 phase transitions inside SF phase,
2,
phase with small U.
U
Cenke Xu, Liang Fu, arXiv: 0911.1782