How can we construct a microscopic
theory of Fermi Arc?
July 4th, 2011＠QC11
T.K. Ng
HKUST
What is a Fermi Arc?
In traditional theory of metal, electrons at zero temperature
occurs energy states with energies   EF
The states with energy   EF forms a closed surface = Fermi
surface
In High-Tc cuprates, it seems (from ARPES expt.) that down to very low
temperatures, electronic states in under-doped cuprates occupied a
non-closed Fermi surface = Fermi arc
What Shall I do in the following?
• I don’t have a theory!
• Phenomenological consideration of how a theory of
(T=0) Fermi arc can be obtained
• Different Phases in k-space – general considerations
based on GL type theory
• An approach based on Spinon-holon combination (?)
The theoretical problem: Is it possible to
construct (theoretically) a zero-temperature
fermion state where the electron Green’s
functions show Fermi-arc behavior?
ARPES expt. measure imaginary part of electron
Green’s function ImG(k,)nF()

Im G (k ,  ) 

Im (k ,  )


2
(   k  Re (k ,  ))  (Im (k ,  )) 2
Fermi-surface is usually represented by sharp pole with weight z in
ImG(k,=0) at T=0, i.e.

Im G(k ,  ) ~ zk (  Ek ( 0))
The theoretical problem: Is it possible to
construct (theoretically) a zero-temperature
fermion state where the electron Green’s
functions show Fermi-arc behavior?
Absence of Fermi surface 
1) Gap developed in that part of Fermi surface? (seems
natural because parent state is d-wave
superconductor?)

2) z k  0 ? Im G(k ,  ) ~ zk (  Ek ( 0))

3) the spectral function is broaden? (Im (k ,   0)  0)
Phenomenological considerations:
(1) leads to Fermi pocket if the Green’s function evolves
continuously in k-space


1
Proof: Let L(k ,  )  G (k ,  )
Thus a Fermi surface is defined by the line of points (I consider 2D here)

Re L(k , Ek  0)  0
(I shall come back to ImL later)
Let me assume that solution to the above equations exist at a
point k0  (k0 x , k0 y ) in k-space. We can form a segment of the Fermi
surface around the point by expanding around this point to obtain




L(k0  k ,   0)  0  k .L(k0 ,   0)  O(k 2 )  0


The condition k .L(k0 ,   0)  0 generates
a line segment in

the plane perpendicular to L(k0 ,   0) which forms part of
the Fermi surface.
Phenomenological considerations:
(1) leads to Fermi pocket if the Green’s function evolves
continuously in k-space


1
Proof: Let L(k ,  )  G (k ,  )
The process can be continued until the line ends on itself or hits the
boundary of the Brillouin Zone i.e. Fermi surface or Fermi pocket!
(cannot stop here)
This is true as long as G changes continuously in k-space
(gapping part of Fermi surface  distorted Fermi surface or Fermi
pocket)
Phenomenological considerations:
(2) & (3) are related (Kramers-Kronig relation)


 1  Im L(k ,  ' )
Re L(k ,  )  C (k )  P 
d '



  '
and

1
  Re L(k ,  ) 

zk  



  Ek

In particular, if Im L(k ,  ) ~   at small 

zk ~ Ek 
1
0
Ek 0
if <1
(or zf is nonzero only if >1)
marginal/non Fermi liquid state

(=0  Im (k ,   0)  0 )
Phenomenological considerations:
(2) & (3) are related (Kramers-Kronig relation)
Therefore, another possibility of Fermi arc is to have Green’s
functions with

Im L(k ,  ) ~  
where >1 at some part of Fermi surface (Fermi liquid state) and
<1 at other parts of Fermi surface (marginal Fermi liquid state)
Or a damping mechanism which gives
>1 and is effective only at part of the
momentum space.
Question: How is it possible if it is
realistic & not coincidental?
Proposal: phase separation in k-space
- different parts of k-space described by different “meanfield” state discontinuity in G possible!
To proceed, let’s consider a general phenomenological G-L type
theory framework
Recall that usual GL theory is characterized by an orderparameter  and the system is in different phases depending on
whether  is zero/nonzero.
Here we imagine a GL theory in k-space where the electron
Green’s functions are characterized by a parameter (k) that may
change when k changes, i.e.



G (k ,  )  G (k , ;  (k ))
with
Proposal: phase separation in k-space
- different parts of k-space described by different “meanfield” state
The parameter (k) is determined by minimizing a GL-type free
energy


 2 b( k )
 4

 2
1 d  

F ([ (k )])   d k a(k ) |  (k ) | 
|  (k ) | c(k ) |  k  (k ) | 
2
2


Notice that because of the gradient term, a state where (k) is nonuniform in k-space is generally characterized by domain wall, or other
types of non-uniform structures which are solutions of the G-L equation
(vortices, Skymions, etc. depending on the structure of  and dimension)
Proposal: phase separation in k-space
- different parts of k-space described by different “meanfield” state
(unrealistic example) (k) = order parameter measuring “strength” of
disorder potential see by electron
Assume (k) goes to zero in the nodal direction but becomes large when
moves to anti-nodal direction
 Electron spectral function broadened by disorder when we move away
from nodal direction!
Proposal: phase separation in k-space
- different parts of k-space described by different “meanfield” state
e.g. (k) = superconductor (pairing) order parameter


 2 b( k )
 4

 2
1 d  

F ([ (k )])   d k a(k ) |  (k ) | 
|  (k ) | c(k ) |  k  (k ) | 
2
2






a(k )  a(T  Tc (k )); b(k )  b; c(k )  c
Tc(k) is negative in the nodal direction, and becomes positive as one
moves to the anti-nodal direction(superconductor with multiple gaps)
However (k) is nonzero even in the nodal direction when we solve the
GL-equation because of “proximity effect” in k-space.
[Good model for students to study, probably do not describes pseudo-gap
state]
Proposal: phase separation in k-space
- different parts of k-space described by different “meanfield” state


e.g. a state where the Green’s function G (k , ; (k ))
property that
have the

G (k ,  ;   0) describes a marginal Fermi liquid state; &

G(k , ;  0 ) describes a normal Fermi liquid state
The parameter (k) is determined by minimizing a GL-type free
energy


 2 b( k )
 4

 2
1 d  

F ([ (k )])   d k a(k ) |  (k ) | 
|  (k ) | c(k ) |  k  (k ) | 
2
2


with
Proposal: phase separation in k-space
- different parts of k-space described by different “meanfield” state


e.g. a state where the Green’s function G (k , ; (k ))
property that
have the

G (k ,  ;   0) describes a marginal Fermi liquid state; &

G (k ,  ;   0) describes a normal Fermi liquid state
Notice that the “true” ground state is a Fermi liquid state in
this model because of proximity effect and the “Fermi arc”
state can only occur only at finite temperature (like the
superconductor model we discuss)
with
Theory of spinon-holon recombination
A model based on t-J model and the concept of spin-charge seperation
Idea: electron = spinon-holon bound pairs
- Fermi liquid state if spinon-holon are well bounded throughout
the whole Fermi surface
- Marginal Fermi-liquid state if some holons remain unbounded to
spinons
Difficulty we face: we can only get either one of the above states
Theoretically: spinon-holon bound state described by an equation of
form

L(k , Ek  0)  0
 Form Fermi pocket instead of Fermi arc
Theory of spinon-holon recombination
A model based on t-J model and the concept of spin-charge seperation
Marginal Fermi-liquid state if some holons remain unbounded to
spinons and are in an almost Bose-condensed state

Im G (k ,  )

“electron” pole
(MF) spinon pole
 poles become branch cuts in the Fermi-pocket
Theory of spinon-holon recombination
A model based on t-J model and the concept of spin-charge seperation
To achieve a Fermi arc state, we need part of the momentum space feels
the presence of unbounded holons

Im G (k ,  )

Im G (k ,  )


Theory of spinon-holon recombination
A model based on t-J model and the concept of spin-charge seperation
Question/challenge:
Can we obtain a microscopic theory with phaseseparation in momentum space?
Thank you very much!