How can we construct a microscopic theory of Fermi Arc?

T.K. NgHKUST

July 4th, 2011 ＠ QC11

The states with energy forms a closed surface = Fermi surface

What is a Fermi Arc?

In traditional theory of metal, electrons at zero temperature occurs energy states with energies FE

FE

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 electronGreen’s function ImG(k,)nF()

22 )),((Im)),(Re(

),(Im),(Im

kk

kkG

k

Fermi-surface is usually represented by sharp pole with weight z in ImG(k,=0) at T=0, i.e.

))0((~),(Im kkEzkG

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) ?

3) the spectral function is broaden?

))0((~),(Im kkEzkG

0

kz

)0)0,((Im k

Phenomenological considerations:

),(),( 1 kGkL

(1)leads to Fermi pocket if the Green’s function evolves continuously in k-space

Proof: Let

Thus a Fermi surface is defined by the line of points (I consider 2D here)

0)0,(Re kEkL

Let me assume that solution to the above equations exist at a point in k-space. We can form a segment of the Fermi surface around the point by expanding around this point to obtain

),( 000 yx kkk

0)()0,(.0)0,( 200 kOkLkkkL

The condition generates a line segment in the plane perpendicular to which forms part of the Fermi surface.

0)0,(. 0 kLk

)0,( 0 kL

(I shall come back to ImL later)

Phenomenological considerations:

),(),( 1 kGkL

(1)leads to Fermi pocket if the Green’s function evolves continuously in k-space

Proof: Let

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:

'

'

)',(Im1)(),(Re

d

kLPkCkL

(2) & (3) are related (Kramers-Kronig relation)

and

In particular, if at small

1

),(Re

kE

k

kLz

~),(Im kL

0~0

1

kE

kkEz

if <1 marginal/non Fermi liquid state

(=0 )(or zf is nonzero only if >1)

0)0,(Im k

Phenomenological considerations:

(2) & (3) are related (Kramers-Kronig relation)

Therefore, another possibility of Fermi arc is to have Green’s functions with

where >1 at some part of Fermi surface (Fermi liquid state) and <1 at other parts of Fermi surface (marginal Fermi liquid state)

~),(Im kL

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 “mean-field” state discontinuity in G possible!

with

Recall that usual GL theory is characterized by an order-parameter 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 maychange when k changes, i.e.

To proceed, let’s consider a general phenomenological G-L type theory framework

))(;,(),( kkGkG

Proposal: phase separation in k-space- different parts of k-space described by different “mean-field” state

242 |)(|)(|)(|2

)(|)(|)(

2

1)])(([ kkck

kbkkakdkF

k

d

Notice that because of the gradient term, a state where (k) is non-uniform 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)

The parameter (k) is determined by minimizing a GL-type free energy

Proposal: phase separation in k-space- different parts of k-space described by different “mean-field” state

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!

(unrealistic example) (k) = order parameter measuring “strength” of disorder potential see by electron

Proposal: phase separation in k-space- different parts of k-space described by different “mean-field” state

242 |)(|)(|)(|2

)(|)(|)(

2

1)])(([ kkck

kbkkakdkF

k

d

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]

e.g. (k) = superconductor (pairing) order parameter

ckcbkbkTTaka c )(;)());(()(

Proposal: phase separation in k-space

- different parts of k-space described by different “mean-field” state

with

describes a normal Fermi liquid state

e.g. a state where the Green’s function have the property that

))(;,( kkG

)0;,( kG

describes a marginal Fermi liquid state; &

);,( 0 kG

The parameter (k) is determined by minimizing a GL-type free energy

242 |)(|)(|)(|2

)(|)(|)(

2

1)])(([ kkck

kbkkakdkF

k

d

Proposal: phase separation in k-space

- different parts of k-space described by different “mean-field” state

with

describes a normal Fermi liquid state

e.g. a state where the Green’s function have the property that

))(;,( kkG

)0;,( kG

describes a marginal Fermi liquid state; &

)0;,( kG

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)

Theory of spinon-holon recombination

Difficulty we face: we can only get either one of the above states

Theoretically: spinon-holon bound state described by an equation of form

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

0)0,( kEkL

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

(MF) spinon pole “electron” pole

),(Im kG

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 kG

),(Im kG

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 phase-separation in momentum space?

Thank you very much!