1. How can we construct a microscopic theory of Fermi Arc? T.K. Ng HKUST July 4 th, 2011 ＠ QC11

2. 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 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

3. 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 (?)

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

5. 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?

6. Phenomenological considerations: (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) 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 The condition generates a line segment in the plane perpendicular to which forms part of the Fermi surface. (I shall come back to ImL later)

7. Phenomenological considerations: (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)

8. Phenomenological considerations: (2) & (3) are related (Kramers-Kronig relation) and In particular, if at small   if  <1 marginal/non Fermi liquid state (  =0  ) (or z f is nonzero only if  >1)

9. 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) 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?

10. 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 may change when k changes, i.e. To proceed, let’s consider a general phenomenological G-L type theory framework

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

12. 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

13. Proposal: phase separation in k-space - different parts of k-space described by different “mean- field” state T c (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

14. 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 describes a marginal Fermi liquid state ; & The parameter  (k) is determined by minimizing a GL-type free energy

15. 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 describes a marginal 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)

16. 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  Form Fermi pocket instead of Fermi arc

17. 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  poles become branch cuts in the Fermi-pocket

18. 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

19. 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!