Examples of Lifshitz topological transition in interacting fermionic systems

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1 Examples of Lifshitz topological transition in interacting fermionic systems Joseph Betouras (Loughborough U. Work in collaboration with: Sergey Slizovskiy (Loughborough, Sam Carr (Karlsruhe/Kent and Jorge Quintanilla (Kent

2 Outline Introduction to Lifshitz transition Dipolar fermions NaxCoO : motivation to consider fluctuations Interacting fermions in D: second order perturbation theory Interacting fermions in D: region of paramagnons Conclusions

3 What is a Lifshitz transition? Lifshitz JETP (1960 Topological transition of the Fermi surface/ no symmetry breaking possibilities: neck or pocket formation/collapsing If x is a controling parameter (distance from the QCP: E.g. in 3d: δω sin g x 5/ for both types

4 Some recent examples/proposals Electron-doped iron arsenic superconductors: Liu et al. Nature (010 Lifshitz transition in NaxCoO: Okamoto et al. PRB RC (010 Zeeman-driven Lifshitz transition in YbRhSi : Hackl, Vojta PRL (011 Lifshitz Transition in the Two Dimensional Hubbard Model; Kuang-Shing Chen, Zi Yang Meng et al. PRB (01

5 Dipolar fermions: Experimental setup Take advantage of dipolar interactions of the fermions at a finite magnetic field V ( R = d [1 3cos 3 R θ] d: dipole moment θ: angle between the vector that gives the relative position of the two dipoles and the external magnetic field J.Quintanilla, S. Carr and JB PRA RC(009 Max attractive: θ=0 Max repulsive: θ=π/ Null: θ=arccos(1/ 3

6 Dipolar Interaction Fourier transform of interaction for four values (0., 0.5, 1, of the anisotropy parameter a α a to justify the use of nearest neighbor interaction for α>: V ( k = V0 cos( k a

7 Hamiltonian reads: H = Model ( + + t c c + t c c + h. c + + i, l i+ 1, l i, l i, l V ci, lci, l+ 1ci, l+ i, l i, l 1 c i, l Parameters: µ / t t t / t V / Effect of interaction: ε * k = t cos( k t * cos( k µ Meta-nematic phase transition: t * = t + V Ω k cos( k n ( k

8 Meta-nematic phase transition

9 Why meta-nematic? Meta-magnetic example: Analogy to Sr3RuO7 First order transition for V>0 At V=0 no longer first order but continuous Lifshitz transition at t = t + µ / Quasi-1D D transition: opposite to the notion of confinement Density of states effect

10 More on meta-nematic By comparing energies we find the true position and size of jump. The jump has a BCS-like form: t * = exp( 1/ bv c S. Carr, J. Quintanilla and JB, PRB (010

11 Landau Theory Renormalized transverse hopping effective order parameter. Similarity to the well known binding energy of a Cooper pair in the presence of arbitrarily weak attractive interaction! Effect of interactions the continuous Lifshitz transition becomes first order.

12 Landau Theory Defining: x = ( t x t 0 QPT * = ( t = t t t QPT QPT / t µ / / t Using the logarithmic divergence of the DOS at van Hove energies at the points ( 0, ± π Requiring the correct physics in absence of V, we obtain the expansion: 1 E x [ln x ] x( x * t = exp( 1/ bv c x 0 av ln x Vbx ln x

13 Collating the results of metanematic transition: Partial Phase Diagram

14 Finite temperature At finite T reduced effect of van Hove singularities Critical end point on metanematic transition is regular: * t ( T T c 1/ S.Carr et al PRB (010

15 Finite-T phase diagram Density Wave dominates the phase diagram but there is still areas where the meta-nematic transition can be seen (even with an exponentially small Tc

16 NaxCoO: motivation for fluctuations Quasi-D metal Alternately stacked CoO and Nax layers (conducting/reservoir In most cases Na is randomly distributed little influence on conducting layers Until now: Na-rich phase (x ~ 0.7 Curie-Weiss metal Na-poor phase (x ~ 0.3 Pauli paramagnetic metal tg band from 3d Co orbitals responsible for most electronic properties splits into a1g and degenerate e g D.Yoshizumi et al. (007 M.L. Foo et al. (004 G. Lang et al. (008 M.Yokoi et al. (005

17 Okamoto,Nishio, Hiroi (010 Lifshitz transition?

18 Specific heat and thermoelectric power

19 explanation by Okamoto et al.: non-rigid band/ resisting occupation! But the Lifshitz transition occurs in the regime of strong magnetic fluctuations: need to understand deeper the coupling to magnetic fluctuations

20 Interacting fermions in D Consider interacting fermions in D with short-range interaction U and with dispersion relation: where and we use the normalization for ε ( k = ε ( k + ReΣ( k; kf F1 =1 0 = k µ = ε ( k = 0 0 m = 1 0 First: to get a feeling use second order perturbation theory (SOPT neglecting the scattering between the two Fermi surfaces.

21 d qdω Self-energy Σ( kf, Ω = 0 = U G( k q, iω χ0( q, iω 3 F + (π χ + 0 = χ χ 01 0 susceptibility of free electrons (Lindhard function Units of energy: k m F1 (at µ = 0, momentum : kf1 (at µ = 0 Question: effect of the formation of the pocket? Σ( k F U, iω = 0 k F log 8π Λ k F Σ( 1, iω = k F 0 : negligible dependence on pocket size Luttinger theorem is respected: 1 1 F n = ( k F k π

22 To locate the chemical potential we need to consider: ε ( kf + δσ( kf, ω = 0 = ε ( kf1 + δσ( kf1, ω = 0; kf or: kf U Λ log 4π kf 1 = υf 1 ( k F1 1 = µ 0 Non-divergent terms containing k F are effectively included in Λ. The actual chemical potential (corrected with the Hartree term is : µ phys = µ U From the form of the transcendental equation: there is a region with 3 solutions for kf 4π k F

23 µ max U = 16π Λ 8π exp( U 1 k F 4π = Λ exp( U 1 Which solution wins? Integrate the grandcanonical potential: d Ω = n dµ phys

24 Exponentially small jump in pocket size for small U Fermi-liquid picture valid ( quasiparticle weight Z independent of kf and damping Exponentially small jump in density Regime of paramagnons By increasing U and using rings + ladders: (known non - analytic term ω logω, ( 1, (, ( 1, (, ( ω χ ω χ ω χ ω χ ω i q U i q U i q U i q i q V + = W. F. Brinkman and S. Engelsberg Phys. Rev. 169, 417 (1968 M. T. Beal-Monod, S- K. Ma, D. R. Fredkin Phys. Rev. Lett. 0, 99 (1968 P. W. Anderson and W. F. Brinkman Phys. Rev. Lett. 30, 1108 (1973 Moriya (1973, Doniach and Engelsberg Phys Rev Lett (1968

25 For small momentum transfer vertex corrections approximately cancel with corrections of weight Z (Hertz and Edwards 1973 Then, self-energy reads: and the Stoner criterion:, (, ( (, ( 3 ω ω π ω i q V i i q k G q d d U i k Ω + + = Ω Σ Λ Ω = Σ log 8 0, ( c k c c k k U k F F F F eff F υ π 1 1/ 1 F U St υ π + =

26 Following same line of thought with SOPT: three possible solution of balance equation one wins and the formation of pocket is 1 st order.

27 Some consequences: Jump in the specific heat before FM By increasing U and reaching Stoner then magnetism drives the Lifshitz 1 st order.

28 Conclusions Lifshitz transitions although very sensitive seem to have been realized in physical systems. Other systems have been also reported lately (heavy fermions, pnictides etc. Dipolar fermions in anisotropic optical lattices show clear meta-nematic transition (discontinuous. Main reason: effect of Van Hove singularities. NaxCoO shows a discontinuous Lifshitz transition accompanied by enhanced magnetic fluctuations. More systematic way to understand effects of fluctuations and FS reconstruction. Paramagnetic fluctuations + special dispersion relation may lead to a discontinuous appearance of a new pocket.