Singularites in Fermi liquids and the instability of a ferromagnetic quantum-critical point

Transcription

1 Singularites in ermi liquids and the instability of a ferromagnetic quantum-critical point Andrey Chubukov Dmitrii Maslov lorida University of Wisconsin Catherine Pepin Jerome Rech SPhT Saclay Dima Kveshchenko KIAS workshop, July 19, 006 University of North Carolina

2 Interest to isotropic models of fermions interacting with low-energy, long-wavelength bosons ermions with ε (k) = v (k - k Landau-overdamped bosons with ) χ (q, Ω) q + γ 1 Ω q Residual spin-fermion coupling g + (c ck+ q bq k + h.c) The most known example a ferromagnetic quantum-critical point

3 The case of a ferromagnetic (Stoner) instability for itinerant electron system erromagnetic phase ermi liquid What is the critical theory?

4 Interest to isotropic models of fermions interacting with low-energy, long-wavelength bosons ermions with ε (k) = v (k - k Landau-overdamped bosons with ) χ (q, Ω) q + γ 1 Ω q Residual spin-fermion coupling g + (c ck+ q bq k + h.c) Other examples: interaction with gauge fields, an instability towards nematic ordering, quarks interacting with gluons,.. P.A. Lee, Nagaosa, Wen; Ioffe & Larkin, Kee & Kim, Lawler & radkin, A.C. & Schmalian The problem is particularly interesting in D =

5 Three issues: What happens with fermions near criticality? (fluctuations are strong and destroy a ermi liquid behavior. What replaces a ermi liquid?) Whether the Landau-overdamped form of the bosonic propagator 1 Ω χ (q, Ω) q + γ survives? (is a second order continuous transition protected against fluctuations?) q Is there a superconductivity near a QCP?

6 What happens with fermions near criticality? (fluctuations are strong and destroy a ermi liquid behavior. What replaces a ermi liquid?) Not an academic issue: Essential for the calculation of the specific heat near a quantum-critical point C L (T) m * T m* diverges at QCP

7 Whether the Landau-overdamped form of the bosonic propagator 1 Ω χ (q, ) q γ survives? (is a second order continuous transition protected against fluctuations?) Ω + q erromagnetic transition is often 1 st order erromagnetic phase ZrZn 1 st order pressure

8 Is there a superconductivity near a QCP? UGe

9 ermions at QCP Eliashberg-type theory (D=) Σ (k, ω) = Σ( ω) + no vertex corrections Altshuler, Ioffe, Millis, Kopietz; Metzner et al; Morr, A.C., fermionic self-energy Σ /3 1/3 ( ω ) = (i ω) ω0 ω = π g E g non ermi liquid at QCP G -1 (k, ω) = i( ω + Σ( ω)) - v (k - k ) i( ω /3 ω 1/3 0 - v (k - k )) G(r, t = 0) G 0 (r, t = 0) / r ~ 1/r

10 Issue of (still) current interest: /3 Is the Eliashberg theory with Σ ( ω) ~ ω stable? Earlier perturbative calculations yes Altshuler, Ioffe, Millis, Metzner, A.C. Recent calculations based on bosonization no (argued that Eliashberg theory is broken below some ) ω min radkin, Lawler Our results: or a charge QCP, Eliashberg theory survives, but controllable calculations are only possible if one extends the model to a large number of fermionic flavors N or SU() symmetric spin case, Eliashberg theory of a ferromagnetic QCP is internally unstable

11 A conventional reasoning for the Eliashberg theory: If G -1 (k, ω) ~ (i ω) /3 v (k - k ) and Then, for the same frequency, χ (q, ω) q + γ 1 ω q typical fermionic momenta typical bosonic momenta /3 k - k ~ ω fermions are much q ~ ω 1/3 faster than bosons (much larger) Quite generally, in this situation, Migdal theorem must be valid

12 Σ /3 1/3 ( ω ) = (i ω) ω0 ω = π g E ermions must remain fast up to ω ~ ω 0 α = g E << 1 If α << 1, vertex corrections and k-dependent self-energy are small Static vertex in the limit of vanishing momentum α 1/ = O( )

13 This is not enough. The theory must satisfy Ward identities, associated with the conservation of the total number of particles, and the total spin χ tot (q = 0, Ω 0) = 0 q, Ω m (q, ) Ω χ0 Ω = 1 = 0 when q = 0 π v q Ω + However, Eliashberg susceptibility m (q, ) Ω χ0 Ω = 1 0 when q = 0 π ( ( )) v q Ω + Σ Ω + Vertex corrections are NOT small when q=0, and Ω is nonzero 0, Ω 1/ 3 δ g Σ ( ω) ω0 = ~ g ω ω

14 inite momentum, zero frequency Q, 0 α 1/ = O( ) vertex corrections are small inite frequency, zero momentum 1/3 0, Ω δ g Σ ( ω) ω vertex corrections 0 = ~ g ω ω are large inite frequency, finite momentum v Q ~ Σ( ω) δ g = g O(1)

15 As a result, the theory is NOT under control: /3 1/3 Σ ( ω) = Σ ( ω) ~ ω ω0 ~ Σ1 Σ ( ω) ~ 3 Σ and so on This is dangerous Example: 0, Ω δ g g Σ ( ω) = ω ~ ω0 ω 1/3 In order by order perturbation theory δ g g ω= 0 = = infinity

16 Another powerful way to do calculations is bosonization. Lawler & radkin 1/3 The bosonization result in D: G(r, t = 0) = G (r, t = 0) (-r/r ) 0 is consistent 0 e /3 with divergent series for the self-energy (i.e., ω does not survive) If /3 Σ( ω) ~ ω G(r, t = 0) G 0 (r, t = 0) / r ~ 1/r One detail bosonization assumes that the curvature of the ermi surface is irrelevant

17 The curvature of the ermi surface is crucial to the physics in D one can extend the theory to N fermionic flavors Σ /3 1/3 ( ω) ~ ω ω0 * ((Log N)/N) Altshuler et al, Rech et al The factor 1/N comes from the curvature of the ermi surface Σ ( ω) ~ Σ1 ( ω) Without the curvature, we would still have, even at N >>1

18 One can improve bosonization to include the curvature Khveshchenko & A.C. Result: G(r) = G 0(r) e -Z(r) Without curvature With curvature Z(r) 1/3 ~ r Z(r ) ~ log r G(r, t η = 0) 1/r at the largest r, => small frequency behavior of the Green s function is consistent with Σ( ω) ( ω) /3 (perturbation theory does not diverge)

19 Intermediate conclusion or D fermions, interacting with a given χ (q, Ω) q + γ 1 Ω q Σ( ω) ( ω) /3 is very likely the exact solution. This was the effect from bosons on fermions. What about the opposite effect? Does the bosonic propagator preserve its form when one includes the corrections due to fermions?

20 A conventinal answer fermions only account for Landau damping of bosons Then, once the fermions are integrated out, d ω 4 S = d Q d ω [Q + ] η + bη Q +... Z=3, Dcr =1 (Dcr + Z =4) Bosonic propagator is not affected by fluctuations in D >1

21 In reality, ferromagnetic transition is 1 st order ZrZn erromagnetic phase pressure 1 st order pressure

22 d ω 4 S = d Q d ω [Q + ] η + bη Q +... Verify this Apply a magnetic field and use the exact formula for the thermodynamic potential in the Eliashberg theory Ω = Ω free + 1 Π d q d Ω (π ) + Π ( log (1+ gπ ) + log (1+ g )) G + G -, Π ++ G + G = = Betouras, Efremov, A.C. Maslov & A.C. χ = Ω H -

23 Diagrams for χ (q) in the Eliashberg theory δχ s pin (Q) = Q 3/ p 1/ Comes from processes in which bosons are vibrating at fermionic frequencies (opposite to Migdal processes)

24 Corrections: go outside Eliashberg theory δχ () spin (Q) Q 3/ p 1/ * log N N Same corrections as for the self-energy Σ log N ( ω) ~ Σ1 ( ω) * N

25 Physics -- Landau damping χ (q, Ω) Ω 1 q Long-range dynamic interaction between fermions Ω/r Ω q A magnetic field changes into, i.e. restores analyticity Ω q + (ih) As a result, the derivative with respect to H is singular χ or the charge susceptibility, measures the response to the change of the chemical potential. Ω / q survives this change, hence the charge susceptibility should be regular.

26 Charge susceptibility + = 0 + = 0 δχ charge (Q) Q = 0(q )

27 χ Back to the static spin susceptibility 3/ 1/ (Q) = (Q Q p ) spin χ spin (Q) Q/p Internal instability of z=3 QC theory in D=

28 Another way to see the instability of a ferromagnetic QCP is to calculate the q=0 susceptibility at a finite T χ 1 T T0 (q = 0, T) = - log T T 0 Or calculate the thermodynamic potential at a finite magnetization Ξ (M) = b M 4 - a M 3 Maslov & A.C., Belitz, Kirkpatrick & Vojta

29 What can happen? χ spin (Q) Ω Ω Q/p M a transition into a spiral state a first order transition to a M Belitz, Kirkpatrick, Vojta, Maslov and A.C., Rech, Pepin, A.C. Both scenarios are possible

30 The instability of the erromagnetic QCP can be also understood in the frameworks of the Hertz-Millis-Morya theory How one obtains Hertz theory Stage one: obtain the spin-fermion model (fermions, collective spin fluctuations, and interacton) (integrate out high energy fermions) spin channel χ 0 (q) q 1 + ξ - low-energy fermions spin-fermion interaction static collective spin fluctuations

31 Stage two: obtain the theory for collective modes (integrate out low energy fermions) Hertz theory d - Ω 4 S = d Q d Ω [Q + ζ + ] η + bη Q +... Ω = Landau damping q Ω b = const + ( v q) vertex is non-analytic in 3 Ω + momentum and frequency

32 = Ω ( Ω + v q) n-1 fermions produce long-range dynamic interaction between collective modes 3/ This dynamic interactions fits back into term in the static susceptibility q

33 The non-analyticity of the spin susceptibility is NOT specific to a ferromagnetic quantum criticality The same effect is already seen in weak coupling perturbation theory, the only difference is that 3/ q is replaced by q as the effect of fermionic self-energy is weak at small coupling. T=0, finite Q δ χ spin (Q) = 4m 3π m U(p 4π ) Q p Q=0, finite T δ χ spin (T) = m π m U(p 4 π ) T E

34 Why only U(p) matters? In perturbation theory, only bubbles with the same spin projection are present. The k bubble is singular and is affected by the magnetic field Π (q p m, ω) = 1- π q - p p + q - p p ω - v p 1/ ω/ p q singularity In the presence of H, p p 1 Ω ~ T ω d q ω p q p q ~ T ω log (H + ω ) Ω χ(t) = - H T

35 or a constant, but arbitrary interaction U -1 δχ (T) T log (1+ Γ) - 1 Γ + Γ δχ Γ = -1 U N(0) 1- U N(0) (T) T Γ at QCP at two - loop order -1 δχ (T) T log ζ ( T log T at QCP) or arbitrary U (q), near a ferromagnetic QCP δχ -1 (T) = - χ -1 Pauli T E ( log (1 + f ) + O(f )) o,s i, s Instability of a ferromagnetic QCP in D Component of the Landau function

36 What would happen if δχ (q, T) was positive? s Self-consistent calculations: χ (q, Ω) (q α + Ω / -1 q ) α =1 χ (q, ω) ( q log q + ω / q ) -1 Σ ( ω) ω log( log ω ) Marginal ermi liquid in D=

37 Conclusions The QCP towards charge (e.g., nematic) ordering is stable in D. The /3 fermionic self-energy is Σ ( ω) ~ ω,, and the theory is controllable at N >>1. A ferromagnetic Hertz-Millis critical theory is internally unstable in D= (and also in D = 3) static spin propagator is negative at QCP up to Q~ p 3 free energy has an term M Either an incommensurate ordering, or 1st order transition to a ferromagnet

38 Collaborators Jerome Rech (Saclay) Catherine Pepin (Saclay) Dima Khveshchenko (U. of North Carolina) Mitya Maslov (U. of lorida) Andy Millis (Columbia) Joseph Betouras (St. Andew) Dima Efremov (TU Dresden) THANK YOU!