Effective Mass Close to a Quantum Critical Point

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C 2003) Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 16, No. 6, December 2003 (°

Effective Mass Close to a Quantum Critical Point I. Grosu1 and M. Crisan1 Received February 25, 2003

The effective mass in heavy fermion materials is calculated, taking into consideration both the antiferromagnetic and ferromagnetic fluctuations. The temperature and the proximity of a quantum critical point effects are considered. The effective mass diverges at the quantum critical point and in the zero temperature limit. KEY WORDS: heavy fermion materials; antiferromagnetic and ferromagnetic fluctuations; quantum critical point.

1. INTRODUCTION

2. EFFECTIVE MASS

The quasiparticle mass m∗ , in a Fermi liquid, is renormalized by interactions [1,2]. Recent measurements on heavy fermion compounds showed that the quasiparticle mass diverges in the approach to a quantum critical point (QCP) [3], a point associated to a quantum phase transition, a transition driven by nonthermal fluctuations. Some of these compounds are non-Fermi liquid systems that exhibit unusual temperature dependencies in their low temperature properties. The electron specific heat divided by temperature, a quantity proportional to the effective mass, shows a singular (logarithmic) temperature dependence. Close to a zero temperature ferromagnetic instability, it was shown [4] that the quasiparticle mass diverges. Example of heavy fermion compounds which are close to ferromagnetic order are Ux Th1−x Cu2 Si 2 [5], Ni x Pd1−x [6], Ce Pd0.05 Ni 0.95 [7]. These compounds present also a non-Fermi liquid character. The quasiparticle mass also diverges in the approach to an antiferromagnetic quantum critical point [3,8,9]. These divergencies cause a breakdown of the Fermi liquid character, because the effective interaction between electrons is very strong. In this paper we analyze the effective mass in the presence of the antiferromagnetic and ferromagnetic fluctuations, close to a QCP.

The effective mass, m∗ , that can appreciably increase due to interactions with low energy excitations, is given by [2,10]: ¯ ¯ ∂ m∗ =1− Re6(ω)¯¯ (1) m ∂ω

1 Department

ω→0

The self-energy 6( p, E iωn ), in the lowest order, is: Z d pE0 g2 X G ( pE0 , iωm) 6( p, E iωn ) = − β m (2π )3 × χ ( pE − pE0 , iωn − iωm)

(2)

where g is the electron-spin fluctuations coupling constant, G( pE0 , iωm) the finite temperature Green’s function, β = 1/T, and χ ( pE − pE0 , iωn − iωm) the spin fluctuations propagator. Using the spectral representation: Z ∞ S( pE0 , E) 0 dE · (3) G ( pE , iωm) = iωm − E −∞ Z ∞ r ( pE − pE0 , E 0 ) dE 0 · χ ( pE − pE0 , iωn − iωm) = iωn − iωm − E 0 −∞ (4) and the sum over Matsubara frequencies: 1 1 1X · β m iωn − E i (ωn − ωm) − E 0 1 tan h (β E/2) + cot h (β E 0 /2) =− · 2 iωn − E 0 − E

of Theoretical Physics, University of Cluj, 3400 Cluj,

Romania.

(5)

981 C 2003 Plenum Publishing Corporation 0896-1107/03/1200-0981/0 °

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Grosu and Crisan

after analytic continuation, and with qE = pE − pE0 , one obtains: Z ∞ Z Z ∞ d pE0 6(ω) ' g 2 dE dE 0 · S ( pE0 , E) (2π)3 −∞ −∞ · r (qE0 , E 0 ) ·

f (E) − 1/2 E + E 0 − ω − iδ

(6)

In the low frequencies limit (ω → 0), using:

where f (E) is the Fermi function, and:

ψ(a + x) = ψ(a) + x · ψ 0 (a)

1 E E 0) r (q, E E 0 ) = − · Imχ (q, π

(7)

After the average over the Fermi surface, the real part of the self-energy becomes: Z ∞ Z g 2 N(0) 2 pF Re6(ω) = − dq · q dE 2π pF2 0 −∞ Z ∞ f (E) − 1/2 (8) dE 0 Imχ (q, E 0 ) · × E + E0 − ω −∞ Using the property: Imχ(q, −E 0 ) = −Imχ(q, E 0 ), Eq. (8) will be rewritten as: 2

g N(0) Re6(ω) = − 2π pF2

Z

2 pF

Z

∞

dq · q

0

dE

Z

0

∞

where N(0) is the density of states. The integral over E will be calculated as follows: · ¸µ Z ∞ 1 1 dE f (E) − I = Re 0−ω 2 E + E −∞ ¶ ½Z D f (E) 1 dE = Re − E − E0 − ω E − (ω − E 0 ) −D ¾¯ Z D ¯ f (E) ¯ − dE (10) 0 E − (ω + E ) ¯ D→∞ −D Using now the result given by Fibich and Horwitz [11]: Z

we obtain:

½ ¶¾ µ 1 iβ E 0 βω · Re iψ 0 − π 2 2π ½ ¶¾ µ iβ E 0 βω 0 1 · Re iψ + =− π 2 2π

I=

Re6(ω) =

Z g 2 N(0)ωβ 2 pF dq 2π 2 pF2 0 Z ∞ dE 0 Imχ (q, E 0 ) ×q 0

f (E) dE E−Ä −D  ³   ψ 12 − = ³  ψ 1 + 2

0

¶¾ µ ½ 1 iβ E 0 + × Re iψ 0 2 2π Using now: ¶¾ ¶ µ µ ½ 1 iβ E 0 1 iβ E 0 + = −Imψ 0 + Re iψ 0 2 2π 2 2π

iβÄ 2π

´ ´

+ ln + ln

³ ³

2π βD 2π βD

´ ´

+

iπ , 2

Ä2 > 0

−

iπ , 2

Ä2 < 0

(11)

(15)

(16)

the effective mass becomes:

Z g 2 N(0)β 2 pF m∗ dq = 1+ m 2π 2 pF2 0 Z ∞ dE 0 Imχ (q, E 0 ) ×q × Imψ 0

iβÄ 2π

(14)

and the effective mass is given by: Z g 2 N(0)β 2 pF m∗ dq = 1− m 2π 2 pF2 0 Z ∞ dE 0 Imχ (q, E 0 ) ×q

0 D

(13)

Now, Eq. (9) becomes:

½ ¶¾ µ iβ E 0 0 1 × Re iψ + 2 2π

0

¸ · 1 dE f (E) − 2 −∞ ¶ µ 1 1 − (9) × 0 E+ E −ω E − E0 − ω × Imχ (q, E 0 ) · Re

where Ä = Ä1 + Ä2 , and ψ(x)—the digamma function, one gets: ¶ ½ µ iβω 1 iβ E 0 − + I = Re ψ 2 2π 2π ¶¾ µ 0 iβω 1 iβ E − − (12) −ψ 2 2π 2π

µ

1 iβ E 0 + 2 2π

¶

The T → 0 limit is obtained using: ¶ µ iz 2π T 1 + '− , z À πT Imψ 0 2 2π T z

(17)

(18)

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Effective Mass Close to a Quantum Critical Point and in this limit: Z g 2 N(0) 2 pF m∗ dq = 1− m π pF2 0 Z ∞ dE 0 Imχ (q, E 0 ) ×q E0 0

2.2. Finite Temperature Effects

(19)

Usually, the susceptibility χ(q, E0 ) has the form: a χ (q, E 0 ) = (20) b − icE 0 and: Z ∞ πa dE 0 (21) · Imχ (q, E 0 ) = 0 E 2b 0 On the other hand: a (22) χ (q, E 0 = 0) ≡ χ(q, 0) = b and the effective mass, in the zero temperature limit, becomes: Z g 2 N(0) 2 pF m∗ =1− dq · q · χ(q, 0) (23) m 2 pF2 0 2.1. Effective Mass Close to a Quantum Critical Point (QCP)

g 2 N(0) m∗ =1− χQ m 2 pF2

Z

2 pF 0

dq · q η + Aq2

The effective mass at finite temperatures is obtained using Eqs. (24) and (17): Z ∞ Z m∗ g 2 N(0)β 2 pF dq · q dω = 1+ m 2π 2 pF2 0 0 ¶ µ χ QCq ω iβω 0 1 · Imψ + × (η + Aq2 )2 + Cq2 ω2 2 2π (28)

2.2.1. Antiferromagnetic Case In this case Cq is independent of q: Cq = C

(26)

With χ Q assumed to be: χ Q = −2 N˜ (a, f) (0), we get: ¯ ¯ µ ¶2 ¯ ¯ m∗ A 1 g 2¯ ¯ ˜ N(0) N(a, f) (0) ln ¯1 + (2 pF ) ¯ = 1+ m 2A pF η (27) where index a and f correspond to antiferromagnetic and ferromagnetic fluctuations respectively. In the η → 0 limit, the effective mass is logarithmic divergent.

(29)

and Eq. (28) becomes: g 2 N(0)βχ QC m∗ = 1+ m 2π 2 pF2 ×

The spin fluctuations propagator is assumed to be of the form [12,13]: χQ (24) χ (q, ω) = η + Aq2 − iCq ω Here η measures the distance to the QCP. (In a QCP, η = 0). In the case of the antiferromagnetic fluctuations Cq = C is independent of q, while in the case ˜ However, in of ferromagnetic fluctuations Cq = C/q. the T = 0 limit one has to use the static form of the susceptibility which is, for both antiferromagnetic and ferromagnetic fluctuations, given by: χQ (25) χ (q, 0) = η + Aq2 The effective mass becomes:

983

Z

2 pF

Z

∞

dq · q

0

ω · Imψ 0 2 (η + Aq )2 + C 2 ω2

µ

dω 0

1 iβω + 2 2π

¶

(30) In order to evaluate the ω-integral, we will use the following approximation: ¶ µ iβω 0 1 + Imψ 2 2π  q 2 − 7ζ (3)βω , ω < π ≡ ωp π β 7ζ (3) q (31) = 2π π 2  − , ω> ≡ ωp βω β 7ζ (3) and we get: Z g 2 N(0)βχ QC 2 pF m∗ = 1+ dq · q m 2π 2 pF2 0 ( " p r β 7ζ (3) η + Aq2 14ζ (3) · 1− × − C2 π 2 C Ã s !) π C 2 × arctan · β 7ζ (3) η + Aq2 2π βC(η + Aq2 ) Ã s !)# ( π C 2 π − arctan · × 2 β 7ζ (3) η + Aq2

−

(32)

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After performing the momentum integral, the effective mass in the presence of antiferromagnetic fluctuations, at finite temperatures, becomes: " p 2 pF2 14ζ (3) g 2 N(0)βχ QC π m∗ = 1+ + − 2 2 2 m C T AC 2π pF ( µ ¶2 µ ¶ ω pC 1 η + 4ApF2 arctan × 2 ω pC η + 4ApF2 µ ¶ η + 4ApF2 η + 4ApF2 1 − arctan + 2ω pC 2 ω pC µ ¶2 ¶ µ ω pC 1 η η − − arctan 2 ω pC η 2ω pC µ ¶¾ 1 η π2 + arctan − ln 2 ω pC 2AC ¯ ¯¸ ¯ 4π ApF2 ¯¯ ¯ ln ¯1 + (33) πη + 2ω pC ¯ with: π ωp = β

s 2 ∼T 7ζ (3)

(34)

Using Eqs. (33) and (34) it is now easy to show, with χ Q = −2 N˜ (a) (0), that in the zero temperature limit, the effective mass is given by: ¯ ¯ µ ¶2 ¯ ¯ 1 g A m∗ =1+ N(0) N˜ (a) (0) ln ¯¯1 + (2 pF )2 ¯¯ m 2A pF η

Fig. 1. m∗ /m as a function of T and with antiferromagnetic fluctuations, for: η = 0.1, A = 1, C = 1, pF = 1, g N(0) = 1. Here limT→0 m∗ /m ' 2.85.

In Fig. 3 we plot m∗ /m as a function of T, but for η = 0, and the same parameters. In this case: limT→0 (m∗ /m) → ∞ (diverges). 2.2.2. Ferromagnetic Case The wave-vector dependence of the factor Cq in Eq. (24), is now given by: Cq =

C˜ q

(37)

(35) which is in fact the result (27), showing the logarithmic divergence of the effective mass in the quantum critical point. At very low temperatures, and in the QCP (η → 0), the effective mass behaves like: ¯ ¯ µ ¶2 ¯ 2π ApF2 ¯¯ 1 g m∗ ¯ ˜ N(0) N(a) (0) ln ¯1 + =1+ m 2A pF ω pC ¯ (36) which is again logarithmic divergent, when T goes to zero. In Fig. 1 we give a qualitative plot of m∗ /m as a function of temperature, for η = 0.1, and for: A = 1, C = 1, pF = 1, g N(0) = 1, and χ Q = −2N(0) (for simplicity). Here: limT→0 (m∗ /m) ' 2.85 (a finite value). In Fig. 2 we plot m∗ /m as a function of η, for the same parameters, and for T = 0 K. Here: limη→0 (m∗ /m) → ∞ (diverges).

Fig. 2. m∗ /m as a function of η and with antiferromagnetic fluctuations, for T = 0, and for the same parameters as in Fig. 1. Here limη→0 m∗ /m = ∞.

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Effective Mass Close to a Quantum Critical Point

985

Fig. 3. m∗ /m as a function of T and with antiferromagnetic fluctuations, for η = 0, and for parameters from Fig. 1. Here limT→0 m∗ /m = ∞.

and we assumed that χ Q = −2 N˜ (f) (0). In the zero temperature limit I1 behaves like:

With this form, the effective mass becomes: Z ∞ Z g N(0)βχ QC˜ 2 pF m dq dω = 1+ m 2π 2 pF2 0 0 ¶ µ iβω ω 0 1 · Imψ + × (η + Aq2 )2 + Cq2 ω2 2 2π ∗

Fig. 4. m∗ /m as a function of T and with ferromagnetic fluctuations, for: A = 1, C˜ = 1, pF = 1, g N(0) = 1, and for: (a) η = 0 (Here limT→0 m∗ /m = ∞); (b) η = 0.3 (limT→0 m∗ /m ' 2.331).

I1 (T → 0) = ω pC˜ ·

2

(38) Using Eq. (31) for the imaginary part of the derivative of the digamma function, we get: µ ¶ m∗ 2 g 2 1 = 1+ N(0) N˜ (f) (0) m π pF ω pC˜ ¸ · 1 (2 pF )3 − · I1 × 3 ω pC˜ µ ¶ 2 g 2 N(0) N˜ (f) (0).I2 + π pF where: Z I1 =

2 pF

0

Z I2 = 0

2 pF

(39)

µ

ω pC˜ dq · q (η + Aq ) · arctan q(η + Aq2 ) 3

µ

dq ·

¶

2

q q(η + Aq2 ) · arctan 2 η + Aq ω pC˜

¶

(40)

(41)

and:

(2 pF )3 3

¯ ¯ ¯ ¯ A π 1 2¯ ¯ ln ¯1 + (2 pF ) ¯ I2 (T → 0) = · 2 2A η

(42)

(43)

and one recovers the result for m∗ /m, given by Eq. (27). In Fig. 4 we give a qualitative plot of m∗ /m in the presence of ferromagnetic fluctuations, as a function of temperature, for the following parameters: A = 1, C˜ = 1, pF = 1, χ Q = −2N(0), g N(0) = 1, and for: (a) η = 0 (when limT→0 (m∗ /m) → ∞), and (b) η = 0.3 (when limT→0 (m∗ m) ' 2.33—a finite value).

3. CONCLUSIONS We calculated the electron effective mass in heavy fermion compounds, taking into consideration the antiferromagnetic and ferromagnetic character of the fluctuations, in the proximity of a quantum critical point. In our model the effective mass diverges in a QCP. The divergence has a logarithmic character, in agreement with many experimental results. Theoretical models that focus on this problem were also developed. Many of them deal with the non-Fermi character of these compounds [14–22]. Even though

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986 the logarithmic dependence of the Cv /T appears for many heavy-fermion compounds, there are several compounds (e.g. CeNi 2 Ge2 , YbRh2 (Si 1−x Gex )2 ) that show a stronger divergence at lower temperatures. Although the definition of a quantum phase transition is valid only for T = 0, very close to the critical point the systems behavior is still determined by the quantum fluctuations and the systems begin to behave classically only if the temperature exceeds the fluctuation frequencies ωf . In our model, the low temperature limit is considered to be in the interval T < ωf . However, even this simple model gives some reliable results, in qualitative agreement with many experimental results on heavy fermion compounds, and further investigations are necessary, in order to explain the stronger divergencies of the effective mass in other heavy fermion compounds.

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