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Spinless fermion model with quantum criticality
ARTICLE IN PRESS
Physica B 378–380 (2006) 66–67 www.elsevier.com/locate/physb
Spinless fermion model with quantum criticality$ P. Schlottmann Department of Physics, Florida State University, Tallahassee, FL 32306, USA
Abstract A simple model of spinless fermions with nested Fermi surface is considered. A lnðTÞ-dependence of C=T and a linear T-dependence of the quasi-particle linewidth is obtained in the neighborhood of the quantum critical point. r 2006 Elsevier B.V. All rights reserved. PACS: 71.27.þa; 71.28.þd; 72.15.Qm; 75.20.Hr Keywords: Non-Fermi liquid; Quantum criticality; Heavy fermions
Non-Fermi liquid (NFL) properties are observed in many heavy fermion systems and frequently attributed to a nearby quantum critical point (QCP). A QCP can arise by suppressing the transition temperature T c of a long-range ordered phase to zero. The nesting of a Fermi surface (FS) in conjunction with the remaining interaction between the carriers after heavy fermions are formed, can give rise to a charge density wave (CDW). The order is gradually suppressed by mismatching the nesting and a QCP is obtained as T c ! 0. We consider here spinless fermions and two different nesting situations: model (A) consists of one electron and one hole pocket separated by a wavevector Q and model (B) two flat parallel FS sheets separated by a vector Q. Model (A): The kinetic energy for model (A) is [1] X H0 ¼ l ðkÞcylk clk , (1) l¼1;2 k
where k is measured from the center of the pocket and 1 ðkÞ ¼ vF ðjkj kF 1 Þ and 2 ðkÞ ¼ vF ðkF 2 jkjÞ. The weak remaining interactions between the heavy quasiparticles is X y c1kþq c1k cy2k0 q c2k0 . (2) HV ¼ V kk0 q $
Work supported by DOE and NSF.
Tel.: +1 850 644 0055; fax: +1 850 644 8630.
E-mail address: [email protected]. 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.027
The leading order corrections to the vertex are the bubble diagrams of the zero-sound type (antiparallel propagator lines), which are logarithmic in the external energy o and summed using the renormalization group V~ ¼ V =ð1 V rF xÞ,
(3)
where x ¼ ln½D=ðjoj þ 2T þ jdjÞ, d ¼ vF jkF 1 kF 2 j=2 is the nesting mismatch parameter and rF is the constant density of states [1]. Within the logarithmic approximation the linear response to a CDW is wc ðQ; oÞ ¼ xrF =ð1 V rF xÞ.
(4)
Hence, if V 40 a CDW can be formed with T c ¼ 12 D exp½1=ðrF V Þ 12 jdj. If T c o0 no long range order has developed. The condition for a QCP is T c ¼ 0, which is obtained by fine-tuning d. In the disordered phase, the g-coefficient of the specific heat is determined by the thermal effective mass m , which is obtained from the fermion self-energy, m ðTÞ g 1 ¼ ¼ 1 þ xðV rF Þ2 =ð1 V rF xÞ. m g0 2
(5)
It increases logarithmically as T is lowered and diverges at the critical point signalling the breakdown of the Fermi liquid [1,2]. In a Fermi liquid the damping of the quasi-particles is proportional to T 2 , while the nesting condition changes this behavior to a quasi-linear dependence in T, which determines the low-T dependence of the resistivity. The
ARTICLE IN PRESS P. Schlottmann / Physica B 378–380 (2006) 66–67
where Im c is the imaginary part of the digamma function. Model (B): For model (B) the kinetic energy is X H0 ¼ vF ðjkx j kF Þ½cy1k c1k þ cy2k c2k , (8)
0.15
0.10 ΓNFL
linewidth G is calculated following a procedure outlined by Virosztek and Ruvalds [3] in the context of high-T c . In the disordered phase, G is given by the imaginary part of the electron self-energy, which can be expressed as a convolution of the charge susceptibility wðq; oÞ with a fermion Green’s function. The most important contribution arises from the nesting vector, i.e. q ¼ Q, leading to the selfconsistency equation Z 1 2 GNFL ðo; TÞ ¼ rF do0 w00 ðQ; o0 ÞV~ 2 0 0 o o o coth , ð6Þ tanh 2T 2T 1 GNFL o 00 þi w ðQ; oÞ ¼ rF Im c þ , (7) 2 2pT 4pT
67
0.05
0.00 0.0
0.2
0.4
0.6 T
0.8
1.0
1.2
Fig. 1. Quasi-particle linewidth for model A (solid line) and B (dashed line) for V rF ¼ 0:2, D ¼ 10 and d ¼ 0:07.
k
where l ¼ 1; 2 denotes the two parallel FS sheets perpendicular to the x-axis. The dispersion is linearized about the two FS at kx ¼ kF . The weak remaining interaction after the heavy quasi-particles are formed is again given by Eq. (2). For leading logarithmic order in V there are bubble diagrams with antiparallel (zero-sound channel) and parallel (Cooper channel) propagator lines. The renormalization group equation for the vertex is now an integrodifferential equation (differential in kx and integrals for the ky and kz variables) [4]. If the flat portion of the FS corresponds to the entire cross-section of the Brillouin zone, then the ky and kz integration can be carried out analytically (using the lattice periodicity) and the zerosound and Cooper channels cancel each other. The model has then the characteristics of a one-component Luttinger liquid. As a consequence of this cancellation the interaction vertex is not renormalized, i.e. V~ ¼ V . The logarithmic derivative of the CDW response function satisfies multiplicative renormalization [5] and we obtain wc ðQ; oÞ ¼ ½expð2rF V xÞ 1=ð2V Þ,
(9)
where x ¼ ln½D=ðjoj þ 2TÞ. For V 40, the susceptibility diverges at o ¼ T ¼ 0 leading to a QCP. The g-coefficient of the specific heat is again obtained from the renormalized self-energy. Since the renormalized vertex is just the bare one, we obtain for the thermal effective mass m ðTÞ=m ¼ g=g0 ¼ xðV rF Þ2 =2.
cancellation between the two channels is only partial and the vertex depends on the transverse momenta variables [4]. Small transverse momentum transfer favors the formation of a CDW, because it maximizes the zero-sound channel. For V 40, the vertex is then divergent and yields a CDW with finite T c . T c can be suppressed by introducing a slight curvature of the FS sheets, so that the two sheets are not separated by Q for all ky and kz . This bending corresponds to a nesting mismatch parameter d. Fine-tuning d can lead to a QCP. The g-coefficient is an average over the FS, so that its T-dependence is between that of Eqs. (5) and (10). The same is true for the quasi-particle linewidth averaged over the FS. As shown in Fig. 1 GNFL is approximately linear in T for both models. Hence, the resistivity and the width of the neutron scattering quasi-elastic peak are also expected to be linear in T. The results are valid in the disordered phase for weak and intermediate coupling. Since the renormalization group does not allow a return to a weak-coupling fixed point once it is strongly coupled, the present approach qualitatively describes the entire precritical regime.
References
(10)
Hence, g increases logarithmically with T as the temperature decreases. The quasi-particle linewidth is given by Eq. (6) with V~ ¼ V . The situation is more complicated if the nested FS is not the entire cross-section of the Brillouin zone. The
[1] [2] [3] [4]
P. Schlottmann, Phys. Rev. B 59 (1999) 12379. P. Schlottmann, Phys. Rev. B 68 (2003) 125105. A. Virosztek, J. Ruvalds, Phys. Rev. B 42 (1990) 4064. A.T. Zheleznyak, V.M. Yakovenko, I.E. Dzyaloshinskii, Phys. Rev. B 55 (1997) 3200. [5] J. So´lyom, J. Low Temp. Phys. 12 (1973) 547.
Physica B 378–380 (2006) 66–67 www.elsevier.com/locate/physb
Spinless fermion model with quantum criticality$ P. Schlottmann Department of Physics, Florida State University, Tallahassee, FL 32306, USA
Abstract A simple model of spinless fermions with nested Fermi surface is considered. A lnðTÞ-dependence of C=T and a linear T-dependence of the quasi-particle linewidth is obtained in the neighborhood of the quantum critical point. r 2006 Elsevier B.V. All rights reserved. PACS: 71.27.þa; 71.28.þd; 72.15.Qm; 75.20.Hr Keywords: Non-Fermi liquid; Quantum criticality; Heavy fermions
Non-Fermi liquid (NFL) properties are observed in many heavy fermion systems and frequently attributed to a nearby quantum critical point (QCP). A QCP can arise by suppressing the transition temperature T c of a long-range ordered phase to zero. The nesting of a Fermi surface (FS) in conjunction with the remaining interaction between the carriers after heavy fermions are formed, can give rise to a charge density wave (CDW). The order is gradually suppressed by mismatching the nesting and a QCP is obtained as T c ! 0. We consider here spinless fermions and two different nesting situations: model (A) consists of one electron and one hole pocket separated by a wavevector Q and model (B) two flat parallel FS sheets separated by a vector Q. Model (A): The kinetic energy for model (A) is [1] X H0 ¼ l ðkÞcylk clk , (1) l¼1;2 k
where k is measured from the center of the pocket and 1 ðkÞ ¼ vF ðjkj kF 1 Þ and 2 ðkÞ ¼ vF ðkF 2 jkjÞ. The weak remaining interactions between the heavy quasiparticles is X y c1kþq c1k cy2k0 q c2k0 . (2) HV ¼ V kk0 q $
Work supported by DOE and NSF.
Tel.: +1 850 644 0055; fax: +1 850 644 8630.
E-mail address: [email protected]. 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.027
The leading order corrections to the vertex are the bubble diagrams of the zero-sound type (antiparallel propagator lines), which are logarithmic in the external energy o and summed using the renormalization group V~ ¼ V =ð1 V rF xÞ,
(3)
where x ¼ ln½D=ðjoj þ 2T þ jdjÞ, d ¼ vF jkF 1 kF 2 j=2 is the nesting mismatch parameter and rF is the constant density of states [1]. Within the logarithmic approximation the linear response to a CDW is wc ðQ; oÞ ¼ xrF =ð1 V rF xÞ.
(4)
Hence, if V 40 a CDW can be formed with T c ¼ 12 D exp½1=ðrF V Þ 12 jdj. If T c o0 no long range order has developed. The condition for a QCP is T c ¼ 0, which is obtained by fine-tuning d. In the disordered phase, the g-coefficient of the specific heat is determined by the thermal effective mass m , which is obtained from the fermion self-energy, m ðTÞ g 1 ¼ ¼ 1 þ xðV rF Þ2 =ð1 V rF xÞ. m g0 2
(5)
It increases logarithmically as T is lowered and diverges at the critical point signalling the breakdown of the Fermi liquid [1,2]. In a Fermi liquid the damping of the quasi-particles is proportional to T 2 , while the nesting condition changes this behavior to a quasi-linear dependence in T, which determines the low-T dependence of the resistivity. The
ARTICLE IN PRESS P. Schlottmann / Physica B 378–380 (2006) 66–67
where Im c is the imaginary part of the digamma function. Model (B): For model (B) the kinetic energy is X H0 ¼ vF ðjkx j kF Þ½cy1k c1k þ cy2k c2k , (8)
0.15
0.10 ΓNFL
linewidth G is calculated following a procedure outlined by Virosztek and Ruvalds [3] in the context of high-T c . In the disordered phase, G is given by the imaginary part of the electron self-energy, which can be expressed as a convolution of the charge susceptibility wðq; oÞ with a fermion Green’s function. The most important contribution arises from the nesting vector, i.e. q ¼ Q, leading to the selfconsistency equation Z 1 2 GNFL ðo; TÞ ¼ rF do0 w00 ðQ; o0 ÞV~ 2 0 0 o o o coth , ð6Þ tanh 2T 2T 1 GNFL o 00 þi w ðQ; oÞ ¼ rF Im c þ , (7) 2 2pT 4pT
67
0.05
0.00 0.0
0.2
0.4
0.6 T
0.8
1.0
1.2
Fig. 1. Quasi-particle linewidth for model A (solid line) and B (dashed line) for V rF ¼ 0:2, D ¼ 10 and d ¼ 0:07.
k
where l ¼ 1; 2 denotes the two parallel FS sheets perpendicular to the x-axis. The dispersion is linearized about the two FS at kx ¼ kF . The weak remaining interaction after the heavy quasi-particles are formed is again given by Eq. (2). For leading logarithmic order in V there are bubble diagrams with antiparallel (zero-sound channel) and parallel (Cooper channel) propagator lines. The renormalization group equation for the vertex is now an integrodifferential equation (differential in kx and integrals for the ky and kz variables) [4]. If the flat portion of the FS corresponds to the entire cross-section of the Brillouin zone, then the ky and kz integration can be carried out analytically (using the lattice periodicity) and the zerosound and Cooper channels cancel each other. The model has then the characteristics of a one-component Luttinger liquid. As a consequence of this cancellation the interaction vertex is not renormalized, i.e. V~ ¼ V . The logarithmic derivative of the CDW response function satisfies multiplicative renormalization [5] and we obtain wc ðQ; oÞ ¼ ½expð2rF V xÞ 1=ð2V Þ,
(9)
where x ¼ ln½D=ðjoj þ 2TÞ. For V 40, the susceptibility diverges at o ¼ T ¼ 0 leading to a QCP. The g-coefficient of the specific heat is again obtained from the renormalized self-energy. Since the renormalized vertex is just the bare one, we obtain for the thermal effective mass m ðTÞ=m ¼ g=g0 ¼ xðV rF Þ2 =2.
cancellation between the two channels is only partial and the vertex depends on the transverse momenta variables [4]. Small transverse momentum transfer favors the formation of a CDW, because it maximizes the zero-sound channel. For V 40, the vertex is then divergent and yields a CDW with finite T c . T c can be suppressed by introducing a slight curvature of the FS sheets, so that the two sheets are not separated by Q for all ky and kz . This bending corresponds to a nesting mismatch parameter d. Fine-tuning d can lead to a QCP. The g-coefficient is an average over the FS, so that its T-dependence is between that of Eqs. (5) and (10). The same is true for the quasi-particle linewidth averaged over the FS. As shown in Fig. 1 GNFL is approximately linear in T for both models. Hence, the resistivity and the width of the neutron scattering quasi-elastic peak are also expected to be linear in T. The results are valid in the disordered phase for weak and intermediate coupling. Since the renormalization group does not allow a return to a weak-coupling fixed point once it is strongly coupled, the present approach qualitatively describes the entire precritical regime.
References
(10)
Hence, g increases logarithmically with T as the temperature decreases. The quasi-particle linewidth is given by Eq. (6) with V~ ¼ V . The situation is more complicated if the nested FS is not the entire cross-section of the Brillouin zone. The
[1] [2] [3] [4]
P. Schlottmann, Phys. Rev. B 59 (1999) 12379. P. Schlottmann, Phys. Rev. B 68 (2003) 125105. A. Virosztek, J. Ruvalds, Phys. Rev. B 42 (1990) 4064. A.T. Zheleznyak, V.M. Yakovenko, I.E. Dzyaloshinskii, Phys. Rev. B 55 (1997) 3200. [5] J. So´lyom, J. Low Temp. Phys. 12 (1973) 547.