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Phase diagram of the one-dimensional extended Hubbard model with hopping dimerization
Physica C 460–462 (2007) 1079–1080 www.elsevier.com/locate/physc
Phase diagram of the one-dimensional extended Hubbard model with hopping dimerization Satoshi Ejima b
a,*
, Florian Gebhard a, Satoshi Nishimoto
b
a Fachbereich Physik, Philipps-Universita¨t Marburg, Renthof 6, D-35032 Marburg, Germany Max-Planck-Institut fu¨r Physik komplexer Systeme, Noethnitzer Strasse 38, D-01187 Dresden, Germany
Available online 3 April 2007
Abstract Using the (dynamical) density-matrix renormalization group method, we calculate the momentum-dependent spectral function and the Tomonaga–Luttinger parameter for the one-dimensional extended Hubbard model with alternating hopping integrals at quarter band filling. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.30.+h; 78.30.Jw Keywords: Dimerized Hubbard model; Tomonaga–Luttinger liquid; Metal–insulator transitions
The Bechgaard salts have attracted much attention over the last thirty years. However, we have not reached a complete understanding of their phase diagram, in particular of the superconducting phase which is located in-between paramagnetic-metallic and spin-density-wave phases. Due to the strong anisotropy, the system can be regarded as being quasi one-dimensional, and the results of recent experiments suggest that the metallic phase is a Tomonaga–Luttinger (TL) liquid. It has been attempted to explain the experimental data within the framework of the Tomonaga–Luttinger-liquid theory for the single-band Hubbard model. However, the material has alternating hopping integrals along the chain and the effect of dimerization must be considered. Therefore, we study the one-dimensional dimerized extended Hubbard model. We use the (dynamical) density-matrix renormalization group method to obtain the TL parameter [1] and the momentum-dependent spectral function A(k, x) [2]. We focus on the region near quarter filling which is relevant for the Bechgaard salts.
*
Corresponding author. E-mail address: [email protected] (S. Ejima).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.316
Our model Hamiltonian is defined by X y X y b ¼ t1 H ð^clþ1r^clr þ h:c:Þ t2 ð^clþ1r^clr þ h:c:Þ l;odd
þU
X l
l;even
X ^nl;" ^nl;# þ V ð^nl nÞð^nlþ1 nÞ;
ð1Þ
l
where the electron transfer-matrix elements t1 and t2 (
1080
S. Ejima et al. / Physica C 460–462 (2007) 1079–1080
where h. . .i0 denotes the ground-state expectation value, g is the broadening, E0 is the ground-state energy, and b kr ¼ ^ckr ð^cy Þ for ARPES (IARPES). Since we apply O kr open-end boundary conditions, we use the quasi-momenta k = pm/(L + 1) for integers 1 6 m 6 L on a chain with Ldimers to express the momentum-dependent operators b kr . In Fig. 1a, we show A(k, x) for t2/t1 = 0.5 and O U = V = 0 on a quarter-filled chain with L = 32 dimers which provides a non-trivial check for the dynamical density-matrix renormalization group method. Apparently, we recover the non-interacting dispersion relation, Eq. (2), which demonstrates the accuracy of our method. Let us now turn to the case U 5 0. For finite dimerization, our model at quarter filling can be effectively mapped onto the single-band Hubbard model at half filling, so that the system is expected to be a Mott–Hubbard insulator. In Fig. 1b, we show A(k, x) of the bonding orbital for t2/ t1 = 0.5 and U/t1 = 10, V/t1 = 0. It is evident that the Mott–Hubbard gap Dc opens at k = ±p/2 and the value Dc 0.65 is consistent with our previous estimate [3]. We
Fig. 2. TL parameter Kq for the infinitesimally doped quarter-filled system as a function of V/t1 for t2/t1 = 0.9, t2/t1 = 0.5 and t2/t1 = 0.1 at U/t1 = 1.
also obtain the so-called ‘shadow band’ of the free electron band _(k), which is one of the significant features in the half-filled Hubbard model. The spectral weight of the shadow band becomes smaller with increasing V/t1, and disappears at a critical nearest-neighbor, V = Vc, where a transition from the Mott–Hubbard insulator to the charge-density-wave (CDW) insulator occurs. Next, we study the TL parameter Kq close to quarter filling [4]. In Fig. 2, we show the TL parameter for U = 1 and various dimerizations in the infinitesimally doped system. Below Vc, the Mott–Hubbard insulator gives K MH q ðt 2 =t 1 Þ ¼ 0:5, independent of V. Above Vc, Kq drops to K CDW ðt2 ¼ t1 Þ ¼ 1=8 for the infinitesimally doped q CDW insulator in the absence of a dimerization. In the presence of a dimerization, we can even achieve K CDW ðV ; t2 =t1 Þ < 1=8. Note that the value of Vc increases q as t2/t1 decreases, i.e., the dimerization opposes the formation of the CDW phase. Our results for the single-particle spectrum and the TL parameter confirm that for small V the low-energy physics of the quarter-filled dimerized extended Hubbard model is essentially the same as that of the half-filled Hubbard model. When V is large and for finite dimerization, we find fairly small values for Kq in the infinitesimally doped CDW insulator. This may be relevant to the Bechgaard salts where experiments suggest Kq 0.2. References
Fig. 1. Spectral function A(k, x) for (a) U = V = 0, g = 0.2t1, L = 32 and (b) U/t1 = 10, V/t1 = 0, g = 0.05t1, L = 64 at quarter filling.
[1] S. Ejima, F. Gebhard, S. Nishimoto, Europhys. Lett. 70 (2005) 492. [2] E. Jeckelmann, Phys. Rev. B 66 (2002) 045114; H. Benthien, F. Gebhard, E. Jeckelmann, Phys. Rev. Lett. 92 (2004) 256401. [3] S. Nishimoto, M. Takahashi, Y. Ohta, J. Phys. Soc. Jpn. 69 (2000) 1594. [4] S. Ejima, F. Gebhard, S. Nishimoto, Phys. Rev. B 74 (2006) 245110.
Phase diagram of the one-dimensional extended Hubbard model with hopping dimerization Satoshi Ejima b
a,*
, Florian Gebhard a, Satoshi Nishimoto
b
a Fachbereich Physik, Philipps-Universita¨t Marburg, Renthof 6, D-35032 Marburg, Germany Max-Planck-Institut fu¨r Physik komplexer Systeme, Noethnitzer Strasse 38, D-01187 Dresden, Germany
Available online 3 April 2007
Abstract Using the (dynamical) density-matrix renormalization group method, we calculate the momentum-dependent spectral function and the Tomonaga–Luttinger parameter for the one-dimensional extended Hubbard model with alternating hopping integrals at quarter band filling. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.30.+h; 78.30.Jw Keywords: Dimerized Hubbard model; Tomonaga–Luttinger liquid; Metal–insulator transitions
The Bechgaard salts have attracted much attention over the last thirty years. However, we have not reached a complete understanding of their phase diagram, in particular of the superconducting phase which is located in-between paramagnetic-metallic and spin-density-wave phases. Due to the strong anisotropy, the system can be regarded as being quasi one-dimensional, and the results of recent experiments suggest that the metallic phase is a Tomonaga–Luttinger (TL) liquid. It has been attempted to explain the experimental data within the framework of the Tomonaga–Luttinger-liquid theory for the single-band Hubbard model. However, the material has alternating hopping integrals along the chain and the effect of dimerization must be considered. Therefore, we study the one-dimensional dimerized extended Hubbard model. We use the (dynamical) density-matrix renormalization group method to obtain the TL parameter [1] and the momentum-dependent spectral function A(k, x) [2]. We focus on the region near quarter filling which is relevant for the Bechgaard salts.
*
Corresponding author. E-mail address: [email protected] (S. Ejima).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.316
Our model Hamiltonian is defined by X y X y b ¼ t1 H ð^clþ1r^clr þ h:c:Þ t2 ð^clþ1r^clr þ h:c:Þ l;odd
þU
X l
l;even
X ^nl;" ^nl;# þ V ð^nl nÞð^nlþ1 nÞ;
ð1Þ
l
where the electron transfer-matrix elements t1 and t2 (
1080
S. Ejima et al. / Physica C 460–462 (2007) 1079–1080
where h. . .i0 denotes the ground-state expectation value, g is the broadening, E0 is the ground-state energy, and b kr ¼ ^ckr ð^cy Þ for ARPES (IARPES). Since we apply O kr open-end boundary conditions, we use the quasi-momenta k = pm/(L + 1) for integers 1 6 m 6 L on a chain with Ldimers to express the momentum-dependent operators b kr . In Fig. 1a, we show A(k, x) for t2/t1 = 0.5 and O U = V = 0 on a quarter-filled chain with L = 32 dimers which provides a non-trivial check for the dynamical density-matrix renormalization group method. Apparently, we recover the non-interacting dispersion relation, Eq. (2), which demonstrates the accuracy of our method. Let us now turn to the case U 5 0. For finite dimerization, our model at quarter filling can be effectively mapped onto the single-band Hubbard model at half filling, so that the system is expected to be a Mott–Hubbard insulator. In Fig. 1b, we show A(k, x) of the bonding orbital for t2/ t1 = 0.5 and U/t1 = 10, V/t1 = 0. It is evident that the Mott–Hubbard gap Dc opens at k = ±p/2 and the value Dc 0.65 is consistent with our previous estimate [3]. We
Fig. 2. TL parameter Kq for the infinitesimally doped quarter-filled system as a function of V/t1 for t2/t1 = 0.9, t2/t1 = 0.5 and t2/t1 = 0.1 at U/t1 = 1.
also obtain the so-called ‘shadow band’ of the free electron band _(k), which is one of the significant features in the half-filled Hubbard model. The spectral weight of the shadow band becomes smaller with increasing V/t1, and disappears at a critical nearest-neighbor, V = Vc, where a transition from the Mott–Hubbard insulator to the charge-density-wave (CDW) insulator occurs. Next, we study the TL parameter Kq close to quarter filling [4]. In Fig. 2, we show the TL parameter for U = 1 and various dimerizations in the infinitesimally doped system. Below Vc, the Mott–Hubbard insulator gives K MH q ðt 2 =t 1 Þ ¼ 0:5, independent of V. Above Vc, Kq drops to K CDW ðt2 ¼ t1 Þ ¼ 1=8 for the infinitesimally doped q CDW insulator in the absence of a dimerization. In the presence of a dimerization, we can even achieve K CDW ðV ; t2 =t1 Þ < 1=8. Note that the value of Vc increases q as t2/t1 decreases, i.e., the dimerization opposes the formation of the CDW phase. Our results for the single-particle spectrum and the TL parameter confirm that for small V the low-energy physics of the quarter-filled dimerized extended Hubbard model is essentially the same as that of the half-filled Hubbard model. When V is large and for finite dimerization, we find fairly small values for Kq in the infinitesimally doped CDW insulator. This may be relevant to the Bechgaard salts where experiments suggest Kq 0.2. References
Fig. 1. Spectral function A(k, x) for (a) U = V = 0, g = 0.2t1, L = 32 and (b) U/t1 = 10, V/t1 = 0, g = 0.05t1, L = 64 at quarter filling.
[1] S. Ejima, F. Gebhard, S. Nishimoto, Europhys. Lett. 70 (2005) 492. [2] E. Jeckelmann, Phys. Rev. B 66 (2002) 045114; H. Benthien, F. Gebhard, E. Jeckelmann, Phys. Rev. Lett. 92 (2004) 256401. [3] S. Nishimoto, M. Takahashi, Y. Ohta, J. Phys. Soc. Jpn. 69 (2000) 1594. [4] S. Ejima, F. Gebhard, S. Nishimoto, Phys. Rev. B 74 (2006) 245110.