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Competition and coexistence of antiferromagnetism and superconductivity in the Hubbard model
Physica C 470 (2010) S947–S948
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Competition and coexistence of antiferromagnetism and superconductivity in the Hubbard model Kenji Kobayashi a,*, Tsutomu Watanabe a, Hisatoshi Yokoyama b a b
Department of Natural Science, Chiba Institute of Technology, Shibazono, Narashino, Chiba 275-0023, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan
a r t i c l e
i n f o
Article history: Accepted 21 October 2009 Available online 27 October 2009 Keywords: Hubbard model Variational Monte Carlo method Superconductivity Cuprate
a b s t r a c t The interplay between antiferromagnetism (AF) and superconductivity (SC) in cuprate is studied for the two-dimensional Hubbard model with a diagonal transfer t0 , using a variational Monte Carlo method. Optimizing an improved wave-function for strongly correlated values of U/t, we construct a phase diagram. It is found that the stable state is sensitive to the value of model parameters: For t0 /t J 0.15 a coexisting state is realized, whose range of doping rate extends as t0 /t increases. In contrast, for t0 / t = 0.3, AF and SC states are mutually exclusive, and the area of pure AF shrinks as U/t increases. Ó 2009 Elsevier B.V. All rights reserved.
The proximity of antiferromagnetic and superconducting phases is a feature universal to all cuprate superconductors, and these two phases are mutually exclusive without coexistence in most experiments. However, recent NMR experiment for a multilayered cuprate argued that the two phases coexist in the outer CuO2 plane [1]. Thus, it is important to study the interplay between antiferromagnetism (AF) and superconductivity (SC), and clarify the stability of coexisting state as a function of U/t, t0 /t and doping rate d. In this work, we study the cooperation or competition between AF and d-wave SC for a Hubbard model on the square lattice with a diagonal transfer t0 (t–t0 –U model), using a variational Monte Carlo (VMC) method, which is useful to reliably treat a wide range of parameters for correlated systems, in linking weak and strong coupling (t–J model) regimes. The following improvements are introduced into the wave function, as in the preceding work [2,3]: (1) coexistence of AF and d-wave singlet gaps that allows a continuous description of their interplay [4–6], (2) band renormalization effect owing to electron correlation by adjusting hopping integrals, and (3) refined doublon–holon correlation factors, which control the effect of Mott transition near half filling more precisely [7,8]. Using optimization VMC techniques, we minimize the energy by searching for the optimal set of variational parameters. We fix the system size (Ns = L L) at L = 12 with periodic–antiperiodic boundary conditions. In Fig. 1, we plot the two quantities which directly measure the magnitude of long-range orders, namely, the staggered magnetiza* Corresponding author. E-mail address: [email protected] (K. Kobayashi). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.10.058
tion m as an indicator of AF order and dx2 y2 -wave SC correlation function for the nearest-neighbor pairing, Pd(r), at the farthermost distance r = (L/2, L/2). In the case of t0 /t = 0 shown in Fig. 1(c), the AF order stabilized by the nesting condition at half-filling (d = 0) becomes weak, as the doping rate increases. However, Pd(r) increases as d increases for small d, and Pd(r) exhibits a dome shape. Both m and Pd(r) have finite values in the underdoped regime, suggesting a coexisting state is realized, while for higher dopings the optimized state with Pd(r) – 0 and m = 0 is indicative of pure SC. Comparing the data for U/t = 16 and 30, both AF and SC orders decrease as U/t increases, and the areas of coexisting and pure SC states shrink. Next, we consider t0 /t dependence. The results for jt0 /tj – 0, also plotted in Fig. 1, exhibit marked electron–hole asymmetry with respect to the stability of the coexisting state. On the electron-doped side (t0 /t > 0), the magnitude of Pd(r) becomes small as jt0 /tj increases. Because the area of Pd(r) – 0 shrinks but that of m – 0 extends, the area of coexistence extends to almost the whole range of finite values of Pd(r) and m, and the pure SC state does not appear. In contrast, on the hole-doped side (t0 /t < 0), SC is enhanced in the highly-doped regime, but becomes weak for small doping, as compared with the t0 /t = 0 case. As a result, for t0 /t = 0.15 appropriate to hole-doped cuprates, the coexisting area shrinks, and the pure SC area extends to d 0.3, as seen in Fig. 1(b). In addition, the pure AF state realized at U/t = 16 disappears for U/t = 30, because the SC order survives for the extremely large value of U/t. For t0 /t = 0.3, however, the SC and AF orders become mutually exclusive and never coexist as seen in Fig. 1(a). As a summary, we draw rough phase diagrams in the d–t0 /t plane for U/t = 16 and 30 in Fig. 2, concluding that the area of coexistence is sensitive
S948
K. Kobayashi et al. / Physica C 470 (2010) S947–S948
0.5
m
0.04
0.2 0.02 0.1
(a) U/t =16
0.3
Pd (L/2,L/2)
0.3
0.06
AF+SC
0.15
t'/t
(a) t'/t = -0.30
0.4
0
0
0.1
0.2
pure SC
0.3
-0.3
0.5
0
m
0.04
0.2 0.02
Pd (L/2,L/2)
0.4 0.3
0.1 0
0 0
0.1
0.3
(b) U/t =30
AF+SC
0.15
Electron doped
no order
0
Hole doped
pure SC pure AF
0.06 0
m
0.04
0.2 0.02
Pd (L/2,L/2)
0.4 0.3
0.2
0.3
-0.3
0.5
(c) t'/t = 0.00
0.1
-0.15
0.3
0.2
pure AF
0.06
t'/t
(b) t'/t = -0.15
Hole doped
-0.15
0
0
Electron doped
no order
0.1
0.2
0.3
doping rate Fig. 2. Phase diagrams constructed within present wave function in d-t0 /t plane for (a) U/t = 16 and (b) U/t = 30.
0.1 0
0 0
0.1
0.2
0.3
0.5
(d) t'/t = +0.15
0.06
Pd Pd
m
0.3
U/t =16 U/t =30
0.04
0.2 0.02
Pd (L/2,L/2)
0.4
0.1 0
0 0
0.1
0.2
to the value of U/t, t0 /t and d. The d dependence of SC and AF orders is similar to that of the t–J model as a whole, but the t–J model has a wide area of the coexisting phase even for t0 /t = 0.3 [9]. In connection with the pseudo-gap problem, we confirmed that the gradient of momentum distribution function at the antinodal point regions (k (p, 0)) mainly dominates the magnitude of the d-wave SC correlation function, and the electronic state at the antinodal point is essential to SC [3]. In conclusion, we have studied the interplay between AF and SC in the Hubbard model. For large U/t, it is found that the d-wave SC and AF ordered states become competing or coexisting, according to the values of model parameters.
0.3
References 0.5
(e) t'/t = +0.30
0.06
m U/t =16 m U/t =30
m
0.3
0.04
0.2 0.02
Pd (L/2,L/2)
0.4
0.1 0
0 0
0.1
0.2
0.3
doping rate Fig. 1. By using the optimized wave function, the expectation values of staggered magnetization m (triangles) and d-wave nearest-neighbor pair correlation function Pd(r) for the farthermost distance r = (L/2, L/2) (diamonds and squares) are plotted as a function of d for different values of t0 /t (=0, ± 0.15, ± 0.3). Data for U/t = 16 and 30 are simultaneously plotted in each panel. Arrows indicate changes of the magnitude of AF and SC orders from U/t = 16 to 30.
[1] H. Mukuda, M. Abe, Y. Araki, Y. Kitaoka, K. Tokiwa, T. Watanabe, A. Iyo, H. Kito, Y. Tanaka, Phys. Rev. Lett. 96 (2006) 087001; H. Mukuda, Y. Yamaguchi, S. Shimizu, Y. Kitaoka, P. Shirage, A. Iyo, J. Phys. Soc. Jpn. 77 (2008) 124706. [2] K. Kobayashi, H. Yokoyama, J. Phys. Chem. Solids 69 (2008) 3274. [3] K. Kobayashi, H. Yokoyama, Physica C 469 (2009) 974. [4] T. Giamarchi, C. Lhuillier, Phys. Rev. B 43 (1991) 12943. [5] A. Himeda, M. Ogata, Phys. Rev. B 60 (1999) R9935. [6] C.T. Shih, Y.C. Chen, C.P. Chou, T.K. Lee, Phys. Rev. B 70 (2004) 220502R. [7] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 59 (1990) 3669. [8] H. Yokoyama, Prog. Theor. Phys. 108 (2002) 59. [9] L. Spanu, M. Lugas, F. Becca, S. Sorella, Phys. Rev. B 77 (2008) 024510.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Competition and coexistence of antiferromagnetism and superconductivity in the Hubbard model Kenji Kobayashi a,*, Tsutomu Watanabe a, Hisatoshi Yokoyama b a b
Department of Natural Science, Chiba Institute of Technology, Shibazono, Narashino, Chiba 275-0023, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan
a r t i c l e
i n f o
Article history: Accepted 21 October 2009 Available online 27 October 2009 Keywords: Hubbard model Variational Monte Carlo method Superconductivity Cuprate
a b s t r a c t The interplay between antiferromagnetism (AF) and superconductivity (SC) in cuprate is studied for the two-dimensional Hubbard model with a diagonal transfer t0 , using a variational Monte Carlo method. Optimizing an improved wave-function for strongly correlated values of U/t, we construct a phase diagram. It is found that the stable state is sensitive to the value of model parameters: For t0 /t J 0.15 a coexisting state is realized, whose range of doping rate extends as t0 /t increases. In contrast, for t0 / t = 0.3, AF and SC states are mutually exclusive, and the area of pure AF shrinks as U/t increases. Ó 2009 Elsevier B.V. All rights reserved.
The proximity of antiferromagnetic and superconducting phases is a feature universal to all cuprate superconductors, and these two phases are mutually exclusive without coexistence in most experiments. However, recent NMR experiment for a multilayered cuprate argued that the two phases coexist in the outer CuO2 plane [1]. Thus, it is important to study the interplay between antiferromagnetism (AF) and superconductivity (SC), and clarify the stability of coexisting state as a function of U/t, t0 /t and doping rate d. In this work, we study the cooperation or competition between AF and d-wave SC for a Hubbard model on the square lattice with a diagonal transfer t0 (t–t0 –U model), using a variational Monte Carlo (VMC) method, which is useful to reliably treat a wide range of parameters for correlated systems, in linking weak and strong coupling (t–J model) regimes. The following improvements are introduced into the wave function, as in the preceding work [2,3]: (1) coexistence of AF and d-wave singlet gaps that allows a continuous description of their interplay [4–6], (2) band renormalization effect owing to electron correlation by adjusting hopping integrals, and (3) refined doublon–holon correlation factors, which control the effect of Mott transition near half filling more precisely [7,8]. Using optimization VMC techniques, we minimize the energy by searching for the optimal set of variational parameters. We fix the system size (Ns = L L) at L = 12 with periodic–antiperiodic boundary conditions. In Fig. 1, we plot the two quantities which directly measure the magnitude of long-range orders, namely, the staggered magnetiza* Corresponding author. E-mail address: [email protected] (K. Kobayashi). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.10.058
tion m as an indicator of AF order and dx2 y2 -wave SC correlation function for the nearest-neighbor pairing, Pd(r), at the farthermost distance r = (L/2, L/2). In the case of t0 /t = 0 shown in Fig. 1(c), the AF order stabilized by the nesting condition at half-filling (d = 0) becomes weak, as the doping rate increases. However, Pd(r) increases as d increases for small d, and Pd(r) exhibits a dome shape. Both m and Pd(r) have finite values in the underdoped regime, suggesting a coexisting state is realized, while for higher dopings the optimized state with Pd(r) – 0 and m = 0 is indicative of pure SC. Comparing the data for U/t = 16 and 30, both AF and SC orders decrease as U/t increases, and the areas of coexisting and pure SC states shrink. Next, we consider t0 /t dependence. The results for jt0 /tj – 0, also plotted in Fig. 1, exhibit marked electron–hole asymmetry with respect to the stability of the coexisting state. On the electron-doped side (t0 /t > 0), the magnitude of Pd(r) becomes small as jt0 /tj increases. Because the area of Pd(r) – 0 shrinks but that of m – 0 extends, the area of coexistence extends to almost the whole range of finite values of Pd(r) and m, and the pure SC state does not appear. In contrast, on the hole-doped side (t0 /t < 0), SC is enhanced in the highly-doped regime, but becomes weak for small doping, as compared with the t0 /t = 0 case. As a result, for t0 /t = 0.15 appropriate to hole-doped cuprates, the coexisting area shrinks, and the pure SC area extends to d 0.3, as seen in Fig. 1(b). In addition, the pure AF state realized at U/t = 16 disappears for U/t = 30, because the SC order survives for the extremely large value of U/t. For t0 /t = 0.3, however, the SC and AF orders become mutually exclusive and never coexist as seen in Fig. 1(a). As a summary, we draw rough phase diagrams in the d–t0 /t plane for U/t = 16 and 30 in Fig. 2, concluding that the area of coexistence is sensitive
S948
K. Kobayashi et al. / Physica C 470 (2010) S947–S948
0.5
m
0.04
0.2 0.02 0.1
(a) U/t =16
0.3
Pd (L/2,L/2)
0.3
0.06
AF+SC
0.15
t'/t
(a) t'/t = -0.30
0.4
0
0
0.1
0.2
pure SC
0.3
-0.3
0.5
0
m
0.04
0.2 0.02
Pd (L/2,L/2)
0.4 0.3
0.1 0
0 0
0.1
0.3
(b) U/t =30
AF+SC
0.15
Electron doped
no order
0
Hole doped
pure SC pure AF
0.06 0
m
0.04
0.2 0.02
Pd (L/2,L/2)
0.4 0.3
0.2
0.3
-0.3
0.5
(c) t'/t = 0.00
0.1
-0.15
0.3
0.2
pure AF
0.06
t'/t
(b) t'/t = -0.15
Hole doped
-0.15
0
0
Electron doped
no order
0.1
0.2
0.3
doping rate Fig. 2. Phase diagrams constructed within present wave function in d-t0 /t plane for (a) U/t = 16 and (b) U/t = 30.
0.1 0
0 0
0.1
0.2
0.3
0.5
(d) t'/t = +0.15
0.06
Pd Pd
m
0.3
U/t =16 U/t =30
0.04
0.2 0.02
Pd (L/2,L/2)
0.4
0.1 0
0 0
0.1
0.2
to the value of U/t, t0 /t and d. The d dependence of SC and AF orders is similar to that of the t–J model as a whole, but the t–J model has a wide area of the coexisting phase even for t0 /t = 0.3 [9]. In connection with the pseudo-gap problem, we confirmed that the gradient of momentum distribution function at the antinodal point regions (k (p, 0)) mainly dominates the magnitude of the d-wave SC correlation function, and the electronic state at the antinodal point is essential to SC [3]. In conclusion, we have studied the interplay between AF and SC in the Hubbard model. For large U/t, it is found that the d-wave SC and AF ordered states become competing or coexisting, according to the values of model parameters.
0.3
References 0.5
(e) t'/t = +0.30
0.06
m U/t =16 m U/t =30
m
0.3
0.04
0.2 0.02
Pd (L/2,L/2)
0.4
0.1 0
0 0
0.1
0.2
0.3
doping rate Fig. 1. By using the optimized wave function, the expectation values of staggered magnetization m (triangles) and d-wave nearest-neighbor pair correlation function Pd(r) for the farthermost distance r = (L/2, L/2) (diamonds and squares) are plotted as a function of d for different values of t0 /t (=0, ± 0.15, ± 0.3). Data for U/t = 16 and 30 are simultaneously plotted in each panel. Arrows indicate changes of the magnitude of AF and SC orders from U/t = 16 to 30.
[1] H. Mukuda, M. Abe, Y. Araki, Y. Kitaoka, K. Tokiwa, T. Watanabe, A. Iyo, H. Kito, Y. Tanaka, Phys. Rev. Lett. 96 (2006) 087001; H. Mukuda, Y. Yamaguchi, S. Shimizu, Y. Kitaoka, P. Shirage, A. Iyo, J. Phys. Soc. Jpn. 77 (2008) 124706. [2] K. Kobayashi, H. Yokoyama, J. Phys. Chem. Solids 69 (2008) 3274. [3] K. Kobayashi, H. Yokoyama, Physica C 469 (2009) 974. [4] T. Giamarchi, C. Lhuillier, Phys. Rev. B 43 (1991) 12943. [5] A. Himeda, M. Ogata, Phys. Rev. B 60 (1999) R9935. [6] C.T. Shih, Y.C. Chen, C.P. Chou, T.K. Lee, Phys. Rev. B 70 (2004) 220502R. [7] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 59 (1990) 3669. [8] H. Yokoyama, Prog. Theor. Phys. 108 (2002) 59. [9] L. Spanu, M. Lugas, F. Becca, S. Sorella, Phys. Rev. B 77 (2008) 024510.