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Phase diagram and BCS–Bose condensation crossover in 1D and 2D Hubbard models
Physica C 364±365 (2001) 131±133
www.elsevier.com/locate/physc
Phase diagram and BCS±Bose condensation crossover in 1D and 2D Hubbard models A.N. Kocharian a,*, C. Yang b, Y.L. Chiang c a
Department of Physics and Astronomy, California State University, Northridge, CA 91330-8268, USA b Department of Physics, Tamkang University, Tamsui 251, Taiwan, ROC c Department of Physics, Chinese Culture University, Taipei 111, Taiwan, ROC
Abstract The ground-state energy, the concentration of double occupied sites and the density of bound electron pairs in lowdimensional attractive Hubbard model are calculated by means of the generalized self-consistent ®eld (GSCF) approach. This approach gives an excellent agreement with the corresponding Bethe-ansatz solution and the quantum Monte-Carlo calculations in a wide range of interaction strength U < 0 and electron concentration n. A simple relationship is found between the order parameter and the density of bound electron pairs, valid for any dimension and in entire parameter space U, n and magnetic ®eld h. The electron pairing and the crossover from the itinerant BCS regime into the Bose condensation regime of local pairs are studied in dependence of both U and n. The GSCF theory correctly displays the separation of the energy gap from the BCS order parameter at n 6 1 as jU j increases. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 71.10.Fd; 74.20. z; 74.20.Fg Keywords: Hubbard model; Bethe-ansatz; Quantum Monte-Carlo; BCS±BC crossover
Unlike the conventional superconductors in which the phase coherence and pairing occur simultaneously at the same temperature, in the underdoped cuprates there exists a separation between the pair binding regime (energy gap) and the superconductivity (coherency between pairs) regime as the doping exceeds an optimal value. The smooth crossover from the itinerant BCS regime into the localized Bose condensation (BC) state from weak to strong coupling superconductivity [1,2] has been a subject of many recent investiga* Corresponding author. Tel.: +1-818-3418938; fax: +1-818677-3234. E-mail address: [email protected] (A.N. Kocharian).
tions in low-dimensional Hubbard lattices [3±6] due to its possible relevance in high temperature superconductivity (HTSC) [7,8]. In this paper we apply the generalized selfconsistent ®eld (GSCF) treatment in one- and twodimensional (1D and 2D) lattices, by introducing the order (correlation) parameter D q with wave vector 0 6 jqj 6 p and the renormalized chemical potential l to study the crossover and ground-state (GS) properties within the attractive Hubbard model in wide range of coupling strength U and electron concentration n [4,5]. Up to date the rigorous results on the Hubbard model are rare except for 1D Bethe-ansatz formalism in the thermodynamic limit, which is free from the ®nite-size eects [9±14]. In one-dimension
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 7 3 0 - 4
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A.N. Kocharian et al. / Physica C 364±365 (2001) 131±133
we compare the GSCF numerical calculations with the Bethe-ansatz solution in entire range of U, n and magnetic ®eld h [4,5,15]. In two-dimensions the exact solution is not known yet, however within the statistical error the quantum MonteCarlo (QMC) approach correctly reproduces the solution [16±21]. Below we compare the predictions of the GSCF theory for the linear and square lattices with those of the Bethe-ansatz [4,5] and constrained-path QMC method [22]. The local pairing is determined by the average concentration of the double occupied sites D. Within the GSCF approach it is given by a simple expression D0 D q 2 =4U 2 , valid for arbitrary dimension, U < 0, h > 0 and 0 6 n 6 1. Parameter D0 n2 =4 s2 is the concentration of unbound electron pairs at U 0 and in the presence of magnetic ®eld h [23]. In Fig. 1 we see the excellent agreement between the Bethe-ansatz (1D) and QMC (2D) results at h 0 and the corresponding GSCF results for the GS energy EGS . A slight deviation exists only at weak coupling [4,5].
Fig. 1. The GS energy EGS =t at h 0 in the GSCF approach (dashed curves) along with the 1D Bethe-ansatz result (solid curves) and the 2D QMC result (circles, triangles and squares) as a function of U =t for various n and as a function of n for various U =t.
The concentration of bound electron pairs is given by the simple expression npair D D0 . In 2 the GSCF approach it reduces to npair D q = 2 4U . The GSCF results clearly show that the order parameter D q , in spite of previous common belief that it imitates the energy gap Egap , closely follows the parameter D and has de®nite physical meaning associated with npair [4,5]. Our understanding is based on the fact, that D q reproduces Egap in the limited space only, while there exists a strict cor2 respondence between npair and D q =4U 2 in the entire parameter space of U, h > 0 and n. Fig. 2 shows the perfect matching between the Bethe-ansatz (1D) and QMC (2D) results for npair and the corresponding result in the GSCF approach. Note that npair increases monotonically with increasing of n and jU j. The position k0 of the minimum of the excitation spectrum Ek distinguishes the itinerant BCS regime (k0 6 0) from the BC regime (k0 0) [4]. At h 0 in the BCS regime Egap D0 , in the BC regime Egap > D0 . Note that at small jU j the critical n for BCS±BC crossover is proportional to jU j in 1D case, while in 2D it approaches to zero much faster (Fig. 3). Thus the itinerant BCS phase occupies larger space U±n in two-dimension. Also
Fig. 2. The concentration of bound electron pairs npair at h 0. The notation is the same as for Fig. 1.
A.N. Kocharian et al. / Physica C 364±365 (2001) 131±133
133
Acknowledgements The authors acknowledge Dr. H.Q. Lin and Dr. Z. Huang for providing us with the corresponding QMC results and the National Science Council of ROC for support (grant no. NSC 90-2112-M-032003). References
Fig. 3. The phase diagram in U ±n plane and the boundaries between the BCS (I) and BC (II) regimes for 1D and 2D cases. The coordination number z 2 for 1D, and z 4 for 2D case.
there is no BC of electrons both in 1D and 2D cases at n 1 no matter how strong is jU j. The BCS±BC crossover in one-dimension at n 6 1 is overall in agreement with the continuum model [3,4]. In summary, the GSCF theory is in good agreement with the Bethe-ansatz solution and the QMC results. The GSCF theory correctly displays the separation of the energy gap from the order parameter in the overdoped HTSC cuprates. The GSCF theory suggests a smooth crossover in the GS from the itinerant BCS-like regime into the localized BC state in one- and two-dimension. The GSCF theory in any dimension and entire parameter space, including the magnetic ®eld h, gives the simple relationship between the inhomogeneous BCS order parameter D q with q 6 0 and the corresponding density of bound pairs npair in the presence of magnetic ®eld [23].
[1] A.J. Leggett, J. Phys. (Paris) Colloq. 41 (1980) C7±19. [2] P. Nozieres, S. Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195. [3] M. Cesas, C. Essebbag, A. Extremera, J.M. Getino, M. de Llano, A. Plastino, H. Rubio, Phys. Rev. A 44 (1991) 4915. [4] A.N. Kocharian, C. Yang, Y.L. Chiang, Phys. Rev. B 59 (1999) 7458. [5] A.N. Kocharian, C. Yang, Y.L. Chiang, Int. J. Modern Phys. B 29±31 (1999) 3538. [6] K. Tanaka, F. Marsiglio, Phys. Rev. B 60 (1999) 3508. [7] V.J. Emery, S.A. Kivelson, Physica C 209 (1993) 597. [8] V.J. Emery, S.A. Kivelson, J. Phys. Chem. Solids 59 (1998) 1705. [9] M. Takahashi, Progr. Theor. Phys. 42 (1969) 1098. [10] M. Takahashi, Progr. Theor. Phys. 44 (1970) 348. [11] M. Takahashi, Progr. Theor. Phys. 43 (1970) 1619. [12] T.B. Bahder, F. Woynarovich, Phys. Rev. B 33 (1986) 2114. [13] K. Lee, P. Schlottmann, Phys. Rev. B 38 (1988) 11566. [14] K. Lee, P. Schlottmann, Phys. Rev. B 40 (1989) 9104. [15] C. Yang, A.N. Kocharian, Y.L. Chiang, Int. J. Phys.: Condens. Matter 12 (2000) 7433. [16] J. Singer, T. Schneider, M.H. Pedersen, Eur. Phys. J. B (1998) 7458. [17] J.M.. Singer, M.H. Pedersen, T. Schneider, H. Beck, H.G. Matuttis, Phys. Rev. B 54 (1996) 1286. [18] S. Zhang, J. Carlson, J.E. Gubernatis, Phys. Rev. Lett. 74 (1995) 3652. [19] S. Zhang, J. Carlson, J.E. Gubernatis, Phys. Rev. B 55 (1997) 7464. [20] M. Guerrero, G. Ortiz, J.E. Gubernatis, Phys. Rev. B 59 (1999) 1706. [21] J. Carlson, J.E. Gubernatis, G. Ortiz, S. Zhang, Phys. Rev. B 59 (1999) 12788. [22] H.Q. Lin, private communication. [23] C. Yang, A.N. Kocharian, Y.L. Chiang, Physica C 364± 365 (2001) 155, these proceedings.
www.elsevier.com/locate/physc
Phase diagram and BCS±Bose condensation crossover in 1D and 2D Hubbard models A.N. Kocharian a,*, C. Yang b, Y.L. Chiang c a
Department of Physics and Astronomy, California State University, Northridge, CA 91330-8268, USA b Department of Physics, Tamkang University, Tamsui 251, Taiwan, ROC c Department of Physics, Chinese Culture University, Taipei 111, Taiwan, ROC
Abstract The ground-state energy, the concentration of double occupied sites and the density of bound electron pairs in lowdimensional attractive Hubbard model are calculated by means of the generalized self-consistent ®eld (GSCF) approach. This approach gives an excellent agreement with the corresponding Bethe-ansatz solution and the quantum Monte-Carlo calculations in a wide range of interaction strength U < 0 and electron concentration n. A simple relationship is found between the order parameter and the density of bound electron pairs, valid for any dimension and in entire parameter space U, n and magnetic ®eld h. The electron pairing and the crossover from the itinerant BCS regime into the Bose condensation regime of local pairs are studied in dependence of both U and n. The GSCF theory correctly displays the separation of the energy gap from the BCS order parameter at n 6 1 as jU j increases. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 71.10.Fd; 74.20. z; 74.20.Fg Keywords: Hubbard model; Bethe-ansatz; Quantum Monte-Carlo; BCS±BC crossover
Unlike the conventional superconductors in which the phase coherence and pairing occur simultaneously at the same temperature, in the underdoped cuprates there exists a separation between the pair binding regime (energy gap) and the superconductivity (coherency between pairs) regime as the doping exceeds an optimal value. The smooth crossover from the itinerant BCS regime into the localized Bose condensation (BC) state from weak to strong coupling superconductivity [1,2] has been a subject of many recent investiga* Corresponding author. Tel.: +1-818-3418938; fax: +1-818677-3234. E-mail address: [email protected] (A.N. Kocharian).
tions in low-dimensional Hubbard lattices [3±6] due to its possible relevance in high temperature superconductivity (HTSC) [7,8]. In this paper we apply the generalized selfconsistent ®eld (GSCF) treatment in one- and twodimensional (1D and 2D) lattices, by introducing the order (correlation) parameter D q with wave vector 0 6 jqj 6 p and the renormalized chemical potential l to study the crossover and ground-state (GS) properties within the attractive Hubbard model in wide range of coupling strength U and electron concentration n [4,5]. Up to date the rigorous results on the Hubbard model are rare except for 1D Bethe-ansatz formalism in the thermodynamic limit, which is free from the ®nite-size eects [9±14]. In one-dimension
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 7 3 0 - 4
132
A.N. Kocharian et al. / Physica C 364±365 (2001) 131±133
we compare the GSCF numerical calculations with the Bethe-ansatz solution in entire range of U, n and magnetic ®eld h [4,5,15]. In two-dimensions the exact solution is not known yet, however within the statistical error the quantum MonteCarlo (QMC) approach correctly reproduces the solution [16±21]. Below we compare the predictions of the GSCF theory for the linear and square lattices with those of the Bethe-ansatz [4,5] and constrained-path QMC method [22]. The local pairing is determined by the average concentration of the double occupied sites D. Within the GSCF approach it is given by a simple expression D0 D q 2 =4U 2 , valid for arbitrary dimension, U < 0, h > 0 and 0 6 n 6 1. Parameter D0 n2 =4 s2 is the concentration of unbound electron pairs at U 0 and in the presence of magnetic ®eld h [23]. In Fig. 1 we see the excellent agreement between the Bethe-ansatz (1D) and QMC (2D) results at h 0 and the corresponding GSCF results for the GS energy EGS . A slight deviation exists only at weak coupling [4,5].
Fig. 1. The GS energy EGS =t at h 0 in the GSCF approach (dashed curves) along with the 1D Bethe-ansatz result (solid curves) and the 2D QMC result (circles, triangles and squares) as a function of U =t for various n and as a function of n for various U =t.
The concentration of bound electron pairs is given by the simple expression npair D D0 . In 2 the GSCF approach it reduces to npair D q = 2 4U . The GSCF results clearly show that the order parameter D q , in spite of previous common belief that it imitates the energy gap Egap , closely follows the parameter D and has de®nite physical meaning associated with npair [4,5]. Our understanding is based on the fact, that D q reproduces Egap in the limited space only, while there exists a strict cor2 respondence between npair and D q =4U 2 in the entire parameter space of U, h > 0 and n. Fig. 2 shows the perfect matching between the Bethe-ansatz (1D) and QMC (2D) results for npair and the corresponding result in the GSCF approach. Note that npair increases monotonically with increasing of n and jU j. The position k0 of the minimum of the excitation spectrum Ek distinguishes the itinerant BCS regime (k0 6 0) from the BC regime (k0 0) [4]. At h 0 in the BCS regime Egap D0 , in the BC regime Egap > D0 . Note that at small jU j the critical n for BCS±BC crossover is proportional to jU j in 1D case, while in 2D it approaches to zero much faster (Fig. 3). Thus the itinerant BCS phase occupies larger space U±n in two-dimension. Also
Fig. 2. The concentration of bound electron pairs npair at h 0. The notation is the same as for Fig. 1.
A.N. Kocharian et al. / Physica C 364±365 (2001) 131±133
133
Acknowledgements The authors acknowledge Dr. H.Q. Lin and Dr. Z. Huang for providing us with the corresponding QMC results and the National Science Council of ROC for support (grant no. NSC 90-2112-M-032003). References
Fig. 3. The phase diagram in U ±n plane and the boundaries between the BCS (I) and BC (II) regimes for 1D and 2D cases. The coordination number z 2 for 1D, and z 4 for 2D case.
there is no BC of electrons both in 1D and 2D cases at n 1 no matter how strong is jU j. The BCS±BC crossover in one-dimension at n 6 1 is overall in agreement with the continuum model [3,4]. In summary, the GSCF theory is in good agreement with the Bethe-ansatz solution and the QMC results. The GSCF theory correctly displays the separation of the energy gap from the order parameter in the overdoped HTSC cuprates. The GSCF theory suggests a smooth crossover in the GS from the itinerant BCS-like regime into the localized BC state in one- and two-dimension. The GSCF theory in any dimension and entire parameter space, including the magnetic ®eld h, gives the simple relationship between the inhomogeneous BCS order parameter D q with q 6 0 and the corresponding density of bound pairs npair in the presence of magnetic ®eld [23].
[1] A.J. Leggett, J. Phys. (Paris) Colloq. 41 (1980) C7±19. [2] P. Nozieres, S. Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195. [3] M. Cesas, C. Essebbag, A. Extremera, J.M. Getino, M. de Llano, A. Plastino, H. Rubio, Phys. Rev. A 44 (1991) 4915. [4] A.N. Kocharian, C. Yang, Y.L. Chiang, Phys. Rev. B 59 (1999) 7458. [5] A.N. Kocharian, C. Yang, Y.L. Chiang, Int. J. Modern Phys. B 29±31 (1999) 3538. [6] K. Tanaka, F. Marsiglio, Phys. Rev. B 60 (1999) 3508. [7] V.J. Emery, S.A. Kivelson, Physica C 209 (1993) 597. [8] V.J. Emery, S.A. Kivelson, J. Phys. Chem. Solids 59 (1998) 1705. [9] M. Takahashi, Progr. Theor. Phys. 42 (1969) 1098. [10] M. Takahashi, Progr. Theor. Phys. 44 (1970) 348. [11] M. Takahashi, Progr. Theor. Phys. 43 (1970) 1619. [12] T.B. Bahder, F. Woynarovich, Phys. Rev. B 33 (1986) 2114. [13] K. Lee, P. Schlottmann, Phys. Rev. B 38 (1988) 11566. [14] K. Lee, P. Schlottmann, Phys. Rev. B 40 (1989) 9104. [15] C. Yang, A.N. Kocharian, Y.L. Chiang, Int. J. Phys.: Condens. Matter 12 (2000) 7433. [16] J. Singer, T. Schneider, M.H. Pedersen, Eur. Phys. J. B (1998) 7458. [17] J.M.. Singer, M.H. Pedersen, T. Schneider, H. Beck, H.G. Matuttis, Phys. Rev. B 54 (1996) 1286. [18] S. Zhang, J. Carlson, J.E. Gubernatis, Phys. Rev. Lett. 74 (1995) 3652. [19] S. Zhang, J. Carlson, J.E. Gubernatis, Phys. Rev. B 55 (1997) 7464. [20] M. Guerrero, G. Ortiz, J.E. Gubernatis, Phys. Rev. B 59 (1999) 1706. [21] J. Carlson, J.E. Gubernatis, G. Ortiz, S. Zhang, Phys. Rev. B 59 (1999) 12788. [22] H.Q. Lin, private communication. [23] C. Yang, A.N. Kocharian, Y.L. Chiang, Physica C 364± 365 (2001) 155, these proceedings.