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Exact magnetic ground-state properties of the one-dimensional Hubbard model
PHYSlCA ELSEVIER
Physica C 341-348 (2000) 245-246 www.elseviernl/Iocate/physc
Exact Magnetic Ground-State Properties of the One-Dimensional Hubbard Model C. Yang a , A. N. Kocharian b and Y. L. Chiang e aDepartment of Physics, Tamkang University, Tamsui, Talwan 251, ROC bDepartment of Physics and Astronomy, California State University, Northridge, CA 91330-8268, USA CDepartment of Physics, Chinese Culture University, Taipei, Taiwan 111, ROC; Department of Physics, Tamkang University, Tamsui, Taiwan 251, ROC T h e exact solution of the one-dimensional H u b b a r d model, b o t h attractive and repulsive, is investigated in dependence of electron concentration n over a wide range of interaction s t r e n g t h U and magnetic field h. The magnetic phase diagram and the ground-state properties, including the total energy, the n u m b e r of double occupied sites and t h e kinetic energy are studied for - c ~ < U < oo and 0 < n < 1.
In search for physical mechanism responsible for the superconductivity and magnetism in many new materials it is instructive to investigate a simplified one-dimensional model with magnetic field. The Bethe-ansatz formalism provides an exact solution for the various physical quantities within the 1D Hubbard model [1-3]. The attractive model in the ground state displays a superconductivity [4] and crossover from the itinerant BCS behavior into the Bose condensation regime of local pairs and may reveal similarities in higher dimensions [5]. In this paper we extend the previous results and study the competition between the electronelectron or electron-hole pairing and the pair breaking effect of the magnetic field through the exact calculation of the ground-state characteristics in entire space of U, n, and h. First we study the spin (magnetic) phase diagram in the n - h plane (Fig. 1). There are the lower hcl(U, n) (bold curves) and the upper he2(U, n) (solid curves) critical magnetic fields for the spin onset and spin saturation respectively, which both depend on n and U/t. For U > 0 the lower critical field he1 is identically equal to zero. These critical magnetic fields he1 and he2 give the boundaries between three different phases: I. zero-spin phase at h < he1 ; II. intermediate phase
0 < s < n/2 at he1 < h < hc2; III. saturated spin phase s = n/2 at h > he2. In the attractive model (U < 0) the lower critical field he1, associated with the spin energy gap, slightly decreases with increasing n at fixed U from its maximum value at n = 0 to its minimal value at n = 1. In contrast, the upper critical field he2 increases remarkably by increasing of n. In the repulsive model (U > 0) the spin energy gap is absent so he1 -= 0 and an infinitesimal h results in s ¢ 0 phase. The critical field he2 also increases by increasing of n (Fig. 1).
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Figure 1. The phase diagram.
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. Pll S0921-4534(00)00462-7
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246
C. Yung et al./Physica C 341-348 (2000) 245-246
The behavior of the total GS energy E h / t vs n (Fig. 2), including the Zeeman energy, is monotonous for U < 0 [6] and nonmonotonous for U > 0. At a sufficiently small magnetic field h < hcl(U, 1) (h = 0 for U > 0) the system is in the phase I for the whole range of 0 < n < 1. In the attractive case at hcl(U, 1) < h < h~l(U,O) the system undergoes a transition from the phase I into the phase II at some critical electron concentration no1 (upward triangle in Fig. 2 at h i t = 1.5, U / t = - 4 , ncl = 0.454) by increasing of n. At a stronger field h~2(U,0) < h < hc2(U, 1) (see Fig. 1) the system undergoes a transition from the phase III into the phase II at some another critical electron concentration n~2 (downward triangles). Finally, at a sufficiently strong field h > hc2(U, 1) the spin of the system is fully saturated (s = n / 2 ) (phase III) for the whole range of n. The curves E h / t vs n are smooth at mentioned phase transitions. The concentration of double occupied sites D for given h / t increases monotonously with n in a wide range of U / t (Fig. 3). In the phase III (n < no2) D = 0, so D appears only at n > nc2 and is strongly suppressed as U increases algebraically. In the phase I and [U[ is large (U < O) D .~ n / 2 [5]. The transition from the phase I into the phase II in D vs n is sharp. The kinetic energy Ekin vs n is a monotonous function in the attractive case (U < 0) and it is nonmonotonous at U > 0. In the phase III Ekin is independent of U / t and the corresponding curves coincide. At phase transition III-II these curves have strongly pronounced fracture. On the other hand, the phase transition I-II is always smooth. The suppression of the spin breaking excitations in the phase I is connected with the presence of spin energy gap at arbitrary U < 0. In contrast, for U > 0 there is always a spin response on an infinitesimal magnetic field (hcl = 0) and there is no spin gap. Our numerical results are in full agreement with analytical expansions at strong and weak interaction limits and provide a solid ground for different self-consistent studies for intermediate region as well [6,7]. The authors wish to acknowledge the National Science Council of R.O.C. for support (Grant
M-05
.
-
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Figure 2. Eh vs n for various U / t (curve indexes).
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.
.
.
.
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Figure 3. The double occupied sites D vs n .
No. NSC89-2112-M-032-010). REFERENCES 1. E. H. Lieb, F. Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. 2. M. Takahashi, Progr. Theor. Phys. 42 (1969) 1098; 44 (1970) 348; 43 (1970) 1619. 3. H. Shiba, Phys. Rev. 6 (1972) 930. 4. M.Bogoliubov, V.E. Korepin, Modern Phys. Lett., B 1 (1988) 349. 5. A. N. Kocharian, C.. Yang, Y. L. Chiang, Phys. Rev. B 59 (1999) 7458; Physica B 259261 (1999) 739. 6. C. Yang, A. N. Kocharian, Y. L. Chiang, Phys. Rev. B, submitted. 7. A.N. Kocharian, C. Yang, Y. L. Chiang, Inter. J. Mod. Phys. B, to be published.
Physica C 341-348 (2000) 245-246 www.elseviernl/Iocate/physc
Exact Magnetic Ground-State Properties of the One-Dimensional Hubbard Model C. Yang a , A. N. Kocharian b and Y. L. Chiang e aDepartment of Physics, Tamkang University, Tamsui, Talwan 251, ROC bDepartment of Physics and Astronomy, California State University, Northridge, CA 91330-8268, USA CDepartment of Physics, Chinese Culture University, Taipei, Taiwan 111, ROC; Department of Physics, Tamkang University, Tamsui, Taiwan 251, ROC T h e exact solution of the one-dimensional H u b b a r d model, b o t h attractive and repulsive, is investigated in dependence of electron concentration n over a wide range of interaction s t r e n g t h U and magnetic field h. The magnetic phase diagram and the ground-state properties, including the total energy, the n u m b e r of double occupied sites and t h e kinetic energy are studied for - c ~ < U < oo and 0 < n < 1.
In search for physical mechanism responsible for the superconductivity and magnetism in many new materials it is instructive to investigate a simplified one-dimensional model with magnetic field. The Bethe-ansatz formalism provides an exact solution for the various physical quantities within the 1D Hubbard model [1-3]. The attractive model in the ground state displays a superconductivity [4] and crossover from the itinerant BCS behavior into the Bose condensation regime of local pairs and may reveal similarities in higher dimensions [5]. In this paper we extend the previous results and study the competition between the electronelectron or electron-hole pairing and the pair breaking effect of the magnetic field through the exact calculation of the ground-state characteristics in entire space of U, n, and h. First we study the spin (magnetic) phase diagram in the n - h plane (Fig. 1). There are the lower hcl(U, n) (bold curves) and the upper he2(U, n) (solid curves) critical magnetic fields for the spin onset and spin saturation respectively, which both depend on n and U/t. For U > 0 the lower critical field he1 is identically equal to zero. These critical magnetic fields he1 and he2 give the boundaries between three different phases: I. zero-spin phase at h < he1 ; II. intermediate phase
0 < s < n/2 at he1 < h < hc2; III. saturated spin phase s = n/2 at h > he2. In the attractive model (U < 0) the lower critical field he1, associated with the spin energy gap, slightly decreases with increasing n at fixed U from its maximum value at n = 0 to its minimal value at n = 1. In contrast, the upper critical field he2 increases remarkably by increasing of n. In the repulsive model (U > 0) the spin energy gap is absent so he1 -= 0 and an infinitesimal h results in s ¢ 0 phase. The critical field he2 also increases by increasing of n (Fig. 1).
le
'
oo
I
0.2
*
[
'
04
I
o6 /?
Figure 1. The phase diagram.
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. Pll S0921-4534(00)00462-7
'
I
(24B
'
1
246
C. Yung et al./Physica C 341-348 (2000) 245-246
The behavior of the total GS energy E h / t vs n (Fig. 2), including the Zeeman energy, is monotonous for U < 0 [6] and nonmonotonous for U > 0. At a sufficiently small magnetic field h < hcl(U, 1) (h = 0 for U > 0) the system is in the phase I for the whole range of 0 < n < 1. In the attractive case at hcl(U, 1) < h < h~l(U,O) the system undergoes a transition from the phase I into the phase II at some critical electron concentration no1 (upward triangle in Fig. 2 at h i t = 1.5, U / t = - 4 , ncl = 0.454) by increasing of n. At a stronger field h~2(U,0) < h < hc2(U, 1) (see Fig. 1) the system undergoes a transition from the phase III into the phase II at some another critical electron concentration n~2 (downward triangles). Finally, at a sufficiently strong field h > hc2(U, 1) the spin of the system is fully saturated (s = n / 2 ) (phase III) for the whole range of n. The curves E h / t vs n are smooth at mentioned phase transitions. The concentration of double occupied sites D for given h / t increases monotonously with n in a wide range of U / t (Fig. 3). In the phase III (n < no2) D = 0, so D appears only at n > nc2 and is strongly suppressed as U increases algebraically. In the phase I and [U[ is large (U < O) D .~ n / 2 [5]. The transition from the phase I into the phase II in D vs n is sharp. The kinetic energy Ekin vs n is a monotonous function in the attractive case (U < 0) and it is nonmonotonous at U > 0. In the phase III Ekin is independent of U / t and the corresponding curves coincide. At phase transition III-II these curves have strongly pronounced fracture. On the other hand, the phase transition I-II is always smooth. The suppression of the spin breaking excitations in the phase I is connected with the presence of spin energy gap at arbitrary U < 0. In contrast, for U > 0 there is always a spin response on an infinitesimal magnetic field (hcl = 0) and there is no spin gap. Our numerical results are in full agreement with analytical expansions at strong and weak interaction limits and provide a solid ground for different self-consistent studies for intermediate region as well [6,7]. The authors wish to acknowledge the National Science Council of R.O.C. for support (Grant
M-05
.
-
h~
15
I
1)
Figure 2. Eh vs n for various U / t (curve indexes).
M:O
M=0.5
-
M=I
.
.
O3
Q 02
Ol
O0 O~
.
.
.
.
.
.
.
.
.
O~ n
Figure 3. The double occupied sites D vs n .
No. NSC89-2112-M-032-010). REFERENCES 1. E. H. Lieb, F. Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. 2. M. Takahashi, Progr. Theor. Phys. 42 (1969) 1098; 44 (1970) 348; 43 (1970) 1619. 3. H. Shiba, Phys. Rev. 6 (1972) 930. 4. M.Bogoliubov, V.E. Korepin, Modern Phys. Lett., B 1 (1988) 349. 5. A. N. Kocharian, C.. Yang, Y. L. Chiang, Phys. Rev. B 59 (1999) 7458; Physica B 259261 (1999) 739. 6. C. Yang, A. N. Kocharian, Y. L. Chiang, Phys. Rev. B, submitted. 7. A.N. Kocharian, C. Yang, Y. L. Chiang, Inter. J. Mod. Phys. B, to be published.