THERMODYNAMIC PROPERTIES OF ONE-DIMENSIONAL HUBBARD MODEL AT FINITE TEMPERATURES

Transcription

1 International Journal of Modern Physics B Vol. 17, Nos. 18, 19 & 20 (2003) c World Scientific Publishing Company THERMODYNAMIC PROPERTIES OF ONE-DIMENSIONAL HUBBARD MODEL AT FINITE TEMPERATURES C. YANG Department of Physics, Tamkang University, Tamsui, Taiwan 251, ROC A. N. KOCHARIAN Department of Physics, California State University, Northridge, CA , USA Y. L. CHIANG Department of Physics, Chinese Culture University, Taipei, Taiwan 111, ROC Received 16 January 2003 The phase diagram and thermodynamic properties of the Hubbard model in one dimension are calculated numerically within the generalized self-consistent approach over a wide range of temperature T, magnetic field h, interaction strength U/t and electron concentration n. The temperature variation of the transverse magnetization, double occupied sites and chemical potential at various n, U/t and h is also analyzed. The wave vector q for the magnetic order parameter in the vicinity of the half-filling in the metallic phase shows a crossover from incommensurate magnetic phase (0 < q < π) into the state with strong antiferromagnetic correlations (q = π) as temperature is increased. Overall our numerical results in the approximate theory at finite temperatures are in good quantitative and qualitative agreement with the Bethe-ansatz results. There are many open questions and unresolved problems with regard to understanding of electron correlations and magnetism in half metallic magnets, quasi-1d magnetic conductors, insulators and superconductors. 1 3 Much progress has been achieved in understanding of the ground-state properties and electron correlations in low-dimensional Hubbard model by comparison of its predictions in generalized self-consistent field (GSCF) approach with exact diagonalization, Bethe-ansatz and quantum Monte-Carlo (QMC) calculations. 4 It is common knowledge that mean-field theory produces continuous broken symmetry ground states, while temperature (Mermin-Wagner theorem) and quantum fluctuations typically prevents the system from forming of the long-range order in one dimension. However, the approximate GSCF theory even in extreme conditions of one dimensionality reproduces qualitatively and in some cases quantitatively the general ground state features of the Hubbard model. 4,5 Therefore there is a reason to hope that, in some circumstances, the short-range correlations within mean-field 3354

2 Thermodynamic Properties of One-Dimensional Hubbard Model 3355 theory can properly capture the thermodynamic properties also at finite temperatures T/t. In this paper the universal generalized self-consistent field (GSCF) is applied in one dimension to study the phase diagram and behavior of the transverse magnetization and the chemical potential of the Hubbard model with magnetic field h in wide range of temperature. Our GSCF approach is straightforward extension of the existing mean-field solutions for incommensurate spiral phase with the wave number 0 < q < π, for general U/t, h, 0 n 1 and all temperatures. 6 9 The variation of the order parameter (+) q 2U k c + q k c k /N latt versus temperature is shown in Fig. 1 for U/t = 6 and h = 0. At n = 1, the parameters q = π and (+) π is equal to the energy gap. At n 1, there is an incommensurate phase 0 < q < π (between the triangles in Fig. 1). At small n we have transverse ferromagnetism (q = 0) with (+) q 0. At n 1, (+) q versus T/t shows slightly non-monotonic behavior and vanishes at some critical temperatures T c depending on n and h/t. The state with strong antiferromagntic correlations (q = π) becomes stable in some range of temperatures and magnetic Fig. 1. The GSCF order parameter (+) q /t versus temperature T/t for various n (figures label the curves) and h/t = 0 and In the regions between the triangles 0 < q < π, on the right hand of these regions q = π. At (+) q = 0 wavenumber q is arbitrary.

3 3356 C. Yang, A. N. Kocharian & Y. L. Chiang Fig. 2. The phase T h diagram of the one-dimensional Hubbard model. The white area corresponds to the normal state with (+) q = 0 and q arbitrary. The light shaded area (I) is an antiferromagnetic state with (+) π with (+) q 0 and 0 < q < π. 0 and q = π. The dark shaded area (II) is a spiral phase field. Away from n = 1 the GSCF theory provides a smooth transition in the absence and presence of magnetic field h from the incommensurate state with the order parameter (+) q 0, where the wavenumber 0 < q < π into antiferromagnetic state with q = π. Figure 2 shows the phase T h diagram in the presence of magnetic field for U/t = 4 and n = 0.8. In general, relatively large doping results in a phase diagram of the metallic state which consists of three different phases. The white area indicates the normal phase with (+) q = 0 and arbitrary q. This phase is stable at relatively large h/t or high temperatures. The separation boundary T c versus h/t between the dark and light shaded areas describes the smooth transition from the normal metallic state into antiferromagnetic phase. There is also a narrow range of h where incommensurate phase one can reach from the normal phase in strong magnetic field by increasing the temperature. Further increase of temperature results in development of the antiferromagnetic correlations. We compared the dependence of the GSCF chemical potential µ (+) versus electron concentration and the Bethe-ansatz result for fixed U/t and h/t. The GSCF results qualitatively well reproduce the exact Bethe-ansatz except in close vicinity of half-filling. In contrast to the exact theory, the GSCF chemical potential

4 Thermodynamic Properties of One-Dimensional Hubbard Model 3357 Fig. 3. Number of double occupied sites D (+) as a function of T/t for various n and h/t. The notation is the same as for Fig. 1. versus n in metallic phase is a non-monotonous function in the ground state and at very low temperatures. At finite temperatures it becomes monotonous. In Fig. 3 the variation of double occupied sites D under the variation of temperature is shown for h = 0 and h/t = In general D increases with temperature as in exact theory. The Hubbard model with U > 0 in the presence of magnetic field h 0 (s 0) can be reinterpreted in terms of the electron-hole pairs for the attractive Hubbard model, using the exact mapping between the repulsive and attractive Hubbard models. Earlier we found the stability for inhomogeneous Larkin Fulde Ferrel Ovchinnikov superconducting phase with finite magnetization s 0 and q 0 and q 0. 4 Based on this analogy we have for U > 0 the transition close to the halffilling case under the variation of n from antiferromagnetic dielectric with q = π into incommensurate metallic state with q π. In summary, the GSCF theory in the ground state and at finite temperatures reproduces qualitatively and sometimes quantitatively the general features of the Bethe-ansatz solution. The GSCF theory predicts the enhancement of antiferromagnetic correlations with temperature in metallic state and properly describes observed instability in behavior of µ (+) for PES and ARPES data in Nd 2 x Ce x CuO 4. 13,14

5 3358 C. Yang, A. N. Kocharian & Y. L. Chiang Acknowledgments The authors acknowledge the National Science Council of R.O.C. for support (grants NSC M and NSC M ). References 1. Yu Irkhin and M. I. Katsnel son, Sov. Physics-Uspekhi 37, 659 (1994). 2. R. Weht and W. E. Pickett, Phys. Rev. B60, (1999). 3. P. Nozieres, Eur. Phys. J. B10, 649 (1999). 4. A. N. Kocharian, C. Yang and Y. L. Chiang, Phys. Rev. B59, 7458 (1999); Inter. J. Modern. Phys. B29 31, 3538 (1999). 5. C. Yang, A. N. Kocharian and Y. L. Chiang, J. Phys.: Condens. Matter B12, 7433 (2000); Physica C , 245 (2000). 6. B. I. Shraiman and E. D. Siggia, Phys. Rev. B46, 8305 (1992). 7. R. J. Shankar, Phys. Rev. Lett. 63, 203 (1989). 8. S. Sarker, C. Jayaprakash et al., Phys. Rev. B43, 8775 (1991). 9. H. J. Schulz, Phys. Rev. Lett. 65, 2462 (1990); ibid. 64, 2831 (1990); ibid. 64, 1445 (1990). 10. T. Deguchi, F. H. L. Essler et al., Physics Reports 331, 197 (2000). 11. Zachary N. C. Ha, Quantum Many-Body Systems in One Dimension, Series on Advances in Statistical Mechanics, 12, (World Scientific, 1996). 12. T. Usuki, N. Kawakami and A. Okiji, J. Phys. Soc. Japan 59, 1357 (1990). 13. P. W. Anderson, Phys. Rev. Lett. 63, 1839 (1990). 14. J. W. Allen, The Hubbard Model (Plenum Press, 1995), p. 357.