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Density-matrix renormalization group calculations of spin chains

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  1. Tez No: 400607
  2. Yazar: FATMA GÖDE
  3. Danışmanlar: PROF. G. A. GEHRING
  4. Tez Türü: Yüksek Lisans
  5. Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1997
  8. Dil: İngilizce
  9. Üniversite: The Unıversıty Of Sheffıeld
  10. Enstitü: Yurtdışı Enstitü
  11. Ana Bilim Dalı: Fizik Ana Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

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Özet (Çeviri)

Critical phenomena were first studied a hundred years ago. Most text books concentrate on critical phenomena which occur at finite temperature. This early stage is known as critical phenomena for classical systems. Other aspects have been discovered in recent years. These are known as critical phenomena for quantum systems that have studied. The Hamiltonian is a function of 'g' at zero temperature where 'g' is a coefficient for total spin in z direction. This study will concentrate on magnetic critical points more than liquid-gas. Two kinds of interactions are important. The first is the Ising model, in which the magnetic moments are assumed to be classical, one-dimensional and capable of only two orientations. The second model is called the Heisenberg model, in which the magnetic moments are related with quantum-mechanical three-component spin operators and energy is proportional to the scalar product of these operators. A phase transition is signalled by a singularity in the free energy at the thermodynamic limit where the system volume becomes infinite while its density remains constant. The free energy is an analytic function of reduced temperature t and magnetic field h. For a finite system, the free energy density is an analytic function of thermodynamic variables and therefore exhibits no singularity. For a system with second order phase transition, we form a renormalization group transformation that maps the reduced temperature t and the external field h for a system of volume V to i and H for another system of volume V . The calculations are based on the singular parts of the free energy. It is an analytic rescaling transformation of macroscopic variables t and h. The theory of critical phenomena will be reviewed in Chapter 1. Next, some explanation about the scaling hypothesis will be given in Chapter 2. For the second order transitions, the finite size scaling form that is derived from leads directly to Kadanoff s scaling hypotheses at the thermodynamic limit and to the scaling laws among the critical exponents. In Chapter 3,1 will discuss renormalization group (RG) theory. This subject is divided into two eras: standard critical phenomena in which the question we have been working on will be explained, and quantum critical behaviour, where the question of how we are studying it will be clarified. Density matrix renormalization group theory (DMRG) will be discussed in Chapter 4. In the last Chapter, the results will be presented analytically and numerically for the antiferromagnetic Heisenberg and transverse Ising model. Books by J. J. Binney [1], Yeomans [2], Stanley [31, Goldenfeld [4], Greiner [5], Davies [6], Kittel [7] have been drawn on during the writing of this thesis.

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