So far, we have proposed a method of expressing finite temperature in the form of ground state and pure state by changing the shape of the distribution function and using the step function. Then, in the second quantized system in this form, the Hamiltonian and the method of giving its eigenvalues are presented. We did this in the form that the creation and annihilation operators, whose matrix elements changed as the statistics changed, acted on the Hamiltonian in the ground state. We also discussed the basics of the parastatistics method used for that. In this paper, we apply this finite temperature expression method , express the eigen- values of the Hamiltonian of the interacting system, and try to express the energy given by the interaction as heat.
The ground state of a two-level system (associated with probabilities
p and $1\ensuremath{-}p$, respectively) defined by a general Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}+\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ is studied. The simple case characterized by $\ensuremath{\lambda}=0$, whose Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}$ is represented by a diagonal matrix, is well established and solvable within Boltzmann-Gibbs statistical mechanics; in particular, it follows the third law of thermodynamics, presenting zero entropy (
SBG=0) at zero temperature (
T=0). Herein it is shown that the introduction of a perturbation $\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ ($\ensuremath{\lambda}>0$) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy
S[p] (with $S[p]\ensuremath{\ne}{S}_{\mathrm{BG}}[p]$), if the Hermitian operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ is represented by a $2\ifmmode\times\else\texttimes\fi{}2$ matrix, defined by nonzero off-diagonal elements ${V}_{12}={V}_{21}=\ensuremath{-}z$, where
z is a real positive number. Hence, this new term in the Hamiltonian, presenting ${V}_{12}\ensuremath{\ne}0$, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature $\ensuremath{\theta}$ ($\ensuremath{\theta}\ensuremath{\propto}\ensuremath{\lambda}z$), which is shown to be a parameter conjugated to the entropy
S. Based on this, one introduces an infinitesimal heatlike quantity, $\ensuremath{\delta}Q=\ensuremath{\theta}dS$, leading to a consistent thermodynamic framework, and by proposing an infinitesimal form for the first law, a Carnot cycle and thermodynamic potentials are obtained. All results found are very similar to those of usual thermodynamics, through the identification $T\ensuremath{\leftrightarrow}\ensuremath{\theta}$, and particularly the form for the efficiency of the proposed Carnot Cycle. Moreover,
S also follows a behavior typical of a third law, i.e., $S\ensuremath{\rightarrow}0$, when $\ensuremath{\theta}\ensuremath{\rightarrow}0$.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains
Introduction The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian. The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.
Introduction In the limit of zero temperature, the partition function is dominated by the ground state of the system, the free energy becomes equivalent to the ground state energy, and the central object of interest is the Hamiltonian of the system H (λ), dependent in general on one or more parameters λ. A ‘quantum’ Hamiltonian, as noted previously, is one which contains non-commuting operators. Here we concentrate on quantum spin Hamiltonians, of which the prime example is the Heisenberg model. Of particular interest is the possibility of a quantum phase transition , i.e. nonanalytic behaviour as a function of coupling λ. A quantum phase transition may exert a dominating influence on the system over a range of low temperatures above T = 0. Such phenomena have figured prominently in theoretical discussions of experiments on the cuprate superconductors, the heavy fermion materials, organic conductors, and related materials (e.g. Sachdev, 1999). In some cases, a correspondence can be found between a quantum system at zero temperature in 1 time and ( d – 1) space dimensions, and an equivalent classical system at finite temperature in d space dimensions, based on the Feynman path integral formalism – see Chapter 9 and Appendix 8 for further discussions. Under this mapping, the temperature kT in the classical system corresponds to a coupling λ in the equivalent quantum system. If a phase transition occurs, the same universal critical exponents are expected to apply in both cases.