This chapter considers the expectation value of an operator (or of products of two operators) over the ground state of the interacting system, when the time-dependent part of the Hamiltonian is switched off. These limitations apply to systems in equilibrium at zero temperature, which include important cases like insulators and semiconductors as well as Fermi liquids, for which the energy gap and the Fermi energy are, respectively, much larger than the available thermal energy. The ensuing formalism for ground-state averages at zero temperature relies on an “adiabatic assumption,” which cannot be applied as it is when excited states are involved in the ensemble averages.
This chapter considers the expectation value of an operator (or of products of two operators) over the ground state of the interacting system, when the time-dependent part of the Hamiltonian is switched off. These limitations apply to systems in equilibrium at zero temperature, which include important cases like insulators and semiconductors as well as Fermi liquids, for which the energy gap and the Fermi energy are, respectively, much larger than the available thermal energy. The ensuing formalism for ground-state averages at zero temperature relies on an “adiabatic assumption,” which cannot be applied as it is when excited states are involved in the ensemble averages.
This chapter considers the factors arising in the zero-temperature frequency representation for the ground state energy. The physical essence of the result, the relation between the symmetry of the non-interacting zero-temperature ground state and the exact ground state. The expectation value of an arbitrary product of operators in the Bose ground state cannot be separated into independent products of contractions and there is no zero-temperature Wick's theorem as in the Fermion case. The point at which our previous derivations for Fermions cannot be generalized to Bosons is the construction of a canonical transformation analogous to such that all the new annihilation operators annihilate the ground state. The Fermion minus signs combined with the factor of N from the color sums do just what is required to generate the correct Bose factors. Zero temperature theory is applied to the dilute Bose gas and to Fermions in one dimension interacting via δ-function forces.
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.
It is shown from first principles that, in spite of the large interatomic forces, liquid 4 He should exhibit a transition analogous to the transition in an ideal Bose–Einstein gas. The exact partition function is written as an integral over trajectories, using the space-time approach to quantum mechanics. R.P. Feynman, 1953 Summary In this chapter we discuss imaginary-time path integrals and the path-integral Monte Carlo method for the calculation of properties of quantum systems at non-zero temperature. We discuss how Fermi and Bose statistics enter, and how to generalize the fixed-node procedure to non-zero temperature. We then discuss an auxiliary-field method for the Hubbard model. The path-integral method can be used to perform ground-state calculations, allowing calculations of properties with less bias than the projector Monte Carlo method. We also discuss the problem of estimating real-time response functions using information from imaginary-time correlation functions. In previous chapters we described two QMC methods, namely variational QMC and projector (diffusion) QMC. Both of these methods are zero temperature, or, more properly, are formulated for single states. In this chapter we discuss the path-integral Monte Carlo (PIMC) method, which is explicitly formulated at non-zero temperature. Directly including temperature is important because many, if not most, measurements and practical applications involve significant thermal effects. One might think that to do calculations at a non-zero temperature we would have to explicitly sum over excited states. Such a summation would be difficult to accomplish once the temperature is above the energy gap, because there are so many possible many-body excitations. In addition, the properties for each excitation are more difficult to calculate than those for the ground state. As we will see, path-integral methods do not require an explicit sum over excitations. As an added bonus, they provide an interesting and enlightening window through which to view quantum systems. However, the sign problem, introduced in the previous chapter, is still present for fermion systems. The fixed-node approximation is used again. An advantage of PIMC is the absence of a trial wavefunction. As a result, quantum expectation values, including ones not involving the energy, can be computed directly. For the expert, the lack of an importance function may seem a disadvantage; without it one cannot push the simulation in a preferred direction.
The effective theory of an open boson or fermion system is studied, which evolves out of equilibrium with time-dependent Hamiltonian H^(t). A measure of nonequilibrium temperature for the open system evolving from an equilibrium is proposed as the time-averaged energy expectation value T(t)=Ti(<H^(t)>Ψˉ/<I^(t)>Ψˉ), where I^(t), the action operator, satisfies the quantum Liouville-von Neumann equation and determines the true density operator. It recovers the result for the adiabatic (quasi-equilibrium) and nonadiabatic (out of equilibrium) evolution from one static Hamiltonian to another and takes into account the particle production due to the intermediate processes.
In this Letter I apply Kohn's theorem to quantum oscillations in cuprates. I show that when combined with Gell-Mann and Low theorem, there is rigorous justification of quantum oscillations as in a Fermi liquid for cuprates. For simplicity, understanding the lack of interaction effects in quantum oscillations can be addressed in the language of a fully translationally invariant system. The Gell-Mann and Low theorem is not perturbative, and applies more generally as long as the adiabatic evolution from an isolated nondegenerate state is the case. Oscillation frequencies are identical to the noninteracting problem. It is the cyclotron gap that justifies Fermi-liquid-based theory of quantum oscillations. This is not to imply that other properties of the cuprates that do not have a gap in the spectrum could not exhibit non-Fermi-liquid behavior, for example, angle-resolved photoemission spectroscopy.
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not linearly proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example in which the applicability of the adiabatic theorem is questionable.