The authors study phase transitions in the isotropic non-degenerate Hubbard Hamiltonian with renormalization group techniques using the epsilon =4-d expansion to first order in epsilon . The functional obtained from the Hubbard Hamiltonian displays full rotational symmetry and describes two coupled fields: a vector spin field, with n components and a non-soft scalar charge field. This coupling is pure imaginary, which has interesting consequences for the critical properties of this coupled field system. The effect of simple constraints imposed on the charge field is considered. The relevance of the coupling between the fields in producing Fisher renormalisation of the critical exponents is discussed. The possible singularities introduced in the charge-charge correlation function by the coupling are also discussed.
The authors study phase transitions in the isotropic non-degenerate Hubbard Hamiltonian with renormalization group techniques using the epsilon =4-d expansion to first order in epsilon . The functional obtained from the Hubbard Hamiltonian displays full rotational symmetry and describes two coupled fields: a vector spin field, with n components and a non-soft scalar charge field. This coupling is pure imaginary, which has interesting consequences for the critical properties of this coupled field system. The effect of simple constraints imposed on the charge field is considered. The relevance of the coupling between the fields in producing Fisher renormalisation of the critical exponents is discussed. The possible singularities introduced in the charge-charge correlation function by the coupling are also discussed.
We derive the free energy functional of an extended Hubbard model Hamiltonian with interatomic interactions. One has to solve a four coupled fields (two vectorial spin fields and two scalar charge fields) problem. Two are potentially soft (one spin and one charge field). We show that as far as the ferromagnetic transition is concerned, universality holds and that introducing a short range interaction between neighbours yields the same free energy functional as for the simple Hubbard model if irrelevant coupling terms are discarded. For the charge ordered transition we show using renormalization group theory arguments that the spin-charge coupling is irrelevant and that the coupling between the two charge fields can give rise to a first or second order transition depending on the details of the metallic band structure and the value of the interaction energy parameters.
We investigate three different types of quantum phase transition occurring in quasi one-dimensional systems theoretically and numerically. First, we study the band-insulator to Mott-insulator transition occurring in charge-transfer complexes, for which the half-filled one-dimensional ionic Hubbard model is considered to be the prototype model. The study is carried out by first deriving an effective spin-one model, and then studying the model numerically using the density matrix renormalization group. We perform a careful finite-size scaling analysis of the mass gaps, order-parameters, and relative susceptibility. We confirm the existence of two quantum critical points. Analysis of the critical exponents confirms that the band-insulator-to-spontaneously-dimerized phase transition belong to the 2D Ising class. The spontaneously dimerized phase undergoes a phase transition to the Mott-insulator which is an infinite-order. Second, we investigate the Mott metal-insulator transition for the half-filled Hubbard model with both nearest-neighbor t and next-nearest-neighbor t′ hopping terms. We study the model using the bosonization approach and density matrix renormalization group simulations. An effective low-energy Hamiltonian that describes the insulator-metal transition is derived. We present results of density matrix renormalization group calculations of spin and charge distribution in various sectors of the phase diagram. The numerical results support the picture derived from the effective theory and give evidence for the complete separation of the transitions involving the spin and the charge degrees of freedom. Finally, we investigate quantum phase transitions phases in low-dimensional fermionic and spin models that go from uniform to spatially inhomogeneous, i.e., dimerized, trimerized, or incommensurate, phases. We propose a new approach based on studying the length dependence of the von Neumann entropy and its corresponding Fourier spectrum for finite segments in the ground state of finite chains. Peaks at a nonzero wave vector are indicators of oscillatory behavior in decaying correlation functions and also provide significant information about certain relevant features of the excitation spectrum; in particular, they can identify the wave vector of soft modes in critical models.
Renormalisation of a field theory model for tricritical behaviour in which the operator phi 6 is essential for thermodynamic stability is discussed in d=4- epsilon dimensions. The crossover scaling form of the equation of state is obtained explicitly to first order in epsilon for both positive and negative values of the four-spin coupling constant. Orthodox scaling, involving Gaussian tricritical exponents is obtained, but is shown to be physically inappropriate in the ordered region of the symmetry plane. A reformulation of scaling, using classical tricritical exponents is possible, but involves an additional parameter p with its own scaling exponent phi p =- 1 / 2 (1- epsilon ). This parameter, introduced by Sarbach and Fisher for the many-component, spherical model limit, is related to the coefficient of (nabla phi ) 2 in the field-theoretic Hamiltonian.
The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter s is coupled to a nonordering additional variable y. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For s2y coupling, the correlation function of y was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}$; $\ensuremath{\alpha}$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\ensuremath{\chi}}_{y}$, has a critical exponent equal to $\ensuremath{\alpha}$, the true specific-heat exponent. When the coupling is s2y2, a weaker singularity of ${\ensuremath{\chi}}_{y}$ appears. The crossover between this behavior and the one typical to s2y coupling is calculated. 〈y〉 has a singular part with an exponent $1\ensuremath{-}\ensuremath{\alpha}$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of y in the constrained case is discussed.
The renormalization-group method is applied to the analysis of the critical singularities in Gamma y (q), the correlation function of a non-ordering parameter y. The coupling between y and s, the order parameter with n components, is s 2 y. When alpha , the specific heat index, is negative, four types of tricritical behaviours are accessible within this Hamiltonian, while the other two may be accessible in other systems in which the non-ordering parameter is a vector. The scaling law gamma y =(2- eta y ) nu is not only trivially fulfilled (zero identically) between the indices characterizing the leading term of Gamma y (q), which is a constant, but is also fulfilled between the corrections to scaling, which are the leading singularities when alpha <0, although these are negligible compared with the regular terms.
In this work we use field-theoretic and renormalization-group methods to study the finite-temperature ferromagnetic phase transition in the three-dimensional Hubbard model. We show that the nature of the ferromagnetic transition may strongly depend on the constraint imposed on the system. For example, if the system is allowed to exchange particles with a reservoir, the stability criterion for the occurrence of a continuous phase transition can be calculated using the mean-field approximation. On the other hand, if the total charge is conserved and charge- and spin-density fluctuations are considered in the theory, a different criterion, quite distinct from the mean-field one, is found, resulting in a fluctuation-induced renormalized Heisenberg tricritical point. The tricritical behavior is studied by calculating the one-loop renormalized free energy and using scaling analysis to determine the tricritical exponents. Finally, the exact solution of the model in the spherical limit and near the ferromagnetic transition is presented, in which case no fluctuation-induced tricritical point is found.