L. De Cesare

Renormalization group and quantum critical phenomena in the large-n limit

The paper is devoted to a sistematic study of the critical behaviour and quantum regime of a wide class of quantum systems on the basis of the Renormalization Group (RG) approach in the large-n limit. It can be considered as a complement to recent perturbative (RG) investigations for the same systems and as a convenient basis for a treatment of the classical-quantum crossover. The differences between “Bosonized systems” and several quantum systems for which a dimensional crossover occurs in the low temperature limit are emphasized, consistently with the perturbative results and a calculation of the quantum non linear scaling field is also realized. Finally, a differential formulation of the quantum RG in the large-n limit is presented and its convenience in investigating some aspects of the problem is pointed out.

Renormalization group approach to quantum bosonized systems

SummaryTo leading order in a perturbative renormalization group analysis we study the critical behaviour of Bose and related quantum systems at temperatureT≠0 andT−0, for which a preliminary investigation has been reported recently. In both cases we specify the meaning of the grand canonical variable which corresponds to the usual critical deviationT−Tc in the canonical Wilson scheme. In particular, forT≠0, when the grand canonical critical exponents are the same as the classical one, we calculate the canonical critical exponents with the use of Fisher’s renormalization relations. The question of dimensional crossover which appears forT→0 is briefly discussed. For the more interesting caseT=0, the critical surface and the exponents associated with the stable nontrivial fixed point are calculated. We discover an unusual behaviour which, nevertheless, can be still included in a universality class in Wilson’s sense.RiassuntoAll’ordine dominante in un’analisi perturbativa del gruppo di rinormalizzazione si effettua uno studio di sistemi quantistici bosonici a temperaturaT≠0 eT−0, per cui è stata fornita recentemente un’indagine preliminare. In entrambi i casi si specifica il significato della variabile gran canonica corrispondente all’usuale deviazione criticaT−Tc nello schema canonico di Wilson. In particolare, nel casoT≠0, in cui gli esponenti critici gran canonici si identificano con quelli di un sistema classico, facendo uso delle relazioni di rinormalizzazione di Fisher, si calcolano gli esponenti critici canonici. Il problema dell’incrocio dimensionale che appare perT→0 è esaminato brevemente. Per il caso più interessanteT=0, si determinano la superficie critica e gli esponenti critici associati al punto fisso non banale stabile. Dai risultati emerge un comportamento insolito che, tuttavia, può essere incluso in una classe di universalità nel senso di Wilson.РезюмеВ главном порядке при пертурбационном анализе группы перенормировки мы исследуем критическое поведение квантовых Бозе-систем при температуреT≠0 иT=0. В обоих случаях мы определяем физический смысл главной канонической переменной, которая соответствует обычному критическому отклонениюT−Tc в канонической схеме Вильсона. В частности, дляT≠0, когда главные канонические критические показатели являются такими же, как для классических систем, мы вычисляем канонические критические показатели с использованием соотношений перенормировки Фишера. Вкратце обсуждается вопрос размерного кроссовера, который возникает дляT→0. Для наиболее интересного случаяT=0 вычисляются критические поверхности и показатели, связанные с устойчивой нетривиальной фиксированной точкой. Мы обнаружили необычное поведение, которое, тем не менее, может быть включено в универсальный класс в смысле Вильсона.

Temperature-dependent scaling fields and quantum criticality within the Wilson renormalization group approach

Within the Wilson Renormalization Group (WRG) approach as applied to d-dimensional quantum systems using appropriate functional representations, we develop a general method for describing the critical behaviour driven by the temperature T when the quantum fluctuations are into play. It consists in solving the quantum RG equations near a zero-temperature fixed point (FP) introducing T-dependent scaling fields which are determined by ordinary first-order differential equations with appropriate boundary conditions. This allows us to explore, in a unified and relatively simple way, the low-temperature properties near a (T = 0)-quantum instability of a great variety of quantum systems (interacting Bose gas, quantum ferroelectrics, itinerant fermion materials, spin models in a tranverse field, granular superconductors, etc.). The crossover from the low-but finite-temperature critical behaviour to the quantum criticality is also described in a natural way in terms of appropriate effective exponents. The predictions appear to be quite quantum systems by means of different approaches. For the interacting Bose gas and other quantum systems lying in the same universality class (bosonic systems), our results give us the opportunity to rexamine some uncorrect calculations for d ≤2 recently appeared in the literature.

Low-temperature quantum critical behaviour of systems with transverse Ising-like intrinsic dynamics

The low-temperature properties and crossover phenomena of d-dimensional transverse Ising-like systems within the influence domain of the quantum critical point are investigated solving the appropriate one-loop renormalization group equations. The phase diagram is obtained near and at d=3 and several sets of critical exponents are determined which describe different responses of a system to quantum fluctuations according to the way of approaching the quantum critical point. The results are in remarkable agreement with experiments for a wide variety of compounds exhibiting a quantum phase transition, as the ferroelectric oxides and other displacive systems.

The spectral-density approach to an antiferromagnetic Heisenberg linear chain

SummaryWe study an antiferromagnetic Heisenberg linear chain (AHLC) by using the spectral-density method (SDM). The values of the thermodynamic quantities are obtained as functions of the temperature; our results are compared with the exact ones obtained by other authors atT=0.RiassuntoSi studia una catena lineare antiferromagnetica di Heisenberg (AHLC) usando il metodo delle densità spettrali (SDM). Il metodo permette di ottenere i valori delle grandezze termodinamiche in funzione della temperatura; si paragonano i risultati ottenuti con quelli esatti ottenuti da altri autori aT=0.РезюмеМы исследуем антиферромагнитные линейные цепочки Гайзенберга, используя метод спектральной плотности. Получаются значения термодинамических величин, как функции температуры. Наши результаты сравниваются с точными значениями, полученными другими авторами приT=0.

Critical behaviour of thin films with quenched impurities.

The critical behaviour of thin films containing quenched random impurities and inhomogeneities is investigated by the renormalization-group method to the one-loop order within the framework of the n-component φ4-model. The finite-size crossover in impure films has been considered on the basis of the fundamental relationship between the effective dimensionality Deff and the characteristic lengths of the system. The fixed points, their stability properties and the critical exponents are obtained and analyzed, using an e=(4−Deff)-expansion near the effective spatial dimensionality Deff of the fluctuation modes in D-dimensional hyperslabs with two types of quenched impurities: point-like impurities with short-range random correlations and extended (linear) impurities with infinite-range random correlations along the small-size spatial direction. The difference between the critical properties of infinite systems and films is demonstrated and investigated. A new critical exponent, describing the scaling properties of the thickness of films with extended impurities has been derived and calculated. A special attention is paid to the critical behaviour of real impure films (D=3).

Dynamical critical exponent for quantum systems with long-range correlated impurities

Abstract The dynamical critical exponent of some quantum systems in the presence of long-range-correlated disorder is calculated with the use of a double renormalization group expansion. The impurity effects on the static critical behaviour in the quantum regime are also outlined.

Spectral density method in classical statistical mechanics

Abstract The spectral density method in classical statistical mechanics is formulated and application to a linear classical spin system is made in the simplest approximation.

Field-induced quantum criticality of systems with ferromagnetically coupled structural spin units

The field-induced quantum criticality of compounds with ferromagnetically coupled structural spin units (as dimers and ladders) is explored by applying Wilsons renormalization group framework to an appropriate effective action. We determine the low-temperature phase boundary and the behavior of relevant quantities decreasing the temperature with the applied magnetic field fixed at its quantum critical point value. In this context, a plausible interpretation of some recent experimental results is also suggested.

Thermodynamic properties of a classical d-dimensional spin-S Heisenberg ferromagnet with long-range interactions via the spectral density method

The thermodynamic properties of a classical d-dimensional spin-S Heisenberg ferromagnet, with long-range interactions decaying as r−p and in the presence of an external magnetic field, is investigated by means of the spectral density method in the framework of classical statistical mechanics. We find that long-range order exists at finite temperature for dd with d>2, consistently with known theorems. Besides, the related critical temperature is determined and a study of the critical properties is performed.