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Journal of Physics Conference Series

Spherical model and quantum phase transitions

V. N. Udodov

Published Nov 1, 2021• 0 citations
Publisher

Abstract

Abstract The spherical Berlin-Katz model is considered in the framework of the epsilon expansion in one-dimensional and two-dimensional space. For the two-dimensional and threedimensional cases in this model, an exact solution was previously obtained in the presence of a field, and for the two-dimensional case the critical temperature is zero, that is, a “quantum” phase transition is observed. On the other hand, the epsilon expansion of critical exponents with an arbitrary number of order parameter components is known. This approach is consistent with the scaling paradigm. Some critical exponents are found for the spherical model in one-and twodimensional space in accordance with the generalized scaling paradigm and the ideas of quantum phase transitions. A new formula is proposed for the critical heat capacity exponent, which depends on the dynamic index z, at a critical temperature equal to zero. An expression is proposed for the order of phase transition with a change in temperature (developing the approach of R. Baxter), which also depends on the z index. An interpolation formula is presented for the effective dimension of space, which is valid for both a positive critical temperature and a critical temperature equal to zero. This formula is general. Transitions with a change in the field in a spherical model at absolute zero are also considered.

DOI
10.1088/1742-6596/2094/2/022027

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Abstract The spherical Berlin-Katz model is considered in the framework of the epsilon expansion in one-dimensional and two-dimensional space. For the two-dimensional and threedimensional cases in this model, an exact solution was previously obtained in the presence of a field, and for the two-dimensional case the critical temperature is zero, that is, a “quantum” phase transition is observed. On the other hand, the epsilon expansion of critical exponents with an arbitrary number of order parameter components is known. This approach is consistent with the scaling paradigm. Some critical exponents are found for the spherical model in one-and twodimensional space in accordance with the generalized scaling paradigm and the ideas of quantum phase transitions. A new formula is proposed for the critical heat capacity exponent, which depends on the dynamic index z, at a critical temperature equal to zero. An expression is proposed for the order of phase transition with a change in temperature (developing the approach of R. Baxter), which also depends on the z index. An interpolation formula is presented for the effective dimension of space, which is valid for both a positive critical temperature and a critical temperature equal to zero. This formula is general. Transitions with a change in the field in a spherical model at absolute zero are also considered.
We investigate a quantum version of the spherical model which is obtained from the classical Berlin-Kac spherical model by a simple canonical quantization scheme. We find a complete solution of the model for short-range as well as for long-range interactions. At finite temperatures the critical behavior is the same as in the classical spherical model whereas at zero temperature we find a quantum phase transition characterized by new critical exponents. Based on a functional-integral representation of the partition function the free energy of the model is shown to be equivalent to that of the nonlinear \ensuremath{\sigma} model in the limit of infinite order-parameter dimensionality. \textcopyright{} 1996 The American Physical Society.
The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a ddd-dimensional model in the spherical approximation confined to a general geometry of the form ${L}^{d\ensuremath{-}{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{L}_{\ensuremath{\tau}}^{z}$ (L(L(L-linear space size and ${L}_{\ensuremath{\tau}}$-temporal size) and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions 1<d<31<d<31<d<3 and $0<~{d}^{\ensuremath{'}}<~d$ a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.
The classical Essam-Fisher and Rushbrooke relationships (1963) that connect the equilibrium critical exponents of susceptibility, specific heat and order parameter are shown to be valid only if the critical temperature is positive. For quantum phase transitions (PT) with zero critical temperature, these relations are proved to be of different form. This fact has been actually observed experimentally, but the reasons were not quite clear. A general formula containing the classical results as a special case is proposed. This formula is applicable to all equilibrium PT of any space dimension. The predictions of the theory are consistent with the available experimental data and do not cast any doubts upon the scaling hypothesis.
A ddd--dimensional quantum model in the spherical approximation confined to a general geometry of the form Ld−d′×∞d′×LτzL^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_τ^{z}Ld−d′×∞d′×Lτz​ (LLL--linear space size and LτL_τLτ​--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1
Abstract Epsilon expansion is an important part of the renormalization group approach; it allows substantiating the scaling paradigm and is used in the modern theory of critical phenomena. It is shown that at a critical temperature equal to zero, the expression for the critical heat capacity exponent already in the first order in epsilon d − = 4 – (d – is the dimension of space) changes its form. This, however, does not indicate a violation of the scaling approach, although the classical Essam-Fisher equation also changes, as does the Rushbrook inequality. The generalized Essam-Fisher interpolation equality and the generalized Rushbrook inequality are given, which are valid for any value of the critical temperature, in particular, for the critical temperature equal to zero. The generalized relations are consistent with the renormalization-group approach, - expansion and scaling paradigm. The fluctuations are considered to be classical (thermodynamic), the conditions are determined when this takes place.
Abstract This work is dedicated to the study of a supersymmetric quantum spherical spin system with short-range interactions. We examine the critical properties both a zero and finite temperature. The model undergoes a quantum phase transition at zero temperature without breaking supersymmetry. At finite temperature the supersymmetry is broken and the system exhibits a thermal phase transition. We determine the critical dimensions and compute critical exponents. In particular, we find that the model is characterized by a dynamical critical exponent z = 2. We also investigate properties of correlations in the one-dimensional lattice. Finally, we explore the connection with a nonrelativistic version of the supersymmetric nonlinear sigma model and show that it is equivalent to the system of spherical spins in the large N limit.