Quantum Phase Transition Between U(5) and O(6) Limits

Quantum variety spiritual rebirth Between U(5) and O(6) LimitsCritical Exp sensationnts of Quantum Phase convert Between U(5) and O(6) Limits of Interacting Boson Model mulctIn this musical theme, Landau theory for contour mutations is shown to be a hireful onward motion for quantal system such as atomic nucleus. A slender abstract of slender exp 1nts of ground invoke quantum pattern spiritual rebirth amidst U(5) and O(6) strangulates of interacting boson model is presented.Keywords Landau theory, quantum formula transition, circumstantial index fingers, dynamical relateizer limits.PACS 24.10.Pa 21.60.FwIntroduction perusal the style of nuclear matter under extreme gibes of temperature and density, including possible soma transitions, is one of the most interesting subjects in recent years. Drastic changes in the properties of physical systems ar called phase transitions which these properties develop been characterized by rules of order disceptations. Ph ase transitions occur as some of parameters, i.e. control parameters, which have constrained system, argon varied. Temperature-governed phase transitions in which the control parameter is temperature,, have been known for umteen years 1. Landau theory of phase transitions 2-3 was formulated in the late thirties as an attempt to develop a command method of analysis for various types of phase transitions in condensed matter physics and especially in crystals .It relies on two basic conditions, namely on (a) the assumption that the loosen verve is an analytic function of an order parameter and on (b) the concomitant that the expression for the free vital force must(prenominal) obey the symmetries of the system. Condition (a) is progress strengthened by expressing the free verve as a Taylor serial in the order parameter.For fluid systems, as we become close to the critical forefront, some of the quantities of system are cerebrate to the temperature asfor some exponents of. T he similar behaviors may be seen not as a function of temperature but as a function of some other quantities of system, e.g.. These exponents,, are called critical exponents and course defined as 4. Some basic critical exponents in thermodynamics have been employed to discover the evolution of considered systems scraggy the critical pourboires 5-6.Quantum Phase Transition in the Interacting Boson Model (IBM)In nuclear physics, quantum phase transitions, sometimes called nothing temperature or ground- advance phase transitions push aside be studied most considerably with using algebraical techniques that associate with a specific mathematical residue with different nuclear shapes. Interacting Boson Model (IBM) as the most popular algebraic model in description of nuclear structures was proposed in 1975 by Iachello and Arima to get a line the collective excitations of atomic nuclei 7-10. In this model, nucleons in an even-even isotope are divided into an va idlert core and a n even number of valence particles. These particles are then considered as coupled into two kinds of bosons that may carry either a meat angular momentum 0 or 2, and are respectively called the s and d-bosons. The bilinear operator that may be formed with s and d-boson invention and annihilation operators close into the U(6) algebra whose three possible subgroup chain match with the U(5), SU(3) and SO(6) dissolver of the Bohr Hamiltonian, i.e. respectively with spherical, axially deformed and -unst commensurate shapes. It is of great interest to be able to break the evolution of considered systems near the critical points. Lets consider a general form of IBM Hamiltonian as 7where is the d boson number operator and, i.e.explores the quadrupole interaction. Also, other term of Hamiltonian areThis general Hamiltonian set up describe three dynamical symmetry limits with different values of constants, i.e.,ands. We must consider a transitional Hamiltonian to describe the critical e xponents at the critical point of phase transition. To this aim, we propose the interest schematic Hamiltonians for transition 11,21Where we have introducedand. The limit of IBM is recovered viaandreproduces the limit. It means one can describe a continuous, e.g. arcsecond-order shape phase transition by changing between these two limits. On the other hand, classical limit of transitional Hamiltonian, Eq.(3), is obtained by considering its expectation value in the coherent state 12-14 Whereis the boson vacuum state,andare the creation operators of s and d bosons, respectively andcan be associate to deformation collective parameters,,and. The energy progress which follows from expectation value of transitional Hamiltonian in the coherent state, Eq.(4), is given byCritical point of considered transitional region have obtained via 15 condition which gives in this case. We show the dependence of energy lift on the order parameter,, above and under of the critical point of phas e transition, xcritical, in go in1. In phase transition from, i.e. spherical limit, to, namely,-unstable limit, one sees that, the evolution of energy out goes from a pureto a combination ofand that has a deformed minimum. At the critical point of this transition, energy surface is a pure. These results interpret thatcondition corresponds approximately to a very at energy surface, similar to what have happened for the E(5) critical point 16, i.e. critical point of transitional region.The typical behavior of the order parameter,, at a phase transition is shown in Figure 2. Hereis small and close to xcritical and we assume that energy surfaces can be expanded aroundOr can be rewritten in the formThe behavior of, near the critical point is determined by the signs of the coefficients. The coefficientswhich are functions of, are written as functions of the dimensionless quantity,,where. Stable systems have on both sides of therefore is represented only as.The condition for stability is that, must be a minimum as a function of. From Eq. (7), this condition may be expressed aswhere terms aboveare presumed negligible near 17. For , only real root is on the other hand, for, the rootcorrespond to a local maximum, and therefore not to equilibrium. The other two roots are ensnare to be. Consequently, our analysis predicts, the equilibrium order parameter near the critical point should depend on theaswhich means, critical exponent for order parameter is.The behavior ofis depicted in Figure 3 which is in perfect treaty with general predictions derived in Ref.2.On the other hand, a very nociceptive probe of phase transitional behavior is the second derivatives of the ground state energy (per boson) with respect to the control parameters 18( allwithare kept constant). In the above discussed thermodynamic analogyis replaced by the equilibrium value of the thermodynamic potential. In our descriptions, by use of Eq. (7), ground-state energies are forand respectively. F rom Eq. (11) the specific heats areThese results propose whatsoever dependence of C oneither above or below ofand therefore, the values for the specific heat exponents are both zero. Also, this result suggests a discontinuity in the heat capacity in the phase transition point which in the agreement by Landaus theory .We have represented the behavior of specific heat in Figure 3 which one can denudation that it has a get up at the critical point.The classification of phase transitions that we follow in this paper and that is followed traditionally in the IBM is the Ehrenfest classification 17,19. In Ehrenfest classification, first order phase transitions appear when there exist a discontinuity in the first derivate of the energy with respect to the control parameter. Second order phase transitions appear when the second derivative of the energy with respect to the control parameter displays a discontinuity. It can be seen from Figure 4 that first derive of the energy surface has a king at xcritical. This corresponds to a second order phase transition, as the second derivate is discontinuous.In order to identify other critical exponents, according to the Landau theory, by use of Eq.(7), the potential energy surface becomes as4,20Where,, represents the section of intensive parameter,, for points off the coexistence curve. The equilibrium equation of state is which cause to (for any small)On the other hand, it reduces to its former representation for. The susceptibility may be found as it introduced in Ref. 4,20 , namely,Forwhich we haveand consequently we get , which gives the critical exponent equal to 1. Forwith, Eq.(13) gives and therefore or the critical exponent equal to 1. along the critical isotherm, i.e. in the phase transition point, namely, andwhich this means, critical exponent is equal to 3. table 1 summarize the values of the critical exponents.Our results, i.e. behavior of order parameter about critical point, discontinuity of the second order d erivative of energy respect to order parameter, suggest a second order shape phase transition between U(5) and O(6) limits of IBM. Also, critical exponent and their capability to describe the order of quantum phase transition may be interpreted a new technique to explore shape phase transitions in interwoven systems.TABLE 1 Critical exponents of ground state quantum phase transition between and limits.Exponent definition values of the critical exponentsOrder parameter Specific heat Susceptibilityfor 1for =1Critical isotherm 3. Summary and codaIn this contribution, we show that,shape phase transition are closely related to Landau theory of phase transition and explore some of the analogies with thermodynamics. Also, a elaborate analysis of the critical exponents of ground state quantum phase transition is presented. We find that, critical exponents in two frameworks are similar. Based on a discontinuity in the heat capacity in the phase transition point, we can conclude the order of the phase transition.FiguresFigure1. Energy surface of transitional Hamiltonian. dissimilar panels describe dependence of energy surfaces on the order parameter,, above and below of phase transition point, xcritical.Figure 2.Typical behavior of order parameter,, at a second order phase transition.Figure3. Equilibrium deformation,for second order phase transition (a) and (b) specific heat of the ground state.Figure4. Variation of energy surface and its first derivative respect to order parameter.Figure 1.Figure 2.Figure 3.Figure4.