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

Quantum Phase Transition Between U(5) and O(6) Limits Critical Exponents of Quantum Phase Transition Between U(5) and O(6) Limits of Interacting Boson Model Abstract In this paper, Landau theory for phase transitions is shown to be a useful approach for quantal system such as atomic nucleus. A detailed analysis of critical exponents of ground state quantum phase transition between U(5) and O(6) limits of interacting boson model is presented. Keywords: Landau theory, quantum phase transition, critical exponents, dynamical symmetry limits. PACS: 24.10.Pa; 21.60.Fw; Introduction Studying the behavior of nuclear matter under extreme conditions of temperature and density, including possible phase transitions, is one of the most interesting subjects in recent years. Drastic changes in the properties of physical systems are called phase transitions which these properties have been characterized by order parameters. Phase transitions occur as some of parameters, i.e. control parameters, which have constrained system, are varied. Temperature-governed phase transitions in which the control parameter is temperature,, have been known for many years [1]. Landau theory of phase transitions [2-3] was formulated in the late 1930s as an attempt to develop a general 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 free energy is an analytic function of an order parameter and on (b) the fact that the expression for the free energy must ob ey the symmetries of the system. Condition (a) is further strengthened by expressing the free energy as a Taylor series in the order parameter. For fluid systems, as we become close to the critical point, some of the quantities of system are related to the temperature asfor some exponents of. The 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 naturally defined as [4]. Some basic critical exponents in thermodynamics have been employed to describe the evolution of considered systems near the critical points [5-6]. Quantum Phase Transition in the Interacting Boson Model (IBM) In nuclear physics, quantum phase transitions, sometimes called zero temperature or ground-state phase transitions can be studied most easily with using algebraic techniques that associate with a specific mathematical symmetry 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 describe the collective excitations of atomic nuclei [7-10]. In this model, nucleons in an even-even isotope are divided into an inert core and an even number of valence particles. These particles are then considered as coupled into two kinds of bosons that may carry either a total 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 creation and annihilation operators close into the U(6) algebra whose three possible subgroup chain match with the U(5), SU(3) and SO(6) solution of the Bohr Hamiltonian, i.e. respectively with spherical, axially deformed and ÃŽ ³-unstable shapes. It is of great interest to be able to describe the evolution of considered systems near the critical points. Lets consider a general form of IBM Hamiltonian as [7] where is the d boson number operator and, i.e.explores the quadrupole interaction. Also, other terms of Hamiltonian are This general Hamiltonian can describe three dynamical symmetry limits with different values of constants, i.e.,ands. We must consider a transitional Hamiltonian to describe the critical exponents at the critical point of phase transition. To this aim, we propose the following schematic Hamiltonians for transition [11,21] Where we have introducedand. The limit of IBM is recovered viaandreproduces the limit. It means one can describe a continuous, e.g. second-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 related to deformation collective parameters,,and. The energy surface which follows from expectation value of transitional Hamiltonian in the coherent state, Eq.(4), is given by Critical point of considered transitional region have obtained via [15] condition which gives in this case. We show the dependence of energy surface on the order parameter,, above and below of the critical point of phase transition, xcritical, in Figure1. In phase transition from, i.e. spherical limit, to, namely,-unstable limit, one sees that, the evolution of energy surface 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 à ¯Ã‚ ¬Ã¢â‚¬Å¡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 around Or can be rewritten in the form The 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 as where 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 found to be. Consequently, our analysis predicts, the equilibrium order parameter near the critical point should depend on theas which means, critical exponent for order parameter is.The behavior ofis depicted in Figure 3 which is in perfect agreement with general predictions derived in Ref.[2]. On the other hand, a very sensitive 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. From Eq. (11) the specific heats are These results propose any 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 Landau’s theory .We have represented the behavior of specific heat in Figure 3 which one can find that it has a jump 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 as[4,20] Where,, represents the contribution 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 derivative 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 complex systems. TABLE 1 Critical exponents of ground state quantum phase transition between and limits. Exponent definition values of the critical exponents Order parameter Specific heat Susceptibilityfor 1 for =1 Critical isotherm 3. Summary and conclusion In 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 detailed 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. Figures Figure1. Energy surface of transitional Hamiltonian. Different 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.

Most prized possesion Essay

Out of all the material items I possess, I would have to call my first car my one true prized possession. Veronica, as I often refer to her is rusty black and has more than a few bruises. Pretty much a replica of the stereotypical teenager’s first car. Possessions like these often are seen as junk to the rest of the world, but in one person’s heart it remains an invaluable treasure. For myself, I was lucky enough to find that one hidden gem in a 1998 Ford Contour. In my eyes it will be forever priceless because it is not the exterior value of the car that matters, but the long road of experience it has carried me on. My car has taught me more than I expected in such a short time, but above all it has given me memories, a feeling of responsibility, and taught me that hard work pays off. As with most material items, certain memories tend to latch onto the object and are able to transport one back to a better time and place just by looking at it. Even though it is only a year old (at least in my possession), it seems as if it already carries a lifetime of memories. It treated me well through a timeless summer, the final summer before college becomes a true reality. Car rides filled with music pulsing, friends laughing, and road trips taken will never be forgotten. It has seen its share of sandy beaches and bright blues skies, as well as torrential downpours and yet has never failed me once. If I ever needed to get away from my sometimes chaotic and frustrating household, the car has allowed me to finally have some form of escape after eighteen long years. So this car has quite literally been with me through everything in this past year, and I will always remember it for the freedom it has granted me for the first time in my life. It even holds promises of even better memories in the future. I can almost guarantee things are going to get hectic at times and circumstances are going to change more than once, but through it all my car will remain the one constant. The tangible mass that is the car is enough to remind me of all I went through to finally get to this point in my life. This fact is what makes my car so special, because no price tag can be put on an unforgettable memory. The newfound responsibility that this car has given me also is what makes it  invaluable. Owning something monumental like a vehicle comes with a certain sense of pride and self-accomplishment, but that same feeling can be taken away in an instant if you do not act responsibly. The freedoms that come with a car unfortunately do not come without their restrictions. There are state laws, parking rules, and probably most important, the parental rules. If these limitations are not taken seriously, many young drivers will find that their feeling of independence is short lived, myself included. I have learned that a great deal of hard work and effort can all be a waste of time if you don’t remain responsible after you have reached your goal. One must become independent and own up to their actions, whether good or bad. This new characteristic of responsibility will carry on to all other aspects of my life and I can thank my experience as a car owner for that. Finally, my car has taught me a valuable lesson of dedication that I will never forget. Never before had I worked so hard to get something that I wanted. Without help from my parents to pay for a car, I spent months and months of job hours busing tables to pay for something I could call my own. Two thousand five hundred dollars later, for the first time in my life I have something that is 100% mine. This feeling of pride and ownership is something I truly value, a feeling I would have never experienced without the vehicle. This over year-long process of saving money and doing hard, manual labor has taught me a lesson that has always seemed abstract up to this point in my life. The car is now tangible proof to me that hard work eventually does pay off and has its rewards if you stay dedicated. It is this lesson that will always stick with my car and with myself and is what makes it one of my prized possessions. In all reality, I know I won’t have the same car for the rest of my life. Eventually I will move on to bigger and better things, as all people seem to do in this new world, and my car will become just another piece of junk on the side of the road. This doesn’t mean the memories and lessons the car has given me will become any less important. They will forever remain special in my mind and even though the car won’t be there to remind me of them, I will take these lessons with me wherever I go.

Does Home Ownership Impair An Individuals Labor Market...

Does home ownership impair an individual s labor market outcome? Some early works suggest that home owners change their locations of residence less often than renters and, thus, home owners are immobile and less flexible. Nickell (1997) found that countries in which citizens are not shackled by the chain of permanent residence (flexible mobility) have relatively high employment rates. This point is also illustrated by Hughes and McCormick (1987), who examined links between housing policies, job mobility and unemployment in the UK. They found that UK housing policies, which seek to reduce spatial mobility, caused an inflexible, inefficient labor outcome in the UK Despite few previous studies on the relationship between home ownership and†¦show more content†¦had the highest unemployment rate in the industrial world, as well as a relatively high home ownership rate. Oswald observed that since 1950s, unemployment rates have risen the fastest in the nations that experienced a relatively high growth in home ownership rate. Also, he found a strong positive correlation between the growth of home ownership and unemployment across the states of the U.S. between 1970 and 1990. He concluded that a higher rate of owner-occupation might therefore be an important factor contributing to higher unemployment rates, a factor which had previously been omitted in empirical studies of unemployment. Oswald s explanation to his observation was that home ownership causes labor immobility, triggering the labor market inefficiency. Selling a home and moving is expensive. Higher moving costs associated with home owners therefore make them less mobile than ren ters. Given this limitation of spatial or labor immobility, an unemployed home owner struggles to find an occupation for which he is ideally suited. This may produce labor inefficiency in the labor market. Although many of macro evidence support Oswald s arugument, Oswald generally drew more opposition than support for his ideas, when scholars used microeconomic data to reject the negative effect of home ownership on employment rate. Flatau et al (2003), Coulson and Fisher (2003, 2007),

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