{'fraction_non_alphanumeric': 0.03205307915148117, 'fraction_numerical': 0.019904681805438744, 'mean_word_length': 4.632631578947368, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Re-entrance in nuclei: competitive phenomena

Using the shell-model Monte Carlo method, we investigate how temperature and rotation affect pairing properties for nuclei in the fp - gds region. The re-entrance of pairing correlations with temperature is predicted at high rotational frequencies. It manifests through an anomalous behavior of the specific heat and level density.


Introduction
Phase transitions, as seen in superconductivity, superfluidity or ferromagnetism, are the result of competition between the order interaction and thermal fluctuations. Ordered phases (less symmetric) usually reside at lower temperatures and transition to disordered phases (more symmetric) at higher temperatures due to thermal fluctuations. Re-entrance (or partial order) phenomena manifests itself in successive phase transitions as a function of temperature or other extensive quantity. For example, re-entrance was discovered experimentally for liquid crystals in 1975 [1], showing a higher symmetry nematic phase occurring at a lower temperature than the lower symmetry smectic phase. Interestingly, the discovery of this phenomena led to many technological applications in electronic displays. Other condensed matter systems that display re-entrant phenomena include ferromagnetic insulators [2] and orbital antiferromagnetism [3].
The properties of nuclei are governed by the nuclear interactions. The singlet-S and triplet-P channels in the nucleon-nucleon interaction generate global nuclear pairing properties. Indeed, a large even-even nucleus will exhibit specific pairing properties, such as a large pairing gap from the ground state to the first excited state [4]. The nuclear Hamiltonian also generates other phenomena such as intrinsic deformation, with an experimental signature of the energy of a band of states increasing like E(J) ∼ J(J + 1) where J is the spin of the state. Pairing and deformation are competitive phenomena in nuclei. One can heat a nucleus thermally and induce a pairing transition. This transition occurs at around a temperature of T = 0.7 − 1.5M eV where thermal fluctuations overcome the effects of pairing [5][6][7][8]. A clearly defined peak occurs in the specific heat at these temperatures. If the nucleus is well deformed, that peak structure can be smeared. Early work on thermally assisted pairing predicted by Kammuri in 1964 [9], describes a local increase of pairing correlations in a rotating nucleus with excitation energy. One can also enhance or suppress structures in a nucleus through cranking, which practically involves adding to the Hamiltonian a time-reversal breaking term ωJ z , where ω is the cranking frequency, and J z is the projection of spin along the axis of rotation. As the cranking frequency increases, pairing within the nucleus should decrease due to spin alignment [10].
The objective of this work is to present a short survey on our studies using the shell-model Monte Carlo method to examine how temperature and rotation affect pairing properties on the N = 40 systems, 68 Ni, 70 Zn, 72 Ge and 80 Zr, which exhibit distinct phase-transitional behavior [10,11]. Interestingly, we have found one system, 72 Ge in which a re-entrant phenomena occurs where at high cranking frequency there is a small window in temperature where pairing actually increases at a critical temperature [10]. This leads to a specific heat that shows a sharp decrease with temperature. We also present early results on current work investigating the phase-transitional behavior of the odd-odd N = Z nuclei, 70 Br, 74 Rb and 78 Y.

The Shell-Model Monte Carlo Method
The shell-model Monte Carlo (SMMC) method [12] allows investigations of nuclei at finite temperatures with the relevant degrees of freedom included. This makes it possible to account for the thermal and quantal fluctuations which are important to describe phase transformations in finite-size systems. The SMMC approach describes nuclear observables at finite-temperature as thermal averages where β = 1/T is the inverse temperature, T r N is the many-body trace at fixed particle number N , and −βH is the imaginary-time many-body propagator. For certain classes of residual nucleon-nucleon interactions [13], like the attractive pairing+quadrupole force employed in this work, the evaluation of observables is exact, subject only to statistical errors related to the Monte Carlo integration. As we are concerned here with a description of collective quadrupole and pairing correlations at relatively low energies, we have employed a pairing+quadrupolequadrupole Hamiltonian where Q 2µ is the mass quadrupole moment operator given by with projection µ, a † jmtz (a jmtz ) creates (destroys) a nucleon of isospin t z in the orbital jm, andã jm = (−1) j+mâ j−m , and the seniority pairing operator P † is defined as The details of our SMMC calculations follow those of Refs. [10,11]. Namely, calculations were performed in the complete (0f 1p − 0g1d2s) model space for protons and neutrons. The singleparticle (s.p.) energies have been determined from a Woods-Saxon potential parametrization of 56 Ni. Using the parameters G = 0.106 MeV and χ = 0.0104 MeV −1 fm 2 , we reproduce the lowenergy spectrum of 64 Ni and 64 Ge. Nuclei are described by valence protons and neutrons outside the closed core of 40 Ca. In order to generate the angular momentum polarization, we consider the RouthianĤ ω =Ĥ − ωĴ z , where the cranking frequency ω (in units of MeV) enters through the cranking term. Our SMMC calculations were performed on Jaguar, a Cray XT5 at Oak Ridge National Laboratory (ORNL) with 18,688 nodes containing dual hex-core AMD Opteron processors, totalling 224,256 cores. Simulations investigated cranking frequencies ω = 0.0 to 0.5 MeV and the temperature parameter β was split into N β ∆β slices with ∆β = 1/32 MeV −1 . Each parameter set was run with up to 15 840 statistical samples, requiring a total of 190,080 cores for 4 hours for a full parameter study of a single nucleus.

Nuclei in the fp-gds region
To understand the competition between nuclear pairing correlations as a function of the rotational frequency and temperature for nuclei, we first apply the SMMC method to look at known nuclei in the f p − gds region where sizable pairing correlations dominate in their ground state without the effects of rotation. Studying the temperature-induced interplay between deformation and pairing in four N = 40 isotones, 68 Ni, 70 Zn, 72 Ge, and 80 Zr shows that shape and pairing contribute to the specific heat of the nucleus [11]. Theoretical studies to identify phase transitions in nuclei have focused on the relationship between pairing correlations and an associated peak in the specific heat C v = dE/d(kT ). To assess the magnitude of pairing correlations, it is convenient to employ the J = 0 pair operator     Fig. 2 shows the corresponding specific heat calculations for the four N = 40 isotones. Only for the full interaction (Q + P ) do peaks appear in the specific heat corresponding to the increase in pairing strength at low temperatures. In the case of 68 Ni, proton pairing correlations are very weak and the peak in the C v is associated with the collapse of the neutron pairing with temperature. For 70 Zn, the two peaks in the specific heat can be associated separately with the proton pairing and neutron pairing transitions to normal phase, which occur at different rates with temperature. The proton and neutron pairing transitions in 72 Ge occur in a similar range of temperatures, resulting in a pronounced single peak in C v . In the strongly deformed 80 Zn, there is an absence of a sharp maxima in C v with a pairing strength equal for proton and neutron that is significantly reduced from the lighter N = 40 isotones.

Re-entrance in 72 Ge
In the absence of rotation, the SMMC calculations [11] gives clear evidence for the breaking of isovector pairs at temperatures around kT c ≈ 0.6 MeV in 72 Ge which is reflected by a noticeable peak in the specific heat C v . To study the effects of rotation, we employ the cranking term from ω = 0.0 to 0.5 MeV. In a model of non-interacting particles, the neutrons in 72 Ge would completely occupy the f p shell. However, correlations induced by the residual interaction make it energetically favorable to scatter neutrons across the N =40 shell gap which is about 2.5 MeV. From Ref. [10], we see the effects of rotating the nucleus. Figure 3 shows single-neutron occupations in the wave function of 72 Ge as a function of rotational frequency at two temperatures: kT = 0.47 meV (slighly above g.s.) and 1.6 MeV (well above T c ). In the g.s. configuration, the total neutron occupation of the gds shell is about 3.5, with about 3 neutrons in the g 9/2 orbital. Upon rotating the nucleus, the s.p. cranking term ωĵ z , representing the combined effect of the Coriolis and centrifugal force, generates angular momentum polarization by lifting the magnetic m-degeneracy of s.p. states.    [14]. The ground state band is identified as isospin T = 1, arising from isovector pn correlations, with T = 0 states favored at higher rotational frequency, or equivalently higher excitation energy at about 1.0 MeV.
To study the effects of rotation and temperature on pn pairing correlations in odd-odd N = Z nuclei, we performed SMMC calculations of 74 Rb and two adjacent odd-odd selfconjugate nuclei 70 Br and 78 Y. 78 Y has a highly deformed J = 0, T = 1 ground state with considerable quenching of the pn pairing compared to 74 Rb [15] and 70 Br is predicted to have a J = 0, T = 1 ground state [16], which has not been verified experimentally. The J = 0 pairing strength seen in Fig.6(b) shows contrasting behavior to that seen in 72 Ge, where a local increase in the pairing strength occurs at low frequency and low temperature rather than at high frequency. A clear indicator of a phase transition, such as the dip seen for 72 Ge, does not appear for these nuclei seen in Fig 6(a). The high frequency specific heat exhibits interesting behavior for 74 Rb and 78 Y, by crossing the low frequency specific heat and remaining unusually high. Understanding of these new behaviors requires additional study.

Conclusions
In summary, using the SMMC technique we explored the interplay between temperature and rotation in f p − gds nuclei and its effect on pairing resulting in phase transitions. Our calculations demonstrate the presence of the partial order phenomenon associated with the reappearance of pairing at high rotational frequencies and intermediate temperatures for 72 Ge. Similar behavior is not currently seen for odd-odd N = Z nuclei, but requires further studies to adequately understand results.