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Thermodynamic properties of the SO(5) theory for the antiferromagnetism and d-wave superconductivity: a Monte Carlo study

The present approach takes into account thermal fluctuations both in the rotation of SO(5) superspins between the AF and SC subspaces, and in the phase variables of SC order parameters. Temperature vs. g-field phase diagrams for null external magnetic field are presented, where the g field is conjugate with the quadratic order parameters and breaks the SO(5) symmetry. The normal(N)/AF and N/SC phase boundaries merge tangentially at the bicritical point into the AF/SC phase boundary. Hysteresis phenomenon is observed at the AF/SC phase transition. Enhancement of AF correlations is observed above the SC critical temperature in systems with AF couplings stronger than SC ones. Its relation with the spin-gap phenomenon is addressed. The SO(5) theory in an external magnetic field is also investigated. At sufficiently large g fields the SC order is established through a first-order freezing transition of the flux-line lattice. Short-range AF fluctuations are larger at cores of flux lines than elsewhere, and decrease continuously to zero with increasing g field. At intermediate g fields, the flux-line lattice of long-range SC order and the long-range AF order coexist. Superlattice spots surrounding the strong AF Bragg peaks at Q=(\pm\pi,\pm\pi) are observed in the simulated structure factor, and are identified with the modulation by the triangular flux-line lattice of SC. The AF phase boundary associated with the continuous onset of long-range AF order drops sharply to the g axis from a finite temperature in the temperature vs. g-field phase diagram.


I. INTRODUCTION
technical details of simulation. In Sec. III, simulation results for the null external magnetic field are presented. There are the AF, AF and SC phase-separation, and SC phases in the phase diagrams. The spin-gap phenomenon is also addressed. Section IV is devoted to reveal the effects of an external magnetic field in the SO(5) theory. Coexistence between the long-range AF order and the FLL of long-range SC order is observed. Vortex cores are found of larger AF components than elsewhere. Summary is given in Sec. V.

II. HAMILTONIAN AND SIMULATION TECHNIQUES
The hamiltonian in the present study is given by defined on the simple cubic lattice. The vector t, of two components and coupling ferromagnetically with nearest neighbors, is for the d-wave SC order parameter; the vector s, of three components and with AF coupling between nearest neighbors, is for the AF order parameter. The interplay between the SC and AF order parameters is introduced by the SO(5) constraint on the superspin: The g factor is a field breaking the SO(5) symmetry into the U (1) and SO(3) subgroups, and is proportional to the doping rate in a loose sense [5].
The following notes on the above hamiltonian seem appropriate at this stage. First, the above hamiltonian can be considered as the Ginzburg-Landau description of the SO(5) theory. Both of the AF and SC order parameters, s and t, are defined in a scale larger than the atomic one, but much smaller than the macroscopic one. In this sense they should be called as the local order parameters. The constraint (2) does not imply the existence of long-range order in the macroscopic scale. The long-range order parameter for the AF component is the staggered magnetization, and that for the SC component is the helicity modulus [16]. Second, although no quantum effect is included explicitly in hamiltonian (1), the competition between the two different long-range orders is taken into account sufficiently.
Therefore, the profound, nontrivial thermodynamic properties of the SO(5) theory can be captured. Third, thermal fluctuations in phase variables of SC order parameters, which are especially important for underdoped high-T c cuprates [17], are taken into account by the first term in the above hamiltonian, and treated using the Monte Carlo technique.
Furthermore, this hamiltonian is easily developed so as to incorporate an external magnetic field for the study of vortex states. Fourth, the superspin amplitude is fixed to unity in the above hamiltonian. The onset of superspin amplitude itself upon cooling can also be taken into account in the mean-field fashion, and is expected to correspond to the so-called pseudo-gap phenomenon [5]. Finally, only the simplest symmetry-breaking field g associated with the quadratic terms of order parameters is included in the hamiltonian. Other symmetry-breaking fields appear when high orders of the order parameters are considered [5,12]. Although an argument on magnitudes of these fields is absent right now, it is reasonably expected that the most important features of the breaking of SO(5) symmetry into the AF and SC subspaces are captured by the g field in (1).
A typical simulation process starts from a random configuration of superspins at a sufficiently high temperature.
The system is then cooled gradually. The equilibrium state at a given temperature is generated using typically 50,000 MC sweeps of update from the state of a slightly higher temperature. In each sweep of update, candidate vectors are generated randomly on the five-dimensional unit sphere for superspins on all sites in the system, and are subject to the standard Metropolis algorithm to determine if they are accepted for the next configuration [18]. After this equilibriation process, statistics on physical quantities is performed over 100,000 MC sweeps. Around transition temperatures, more than 10 6 MC sweeps are spent in order to make sure of sufficient equilibriation and statistics.
The system size for simulations in null external magnetic field is L 3 = 40 3 , with periodic boundary conditions in all crystal directions. As the SO(5) superspins are continuous in five dimensions, and the system is of three dimensions in crystal space, a thorough analysis of finite-size effects on simulation results, which is important for determining the relevant critical and bicritical exponents in high precisions, is extremely time consuming. Only for several chosen parameter sets, larger systems have been simulated in order to make sure that the main properties derived from the present simulations do not suffer from finite-size effects. Systematic errors (finite-size effects) are therefore not estimated for data presented in this paper. Statistical errors are comparable to sizes of marks in figures as far as not specified. The AF coupling in the ab plane J AF ab ≡ J is taken as the energy unit, and temperature is measured by J/k B throughout the present paper. observed in the isotropic system is broken. Nevertheless, as indicated in the inset of Fig. 2, in the ab plane the AF correlation length is equal to the SC correlation length when the g field is fixed at the bicritical value. This agreement is not trivial in contrast with the isotropic system. The SC correlation length in the c axis is much smaller than the other correlation lengths.
C. Strongly anisotropic system: In order to simulate real high-T c cuprates, the AF exchange coupling should be taken much stronger than the effective SC coupling, and both AF and SC couplings are much weaker in the c axis. The temperature dependence of the AF staggered magnetization and the helicity modulus of the SC components [20] are shown in Fig. 3 for the system of couplings J SC ab = 0.1J and J SC c = J AF c = 0.01J at the symmetry breaking field g = 1.96J. Since the helicity modulus is proportional to the superfluid density [16], it is clear that the long-range SC order is established below the critical temperature T c ≃ 0.115J/k B . As shown in the same figure, the AF correlation length in the ab plane, ξ AF ab , increases at first as temperature is reduced, and then is suppressed as temperature approaches T c . The maximal ξ SC ab is taken at the temperature T sg ≃ 0.15J/k B . The weight of AF components, s 2 , decreases monotonically in the whole cooling process and shows a sharp decline among T sg and T c . Therefore, the enhancement of ξ AF ab above T sg is clearly the result of reduction of thermal fluctuations; the suppression of ξ AF ab below T sg is because of the loss of the AF order in its competition with the SC order. This peculiar behavior occurs because the AF coupling in the ab plane overwhelms over the SC one, while the SC groundstate is established by the large g field. Temperature dependence of the internal energy and the specific heat for this system are depicted in Fig. 4. No feature can be found around T sg in these two thermodynamic quantities. Therefore, the only phase transition takes place at T c , and T sg corresponds merely to a crossover. The SC correlation length in the ab plane, ξ SC ab , diverges when temperature approaches T c in Fig. 3, as usually in a thermodynamic second-order phase transition.
It is found experimentally that the spin-lattice relaxation rate assumes its maximum at a temperature well above the SC critical point [2]. The present simulation results indicate that this spin-gap phenomenon can be explained by the competition among the long-range SC and AF orders, and thermal fluctuations. Since the enhancement of AF correlations above the SC critical point is observed in the strongly anisotropic system of Fig. 3, but not in isotropic and slightly anisotropic systems of Figs. 1, and 2, it becomes clear that in order to observe the spin-gap behavior, the system should have SC couplings much weaker than AF ones, as in real high-T c cuprates.
The latent heat associated with the first-order transition between the AF and SC phases is approximately Q ≃ 0.05J, and decreases to zero as the bicritical point is approached. The spin-gap like phenomenon is observed in the region g b < g < 2.2J. For g > 2.2J, AF correlations are suppressed by SC components at all temperatures. The experimental fact that spin-gap behaviors are observed only in the underdoped region of high-T c cuprates may be explained by the present simulation result. The ratio between the spin-gap temperature and the SC critical point is T sg /T c ≃ 1.6 at g = 2J, which counts well the experimental observation [2].
In Fig. 5, the spin-gap temperature T sg decreases as the bicritical point is approached. This might seem curious at a first glance, since it is clear from hamiltonian (1) that the larger the g field the smaller the AF components. Shown in Figs. 6 (A) and (B) are the temperature dependence of the AF correlation length and the staggered susceptibility at several g fields. Although both of them are monotonically suppressed by increasing g field when temperature is fixed, the temperature where they take maxima, T sg , increases with the g field, as clearly seen in Figs. 6. It is noted that the spin-gap temperature increases with decreasing doping rate in experiments. The present theory therefore conflicts with experimental observations in this aspect.
The SC correlations are suppressed in the normal state above the AF phase boundary in the present system. In this sense, there is no counterpart of the spin-gap temperature above Néel points. However, it is interesting to observe in Fig. 7 that for the g field in a certain region below the bicritical value, the SC weight, t 2 , takes maximum at a temperature above the corresponding Néel point. The temperature associated with the maximal SC weight, denoted by T p in Fig. 5, may be identified with the pairing temperature [5]. There is no feature in the internal energy and the specific heat around this crossover temperature.

IV. PHASE DIAGRAM AND VORTEX STATES FOR H > HC1
A. Model hamiltonian and phase diagram An external magnetic field penetrates into a type-II superconductor via thin flux lines associated with flux quanta for H larger than the lower critical field H c1 . SC is broken along the flux lines. High-T c superconductors are extremely type-II with very large Ginzburg-Landau numbers κ ∼ 100. Research of the vortex states in high-T c SC has been growing into a vivid field of condensed matter physics and statistics. The most important feature of the vortex states is that the Abrikosov FLL melts into FL liquid via a first-order phase transition [21,22,20].
In the scheme of the SO(5) theory, the free energy of a vortex state can be reduced by rotating the superspins from the SC subspace into the AF subspace at the flux-line cores [5]. The possibility of AF cores of flux lines in the SO(5) theory was first addressed by Arovas et al. [8]. Recently, Alama et al. discussed the κ dependence of the core state [15]. In these studies, the Abrikosov mean-field theory was developed so as to incorporate the AF components.
However, the Abrikosov mean-field theory for the vortex states is not appropriate for the high-T c SC since it only takes into account the amplitude of SC order parameter, and cannot treat thermal fluctuations in the phase variables, which are essentially important for determining the phase diagram of the vortex states in high-T c SC [20].
The hamiltonian for the SO(5) theory in the presence of an external magnetic field may be given as following [13]: where |t| and ϕ are the amplitude and phase of the SC order parameter. The same constraint (2) is applied.
Fluctuations of the magnetic induction are neglected. This approximation is justified when the separation between vortices is larger than the SC correlation length and much smaller than the penetration depth in the ab plane, a condition satisfied in large portion of H − T phase diagrams of high-T c superconductors. The Josephson coupling should also be dominant over the electromagnetic coupling. The first term in the above hamiltonian, known as the fully frustrated 3D XY model, has been used successfully for explaining many important thermodynamic properties of the vortex states in high-T c SC [20,23].
In order to simplify the situation, the case of an external magnetic field parallel to the c axis is addressed in the present paper [23]. The vector potential is given by A = (−yB/2, xB/2, 0) with B = f φ 0 /l 2 ab . Here, φ 0 is the flux quantum, l ab the unit length in the ab plane, and f the average number of flux in each square unit cell in the ab plane. The data shown in the following are for f = 1/25, corresponding to the inter-vortex distance of In the region 1.0J ≤ g ≤ 1.32J, the long-range AF and SC orders coexist at low temperatures. The temperature dependence of the helicity modulus along the c axis, the staggered magnetization, and the specific heat are shown in Fig. 9 for g = 1.1J. The N/AF transition at T N is a thermodynamic second-order phase transition, above the first-order onset of the long-range SC order at T m . Although suppressed by the SC order in certain degree, the staggered magnetization survives to groundstate.
The AF phase boundary drops sharply from [g, T ] = [1.32J, 0.59J/k B ] in Fig. 8. The phase transition at this almost vertical part of AF phase boundary is investigated by tuning the g field at a fixed temperature, in addition to the cooling process mentioned in Sec. II. The turning point on the N/AF phase boundary in Fig. 8 and the tricritical point in Ref. [13] are at the same temperature.

B. AF vortex cores
The vortex cores in the SC phase of the SO(5) theory are different from those without AF competition studied up to date. The structure factors S(q ab , z = 0) for vortices, s 2 , s ab , and s c for g = 1.5J in the FLL phase are displayed in Figs. 10. The Bragg peaks in the structure factor Fig. 10(A) for the vortex correlations are from the triangular FLL.
One also finds Bragg peaks in structure factor Fig. 10(B) for the AF amplitudes at the same wave numbers of Fig.   10(A). This coincidence indicates clearly that cores of flux lines are of larger AF weights than elsewhere. The g-field dependence of the Bragg-spot height for AF weights in the FLL phase, such as those in Fig. 10(B), is investigated when temperature is fixed. As shown in Fig. 11 for T = 0.3J/k B , S(q ab = q max , z = 0) decays with increasing g field in a power law S ≃ p/g q with p = 0.8 ± 0.05 and q = 3 ± 0. For null external magnetic field, there is a bicritical point in the temperature vs. g-field phase diagram, at which the second-order N/AF and N/SC phase boundaries merge tangentially into the first-order AF/SC phase boundary.
Hysteresis phenomenon is observed at the first-order AF/SC phase transition, which may suggest a phase-separation region in the phase diagram.
In systems with much stronger AF couplings than SC ones while the SC groundstate is achieved by g fields larger than the bicritical value, AF correlations are enhanced at a crossover temperature above the SC critical point. The origin of this enhancement in AF correlations is clarified to be the competition among the long-range AF and SC orders and thermal fluctuations. When the g field becomes too large, this crossover fades away since AF correlations are