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A Microscopic Derivation of the SO(5)-Symmetric Landau-Ginzburg Potential

We construct a microscopic model of electron interactions which gives rise to both superconductivity and antiferromagnetism, and which admits an approximate SO(5) symmetry that relates these two phases. The symmetry can be exact, or it may exist only in the long-wavelength limit, depending on the detailed form of the interactions. We compute the macroscopic Landau-Ginzburg free energy for this model as a function of temperature and doping, by explicitly integrating out the fermions. We find that the resulting phase diagram can resemble that observed for the cuprates, with the antiferromagnetism realized as a spin density wave, whose wavelength might be incommensurate with the lattice spacing away from half filling.


Introduction and Summary
Zhang has recently proposed [1] that the superconducting and antiferromagnetic phases of the high-T c cuprates might be related to one another by an approximate SO (5) symmetry of the electronic Hubbard Hamiltonian. This proposal incorporates in a fundamental way the connection between antiferromagnetism (AF) and superconductivity (SC) which is observed in these systems.
Recently, however, the foundations of this picture have come under attack, with criticism directed against the existence of an approximate SO(5) symmetry of the Hubbard Hamiltonian [2] (for a reply, see [3]), as well as against the very possibility, in principle, of relating the antiferromagnetic and superconducting phases by the rotation of a finite-dimensional order parameter [4]. This motivates studying whether and under what conditions an approximate SO(5) symmetry can emerge from a microscopic picture of electron dynamics.
In this paper we investigate a model of electron dynamics which can naturally incorporate an approximate SO(5) symmetry. 1 Here ψ p = ψ p↑ ψ p↓ is the electron field, while φ q and n q are Hubbard-Stratonovich auxiliary scalar fields which, when integrated out, produce the two four-Fermi interactions. ε p denotes the electron single-particle dispersion relation, µ is the chemical potential that measures deviations from half-filling, and σ = {σ i , i = 1, 2, 3} are the Pauli matrices. Q is 1 A broader class of similar models has recently been proposed in ref. [5].
a fixed nesting vector which will be defined below. a 0 φ and a 0 n are positive constants which could, if one wished, be absorbed into the definitions of f and g by a redefinition of φ and n; however it is convenient to keep them for the purposes of renormalization in §4.
Our goal in this work is twofold: first, to find the conditions under which this model exhibits an approximate SO(5) symmetry, and to see how the symmetry-breaking effects manifest themselves at temperatures much less than the Fermi energy. We find the symmetry to be possible, and it can arise "accidentally"at long wavelengths even if it is not important for the microscopic dynamics. We also find that symmetry-breaking effects are marginal, for weak couplings, in the sense that they increase only logarithmically as one scales into the infrared. Second, we wish to compute the effective Landau-Ginzburg (LG) potential, so as to determine how the phase diagram depends on the microscopic electronic couplings. The result is that the AF phase can have the phenomenologically desired shape in the temperature-doping plane, preferring low temperatures and zero doping. Moreover the SC phase, while also preferring small doping, is not suppressed as quickly by increased doping as is the AF phase. The result is that SC becomes energetically favorable to AF at some lower critical doping, and persists until some upper critical doping. These results are consistent with but more predictive than the general effective-field-theory description of [1] and [6]. For example they predict the absence of a mixed AF/SC phase away from zero doping when the fundamental Hamiltonian is SO(5)-invariant. They also agree with the observed phase diagram of the high-T c systems, although we do not attempt here a detailed discussion of the applicability of these models to the cuprates.
Our analysis of the model defined by eqs. (1)-(3) is motivated by the renormalizationgroup (RG) approach to understanding superconductivity and antiferromagnetism as BCS and spin-density-wave instabilities within Fermi liquid theory [7]. In the RG language, a Fermi liquid is understood as a regime of scales for which the dominant quasiparticle degrees of freedom are weakly-coupled, degenerate fermions carrying the same quantum numbers as the underlying electrons. Under such circumstances almost all of the quasiparticle self-interactions are irrelevant in the RG sense: they become less and less important as one integrates out high energy modes to obtain an effective Hamiltonian valid near the Fermi surface.
The only exceptions to the rule that interactions are irrelevant in the infrared are certain four-Fermi terms, which can be marginally relevant for special kinds of electron kinematics. An important example is any pairwise attraction between electrons (or holes) having opposite momenta. This kind of interaction grows logarithmically in the infrared, eventually becoming strong enough to trigger the BCS instability at sufficiently low energies. A second exception exists when the quasiparticle Fermi surface (FS) is nested. 2 In this case the attraction between electrons whose momenta sum to the nesting vector, Q, is also marginally relevant, potentially triggering an instability towards the formation of a condensate which is modulated in space with wavevector Q. The result is a charge-or spin-density wave depending on the electric charge and spin of the attractive channel.
We have both types of instability in mind when using the Hamiltonian in eqs. (1)-(3).
It is assumed that the quasiparticle energies ε p are time-reversal invariant, ε −p = ε p , and have a FS with a nesting vector Q, defined by the property ε p+Q = −ε p for p near the FS.
(Our convention is to define the FS at half filling to be the zero of energy.) An example of a dispersion relation with these properties in two dimensions is that of the extended Hubbard model, where a is the lattice spacing and the nesting vector is Q = ± π a , ± π a . However the general analysis that follows in §4 does not depend on using this explicit form. Notice that because antiferromagnetism arises here as a spin-density wave, its periodicity need not be precisely commensurate with the lattice spacing, particularly away from half filling. Such incommensurate antiferromagnetism is particularly interesting in view of the splitting of the AF peaks which is seen in recent neutron-scattering data [8].
Since eqs. (1)-(3) involve an attractive interaction in the spin-triplet channel, the nesting produces a spin-density-wave instability. The Hubbard-Stratonovich formulation of the Hamiltonian, eqs. (1)-(3), is particularly convenient for following these instabilities of the system since they may be studied by examining the minima of the potential for the modes φ 0 and n Q , which is induced when the electrons are integrated out. Depending on which of these modes condenses most strongly, the resulting ordered phases are superconductors or antiferromagnets, possibly both.
Our presentation is organized as follows. In the next section, §2, the conditions for the Hamiltonian, eqs. (1)-(3), to exhibit SO(5) symmetry are derived. We find that the symmetry is present in the long-wavelength limit, q → 0, if the coupling functions satisfy the relation f 2 = g 2 . In §3 we integrate out the electrons to obtain the long-wavelength effective potential for the order parameters, φ 0 and n Q . The expressions obtained in §3 are evaluated in §4, in the limit of degenerate electrons, i.e. with all couplings and dispersion relations linearized around the FS. We present explicit expressions in this limit for the coefficients of the Landau-Ginzburg theory, as functions of the fundamental fermion couplings, temperature and chemical potential. In some parts of the phase diagram the usual expansion of the free energy to quartic order in the fields is unbounded from below, forcing us to use the full potential. From these results it is straightforward to determine the main features of the phase diagram. We find the dependence on T and µ can have the desired form if the AF interaction in eq. (2) is chosen to be somewhat (∼ 10 − 20%) stronger than the SC interaction of eq. (3). In §5 we examine the validity of the degenerateelectron approximation by recomputing the quadratic and quartic terms of the effective potential without linearizing about the FS. We do so using specific choices of ε, f and g that have been previously suggested [1,5,9] in connection with d-wave superconductivity, to show that the approximations used in §4 are indeed reliable. §6 concludes with a brief recapitulation of our results.

The SO(5) Symmetry
In this section we identify the conditions under which the Hamiltonian, eqs.  The electron part of the corresponding conserved charges (electric charge and spin) are The nontrivial requirement for SO (5) invariance is the existence of the electricallycharged, spin-triplet off-diagonal operators, Π, whose real and imaginary parts fill out the generators of SO(5) in addition to Q and S. Motivated by Zhang's [1] choice, we take its action on electrons to be given by where h p is a function to be determined. As is easily verified, this has the desired commutation relations with Q and S, and satisfies Π a , Π b = 0. In addition, if the function h p shares the same properties as does ε p : i.e. h p+Q = −h p , and h −p = h p , then: Eq. (7) shows that {Q, S, Π, Π * } satisfy the SO(5) commutation relations provided we choose |h p | = c, where c is an appropriately chosen constant. This is only consistent with the property h p+Q = −h p if h p = ±c for p inside the FS, and h p = ∓c outside.

2.2) Conditions for Symmetry
We next determine the conditions under which the Hamiltonian, eqs. (1)-(3), commutes with the Π. A nonvanishing chemical potential explicitly breaks SO(5) down to SO(3) × SO(2), so we set µ = 0 here. Furthermore, we ignore for now the scalar part of H 0 , which is invariant provided φ and n transform together in the fundamental representation of SO (5). The electron part of the free Hamiltonian then satisfies: This vanishes by virtue of the Fermi statistics of ψ, provided that ε p shares the previously advertised symmetry properties of h p : i.e. if ε p+Q = −ε p and ε −p = ε p .
For notational convenience, we define the operators appearing in H int by V b (q) and The operators V a and W rotate into one another as a five-dimensional vector of SO (5) provided the coupling functions satisfy the conditions Of particular interest for many purposes, particularly the phase diagram, is the long-wavelength limit, q → 0. In this limit we have Π a , V b (0) = −4δ ab W * (0) and Π a , W (0) = −4 V a (0), provided the functions f p ≡ lim q→0 f (p, q) and g p ≡ lim q→0 g(p, q) satisfy the properties f p = f −p , g p = g −p = −g p+Q and f p h p = g p and g p h p = f p . Given our earlier choice that h p = ±c, with opposite signs inside and outside the FS, we see that W (0) and V b (0) fill out the fundamental representation of SO(5) provided that the coupling functions satisfy f p = sign(ε p )g p .
Finally, so long as W and V b transform as a 5 of SO (5), the entire Hamiltonian, (5) is taken also to rotate the five scalars, 2 and n Q , into one another as a 5. Notice that although the demands made on f and g for exact SO(5) symmetry are quite strong, the symmetry can emerge for small q under much weaker conditions, and this is sufficient for the invariance of the phase diagram.

Integrating out the Electrons
We now turn to the derivation of the effective theory which governs the long-wavelength scalar modes, φ 0 and n Q . The resulting effective potential will be subsequently used to identify the phase diagram of the system, and to see to what extent it exhibits SO (5) invariance. Since the effective potential must be symmetric under U (1) transformations of φ 0 and under SO(3) rotations of n Q , for notational simplicity we shall take φ 0 ≡ φ real and (n Q ) i = n δ i,3 .

3.1) Preliminary Issues
The thermodynamic variables we wish to follow in the LG potential are the values of the fields φ and n themselves, as well as the temperature, T , and the electron doping away from half-filling, x. We therefore ignore all scalar modes apart from φ and n, and compute the effective potential for these variables. In order to follow the doping we also introduce a chemical potential, µ, which measures the change in density away from the free-particle FS (which is defined here by ε p = 0).
We now integrate out the electrons, using the Hamiltonian, eqs. (1)-(3), supplemented by the chemical potential. To incorporate the Fermi sea we exchange the electron field, ψ p for p inside the FS, for a hole field, χ p = σ 2 ψ * −p , in the usual way, permitting the use of vacuum propagators for the integration over the fermion fields.
For the purposes of performing the functional integral over ψ p and χ p , it is convenient to group ψ p , ψ * p and their counterparts having momentum −p and p − Q into the following 16-component vector: When Ψ p is used as the field, all sums over p must be restricted to the original Brillouin zone, modulo the transformations p → −p and p → p + Q.
With this choice the quadratic part of the Hamiltonian may be written and A and B denoting the following 8 × 8 matrices: and In these last expressions, each entry of the matrices A and B are 2 × 2 matrices in spin space.

3.2) The Functional Determinant
Standard methods may now be used to compute the functional integral over the fermion fields. Using the matrix form for the Hamiltonian, eqs. (11)-(13), the result for the one-loop, finite-temperature contribution to the free energy density, F 1 , in d spatial dimensions is: where, as usual, β = 1/kT and the sum is over the Matsubara frequencies, ω j = (2j +1)πβ.
We have switched to continuum momenta, and restored the integration region to the full Brillouin zone (so that it is no longer modded out by p → −p and p → p + Q). The eigenvalues of the Hamiltonian ∆ are given by Even before undertaking a detailed evaluation of the free energy (14), we can prove an interesting result: in the SO(5) limit where f 2 p /a 0 n = g 2 p /a 0 φ , and for nonzero doping, µ = 0, the ground state must have φ = 0 or n = 0 (possibly both) and hence is never a mixed phase of AF and SC. This follows from the form of the energy eigenvalues (15) in the conditions for minimizing the free energy. Adding F 1 to the tree-level free energy F 0 = a 0 φ φ 2 + a 0 n n 2 , these conditions are where X stands for either of the fields φ or n. The derivatives of the eigenvalues have the form When f 2 p = g 2 p and a 0 φ = a 0 n = a 0 , the minimization conditions can be written in the form where it is easily shown that G and µH are positive definite functions. Therefore in this limit there are never simultaneous solutions where both φ and n are nonvanishing, except when µ vanishes.
When evaluating these expressions in the next section, we focus on the degenerate limit, for which kT is much less than the Fermi energy at zero doping. For example, using the explicit dispersion relation given by eq. (4), this corresponds to neglecting powers of kT /t. In this limit only states near the Fermi surface are important in the integrals, and it is useful to decompose the momentum p into a component k on the FS and a component ℓ perpendicular to the FS. We may then linearize all quantities in ℓ, restrict the coupling functions g and f to momenta lying in the FS, and obtain for the dispersion relation generally being a function of momentum, k.
Eqs. (14) and (15) are this section's main results. We now extract their implications for the LG potential, 3 first by examining them in detail in the vicinity of zero fields, φ = n = 0, and then by computing the potential for arbitrarily values of the fields, in certain directions in the φ-n plane.

The Landau-Ginzburg Theory
Eqs. (14) and (15) define the effective potential for the SC order parameter, φ, and the AF order parameter, n. Since this potential is analytic 4 at φ = n = 0, an expansion to quartic order in the fields gives a good indication of when a condensate will form for either φ or n. However we will find that the quartic terms do not always stabilize the potential, and it is necessary to know how the full potential looks away from φ = n = 0. In this case we will use the full free energy in order to determine the phase diagram of the system.

4.1) Analytic Expressions in the Quartic Approximation
We start by approximating the free energy with a quartic potential, whose form we take to be where the ellipsis denotes terms involving more than four powers of φ or n. Using Eqs. (14) and (15) and defining we find that the quadratic coefficients are given by the following integrals: where a 0 φ and a 0 n are the "bare" values of these quantities that appear in the original Hamiltonian, eqs.  3), are no longer appropriate, whichever is lower. If these approximations were exact, then Λ would be the actual maximum energy available on the lattice. In any case, we can consider Λ to be of order the Fermi energy.
The divergent dependence on Λ as Λ → ∞ can be absorbed into a renormalization of a 0 φ and a 0 n .
The quartic coefficients are given similarly by the following expressions, which since they are convergent allow us to take the cutoff Λ to infinity if we wish: The approximation Λ → ∞ is only valid if T and µ are much smaller than the Fermi energy.
Because we work in the degenerate limit, near the FS, and at zero field, the shape of the coefficient functions as k varies over the FS enters only as an overall normalization factor. As expected from our previous discussion, the SO(5) symmetry is manisfestly present in the limit µ = 0 and f 2 /a 0 n = g 2 /a 0 φ .

4.2) Instability Towards Condensation
Before turning to the numerical evaluation of these integrals, it is instructive to first explore the physical implications of the renormalization itself. Because the sign of the loop contributions to a φ and a n is negative, these quantities decrease as the temperature is decreased. In the limit of large βΛ, it is easy to see that the one-loop contributions ∆a X = a X − a 0 X have the limiting behavior ∆a φ ∼ −2 N 0,2 ln(βΛ); ∆a n ∼ −2 N 2,0 ln(βΛ).
Even if a φ (T ∼ Λ) and a n (T ∼ Λ) are initially chosen to be positive, so that F is minimized by φ = n = 0, eventually their decrease with decreasing T can drive them negative, causing an instability towards the development of nonzero condenstates for φ or n.
Eq. (26) gives the evolution of the SO(5)-breaking combination of the electron couplings: This shows that any SO(5) breaking present in the original Hamiltonian due to f 2 = g 2 will be enhanced in the effective GL Hamiltonian by a potentially large logarithm of the form ln(βΛ), where Λ is of the order of the Fermi energy. For example, suppose the AF pairing is taken to be s-wave, f = f 0 , whereas the SC pairing is d-wave, corresponding to g = g 0 (cos(ak x ) − cos(ak y )). Here f 0 and g 0 are arbitrary constants. In general N 2,0 and N 0,2 differ from each other, and although g 0 /f 0 can be tuned to make the difference vanish, there seems to be no symmetry principle that would require it to take this special value. Moreover, even if such a choice were made, it would still not insure that the quartic couplings would have SO(5) symmetry, i.e., N 4,0 = N 0,4 = N 2,2 . In §5 we will say more about the extent of SO(5)-breaking that arises from this choice of coupling functions.

4.3) Numerical
Results near φ = n = 0 Figures (1) and (3)   like ln(T /Λ), so it is always possible to choose the bare coefficients a 0 n or a 0 φ in such a way that a n or a φ is positive for high temperatures and changes sign at some T c , triggering the instability toward condensation. As the doping is increased, a n increases much more rapidly than a φ . It follows that if the AF phase is energetically preferable to the SC phase at µ = 0, the system will be AF at zero doping, and then make the transition to SC at some small doping, provided that a φ is still negative at this value of µ.
With applications to the high-T c cuprates in mind, we imagine Λ ∼ 0.1 eV ∼ 10 3 K and so we adjust the bare coupling a 0 n to ensure that the AF transition (Néel) temperature, T N , at µ = 0 is 1/10 of the cutoff scale, β N Λ = 10. From fig. 1 it can be seen that at a temperature T = T N /10, for example, we have βΛ = 100 and so the AF phase is quenched at a critical doping of x c ∼ µ c /Λ = 10 %. This is the doping at which a n (µ c ) βΛ=100 becomes equal to a n (0) βΛ=10 . The coefficients (a φ −a 0 φ )/N 0,2 (dashed) and (a n −a 0 n )/N 2,0 (solid) as a function of µ/Λ, for βΛ=10, 20, 50, 100 and 200.
The above example ignores the SC order parameter, which would be valid if a 0 φ was taken to be much larger than a 0 n . If on the other hand SO(5) symmetry was realized, so that a 0 φ = a 0 n and N 0,2 = N 2,0 , then at µ = 0 the ground state would be an arbitrary admixture of AF and SC phases since both have the same energy. In this case, the system will always make a transition to the pure SC phase at the critical doping x c . Since experiments indicate that at µ = 0 the cuprates are pure AF, the SC phase must be suppressed by having a 0 φ /N 0,2 > a 0 n /N 2,0 , so that SC is energetically disfavored until the critical doping is reached. In this situation, the phase diagram must have a form similar to figure 2. Of course, this prediction does not take account of quantum fluctuations within the effective theory itself. Such higher-loop effects are believed to be especially important near the bicritical point, where the SC phase first emerges [1]. They may have the effect of driving this point downward, leaving a gap between the SC and AF phases at temperatures somewhat above the existence of the bicritical point.
To determine exactly how much SO(5) breaking is needed in the quadratic terms in order to suppress SC at zero doping, one must compare the local minima of the potential along the φ = 0 and n = 0 axes and insure that the minimum with n = 0 is the deeper of the two. This issue can usually be settled by looking at the higher order terms in the AF SC T Figure 2 Schematic representation of the phase diagram for the present Hamiltonian, with SO(5) symmetry broken to favor the AF condensate. Quantum effects not considered here may be able to drive the bicritical point downward.
potential. To this end we have computed the quartic coefficients in the dimensionless forms b φφ T 2 /N 0,4 , b nn T 2 /N 4,0 and b nφ T 2 /N 2,2 , and plotted them in fig. 3.
The fact that b φφ is only weakly dependent on the doping can be see by observing that when Λ → ∞, µ can be completely removed by shifting the integration variable from ε to ξ. The low temperature form of the quartic couplings can be further understood analytically through the identities Eq. (29) is valid so long as it is combined with a function that is nonsingular at ε = 0, in which case its own 1/ε singularity can be integrated over using the principal value. Both  The coefficients b φφ T 2 /N 0,4 , b nn T 2 /N 4,0 and b nφ T 2 /N 2,2 as a function of µ/Λ, for Λ/T =10, 20, 50, 100 and 200. The curves approach an asymptotic limit as Λ/T →∞.

4.4) Numerical Results for Arbitrary φ and n
In computing the full expression for F 1 in eq. (14), we can no longer factorize the momentum integral into a product of the form d d−1 k G(k) dεH(ε) because the integrand is not a simple polynomial in f p and g p . However we can still treat it similarly to our previous computations if we consider the special case where f 2 and g 2 are constant. We do not expect this assumption to radically change the features of the more general case, since the k-dependence of f and g has played no essential role in the preceding discussion.
Constant g 2 still describes d-wave SC and is consistent with SO(5) symmetry, so long as g changes sign across the nesting surface, e.g. g p = sign(ε p ).
With this assumption, the free energy reduces to a single integral over the energy.
We have evaluated it numerically for the relevant ranges of doping, temperature, and order parameters. The results are most conveniently expressed in terms of the following variables, for which the cutoff, Λ, is scaled out: In the last line we subtracted a constant to make ∆F vanish at φ = n = 0. Then the full effective potential can be written in terms of The rescaled effective potentialF for six values of the doping parameter, x (horizontal rows) and along three directions in the φ-n plane (vertical columns).F is plotted as a function ofφ 2 , (φ 2 +n 2 )/2, andn 2 , in the three respective columns. Each graph is shown for the range of temperatures given by βΛ=4, 10, 20, and 200, withF always increasing with temperature.

4.5) Terms with Derivatives
So far we have presented the one-loop free energy only for spatially constant order parameters. If φ(x) and n(x) are not constant, there will be additional terms depending on their spatial derivatives. Of these, the most interesting are the (∇φ) 2 and (∇n) 2 terms, since they are needed to compute quantum corrections coming from within the effective long-wavelength theory (two-loop and higher corrections).
By calculating the relevant Feynman diagrams, it is straightforward to show that the quadratic terms in F 1 for φ q and n q with an arbitrary momentum q can be obtained from our previous formulas (21)-(22) by making the replacements where ξ p = ε p − µ and p is the loop momentum to be integrated over. Expanding in the external momentum q gives the expansion in derivatives of the fields. From this prescription it is clear that SO(5) symmetry will be realized in the derivative terms so long as f 2 (p, q) = g 2 (p, q) and µ = 0, since the symmetry is then manifest in the energy eigenvalues λ ± that appear in (15).
The resulting integral over p depends on the exact form of the dispersion relation ε p and the coupling functions f p , g p , so we will defer further evaluation until the next section where we consider a specific choice of these functions.

Exact Results in a Specific Model
Our analysis so far has relied upon the approximation that the most important contributions to the effective potential are coming from electronic excitations near the Fermi surface. This approximation presupposes the Fermi energy to be much greater than the temperature and the chemical potential. On the other hand this hierarchy of scales may only be a factor of 10 in the actual cuprate systems we are interested in. It is therefore interesting to corroborate our conclusions without assuming that F 1 is dominated by the contributions from near the FS. In this section we will exactly evaluate the first few terms in the small-field expansion of F using a semi-realistic lattice dispersion relation and pairing functions, which have been previously suggested in the literature [1,5,9]. It will be shown that the results are in good qualitative agreement with those based on the RG approach.

5.1) Method of Solution
For our specific model, we take the lattice dispersion relation (4)  The effective potential at q = 0 is still given by the general expression (14), but now the momentum integrals range over the Brillouin zone, −π/a < p x , p y < π/a, and there is no need to introduce any cutoff because the energy is explicitly bounded by −4t < ε < 4t.
As before, we expand F to quadratic and quartic order in the fields. Defining ξ p = ε p − µ, this gives Next one would like to reexpress the momentum integrals as an integral over the energy times an integral over the Fermi surface. This can be accomplished using the identity where u = cos(ap x ) and v = cos(ap y ). The integrals over u and v can be done exactly, leaving an energy integral times a density of states function, The density of states N m is given in terms of complete elliptic functions K and E; 5 definingε = ε/4t and α = (1 + |ε|)/(1 − |ε|), the first few are The density of states N 0 , N 2 and N 4 as a function ofε = ε/4t.
near ε = 0 in N m is too weak to compensate for the SO(5)-breaking differences between N 0 and N 2 away from the FS, as we will presently show.

5.2) Expansion near
Now the expressions for the coefficient functions in the small-field expansion of F 1 can be written in a way which looks very similar to our previous results (21)-(25). Let us first where a is the lattice spacing, and m is the quasiparticle mass in the vicinity of the FS, where the dispersion relation can be approximated as ε = p 2 /2m. We then obtain The dependence of these coefficients on doping and temperature is shown in Figures 6 and 7, which can be compared to Figures 1 and 3 from our model-independent analysis. In Figure 6, which shows the 1-loop contributions to the quadratic coefficients, one sees that our previous result for a n is in good agreement with the exact computation in the present model. This is not suprising because f p is treated as a constant in both cases, and the only difference between the two is the lattice dispersion relation in the present computation versus the approximation ε ≈ v F ℓ in §4. However the difference between the two approaches is quite apparent in a φ , where g p is relevant: a φ increases with doping faster in the exact computation than in the approximation of factoring out the FS integration. Nevertheless the important qualitative feature needed to obtain the phase diagram of Figure 2 persists: a φ increases more slowly as a function of doping than does a n .
Similar remarks apply in comparing the quartic coefficients in the two treatments.
The difference in the dispersion relation causes a small change in the T -and µ-dependence of b nn , the coefficient that does not depend at all on g p . Nevertheless in both analyses, b nn and b nφ have the same behavior of initially falling very sharply from positive to negative values as the chemical potential µ is increased from µ = 0, and gradually becoming zero as µ → 4t. b φφ falls much faster with x in the exact computation than in the previous one, but still more gradually than b nn or b nφ . Again, this can be understood analytically in the low-temperature limit using (28)-(29), which gives b nn = b nφ = 0 and b φφ = (7ζ(3)β 2 m/π 4h2 )N 4 (µ/4t) for µ ≫ 1/β. The qualitative agreement between the two approaches gives us confidence that the shape of the phase diagram in Figure 2 is indeed a robust prediction of the model, in the one-loop approximation. µ /Λ ∆ /κ a n ∆ /κ a φ

Figure 6
The exactly computed coefficients (a n −a 0 n )/κ (solid) and (a φ −a 0 φ )/κ (dashed) as a function of µ/Λ, for the same values of βΛ as in Figures 1 and 3. The exactly computed coefficients b nn T 2 /κ, b nφ T 2 /κ and b φφ T 2 /κ as a function of µ/Λ, for the same values of βΛ as in Figures 1, 3 and 6.

5.3) SO(5)-breaking due to
One interesting issue we can explore within the exact computation is the extent to which SO(5) symmetry is broken by the assumption of d-wave SC pairing versus s-wave AF pairing. In Fig. 6 one sees that, despite our tuning the normalization of g p so as to preserve SO(5) near the FS, |∆a n | > |∆a φ | even at zero doping. We could have tuned g p such as to force ∆a n = ∆a φ at some temperature T 0 , but the equality would hold only for T = T 0 . In fact we find that the ratio ∆a φ /∆a n scales linearly with ln(β/4t): ∆a φ ∆a n µ=0 ∼ = 0.20 + 0.064 ln(β/4t).

5.4) Derivative Terms in the Free Energy
Using the above techniques and the procedure discussed in §4.5, we can evaluate the lowest derivative terms in the quadratic part of F 1 , Defining κ ′ = β 2 t/(4πh) 2 ,ε = ε/4t,μ = µ/4t, and going to the limit of large βt, we can write the coefficients as To obtain (54) we used the identity (28), which is however only valid for µ ≫ 1/β. For µ < ∼ 1/β, the logarithmic singularities in N 0 and N 2 are integrated over to obtain result that is finite as µ → 0. The identity (28) gives a vanishing result when applied to (55), meaning that c n is of order 1/βt compared to c φ , and thus c n must be computed numerically to find the leading behavior in the large βt limit. Again, these statements hold only for µ ≫ 1/β due to the singular behavior of the integrand at ε = 0. For µ < ∼ 1/β, we expect that c φ ∼ c n , just as was found for the coefficients of the quartic terms.
These derivative terms will not be of further use in the present preliminary study, but may be useful in computing the higher order quantum corrections which we leave for future investigation.

Conclusions
In summary, our calculations lead us to the following conclusions: • Microscopic SO(5) Invariance: We find, by construction, that microscopic models can indeed give rise to SO(5) invariance which relates the d-wave superconducting and antiferromagnetic phases. This symmetry can emerge accidentally at long wavelengths, and so be relevant for the effective potential which determines which phase is energetically preferred (in a homogeneous system). Being a long-wavelength effect, the symmetry can be easily hidden in treatments which include all scales together.
• Features of the Phase Diagram: Using our model we have computed the Landau-Ginzburg free energy whose minimization determines the phase diagram in the temperature-doping plane. We find that the coefficients of the Landau-Ginzburg potential have the right qualitative dependence on temperature and chemical potential to allow for antiferromagnetism at half filling, but which is destroyed as the system is doped. It can also describe a superconductor which is favoured only for a range of nonzero dopings.
• Significance of the SO(5) Symmetry: The class of models we considered encompasses SO(5) symmetry as a special case among the most general possible coupling functions. We find that if the symmetry is broken in the microscopic theory, there is no special tendency for it to be restored or further broken in the effective long-wavelength theory. The phase diagram is only of the desired form if there is a certain level of SO(5)-breaking, whose fundamental origin is unknown. In the specific model considered in §5, the symmetry appears to be badly broken, since f 2 p and g 2 p are quite different functions of momentum, yet by tuning their relative sizes it appears possible to obtain the correct phase diagram.
At present it is therefore not yet clear whether nature demands an approximate SO (5) symmetry as the correct way to understand the relation between the AF and SC phases in cuprates, or whether SO(5)-symmetric models simply fall into the same universality class as the correct theory.