Magnetic Instabilities and Phase Diagram of the Double-Exchange Model in Infinite Dimensions

Dynamical mean-field theory is used to study the magnetic instabilities and phase diagram of the double-exchange (DE) model with Hund's coupling J_H>0 in infinite dimensions. In addition to ferromagnetic (FM) and antiferromagnetic (AF) phases, the DE model supports a broad class of short-range ordered (SRO) states with extensive entropy and short-range magnetic order. For any site on the Bethe lattice, the correlation parameter q of a SRO state is given by the average q=<sin^2(theta_i/2)>, where theta_i is the angle between any spin and its neighbors. Unlike the FM (q=0) and AF (q=1) transitions, the transition temperature of a SRO state (T_{SRO}) with 0<q<1 cannot be obtained from the magnetic susceptibility. But a solution of the coupled Green's functions in the weak-coupling limit indicates that a SRO state always has a higher transition temperature than the AF for all fillings p<1 and even than the FM for 0.26\le p \le 0.39. For 0.39<p<0.73, where both the FM and AF phases are unstable for small J_H, a SRO phase has a non-zero T_{SRO} except close to p=0.5. As J_H increases, T_{SRO} eventually vanishes and the FM dominates. For small J_H, the T=0 phase diagram is greatly simplified by the presence of the SRO phase. A SRO phase is found to have lower energy than either the FM or AF phases for 0.26\le p<1. Phase separation (PS) disappears as J_H-->0 but appears for J_H\neq 0. For p near 1, PS occurs between an AF with p=1 and either a SRO or a FM phase. The stability of a SRO state at T=0 can be understood by examining the interacting DOS,which is gapped for any nonzero J_H in an AF but only when J_H exceeds a critical value in a SRO state.


Abstract
Dynamical mean-field theory is used to study the magnetic instabilities and phase diagram of the double-exchange model with Hund's coupling J H > 0 in infinite dimensions. In addition to ferromagnetic (FM) and antiferromagnetic (AF) phases, the DE model also supports a broad class of short-range ordered (SRO) states with extensive entropy and short-range magnetic order. For any site on the Bethe lattice, the correlation parameter q of a SRO state is given by the average q = sin 2 (θ i /2) , where θ i is the angle between any spin and its neighbors. Unlike the FM (q = 0) and AF (q = 1) transitions, the transition temperature of a SRO state with 0 < q < 1 cannot be obtained from the magnetic susceptibility. But a solution of the coupled Green's functions in the weak-coupling limit indicates that a SRO state always has a higher transition temperature than the AF for all fillings p below 1 and even has a higher transition temperature than the FM for 0.26 ≤ p ≤ 0.39. For 0.39 < p < 0.73, where both the FM and AF phases are unstable for small J H , a SRO phase has a non-zero transition temperature except close to p = 0.5. As J H increases, the SRO transition temperature eventually vanishes and the FM phase dominates the phase diagram.
For small J H , the T = 0 phase diagram of the DE model is greatly simplified by the presence of the SRO phase. A SRO phase is found to have lower energy than either the FM or AF phases for 0.26 ≤ p < 1. Phase separation disappears as J H → 0 but appears for any non-zero coupling.
For fillings near p = 1, phase separation occurs between an AF with p = 1 and either a SRO or a FM phase. The stability of a SRO state at T = 0 can be understood by examining the interacting density-of-states, which is gapped for any nonzero J H in an AF but only when J H exceeds a critical value in a SRO state. With the renewed interest in manganites [1] and the revitalized study of dilute magnetic semiconductors [2], attention has once again focused on the double-exchange (DE) model and its variants. Applied to the manganites, the DE model is customarily formulated with one local moment per site and a Hund's coupling J H taken to be much larger than the electron bandwidth W . In studies of dilute magnetic semiconductors, on the other hand, the local moments are sparse and the exchange coupling is comparable to W . While the full phase diagram of the DE model as a function of J H and electron filling p has been studied by several authors [3,4,5,6], the properties of the DE model for small coupling constant remain very much in doubt. In particular, there have been conflicting claims about the presence of phase separation (PS) [5] and incommensurate phases [3,4,6] for small J H , when the ordered FM and AF phases are magnetically frustrated by a RKKY-like interaction between the local moments [7]. This paper uses DMFT to evaluate the magnetic instabilities and T = 0 phase diagram of the DE model. Developed in the late 1980's by Müller-Hartmann [8] and Metzner and Vollhardt [9], DMFT exploits the fact that the self-energy becomes independent of momentum in infinite dimensions, where DMFT becomes formally exact. Even in three dimensions, DMFT is believed to capture the physics of correlated systems including the narrowing of electron bands and the Mott-Hubbard transition [10]. Although DMFT has been widely applied to the DE model [4,5,11,12,13,14,15,16,17,18], until now there has been no complete treatment of the phase instabilities and T = 0 phase diagram of the DE model for arbitrary J H and p.
We shall study a system with a bare semicircular density-of-states (DOS) given by N 0 (ω) = (8/πW 2 )Re W 2 /4 − ω 2 . In real space, this DOS belongs to an infinitedimensional Bethe lattice or an infinite Cayley tree with no closed loops [19]. A finitedimensional Bethe lattice with coordination number z c = 4 is sketched in Fig.1. Although the Bethe lattice lacks translational symmetry. it is quite convenient for calculations. As shown in Fig.1, the Bethe lattice can be partitioned into A and B sublattices so that both ferromagnetic (FM) and antiferromagnetic (AF) long-range order are possible. Due to the bounds ±W/2, the DOS of the Bethe lattice more closely resembles the DOS of two-and three-dimensional systems than does the unbounded DOS of the hypercubic lattice. Indeed, pathological results have been obtained on a hypercubic lattice due to the asymptotic freedom of the quasiparticles in the tails of the Gaussian DOS [16]. As we shall see, the Bethe lattice also has the advantage that analytic results are possible in the limit of small J H , precisely the regime where controversies persist.
The high-temperature, non-magnetic (NM) phases of the Heisenberg and DE models have a correlation length ξ that vanishes as z c → ∞. By contrast, the short-range ordered (SRO) states introduced in an earlier paper [20] possess some of the same characteristics as spin glasses: local magnetic order and exponentially decaying magnetic correlations [21]. The SRO states are characterized by the correlation parameter q = sin 2 (θ i /2) , where θ i are the angles between any spin and its neighbors. Overall, the neighboring spins describe a cone with angle 2 arcsin( √ q) around the central spin. This disordered state reduces to the FM and AF phases when q = 0 and 1, respectively. Unlike the FM and AF phases, a SRO state with 0 < q < 1 has extensive entropy and short-range but not long-range magnetic order. Since it is not possible to associate a wavevector with any SRO state, its transition temperatures where c † iα and c iα are the creation and destruction operators for an electron with spin α at site i, J H > 0 is the Hund's coupling, s i = c † iα σ αβ c iβ /2 is the electronic spin, and S i = Sm i is the spin of the local moment. Repeated spin indices are summed. The local moment will be treated classically, which is only a fair approximation for the magnanites with spin S = 3/2 but a better approximation for dilute magnetic semiconductors with S = 5/2. For large z c , the hopping energy t scales as 1/ √ z c and the local effective action on site 0 may be written where J = J H S, ν n = (2n + 1)πT ,c 0α (iν n ) and c 0α (iν n ) are now anticommuting Grassman variables, m is the orientation of the local moment on site 0, and G 0 (iν n ) is the bare Green's function containing dynamical information about the hopping of electrons from other sites onto site 0 and off again.
Because S eff (m) is quadratic in the Grassman variables, the full Green's function G(m, iν n ) αβ at site 0 for a fixed m may be readily solved by integrating over the Grassman variables, with the NM result where G 0 (iν n ) αβ = δ αβ G 0 (iν n ) is diagonal above T C . Averaging over the orientations of the If P (m) = Tr f exp(−S eff (m)) /Z is the probability for the local moment to point in the m direction, then the average over m is given by C(m) m ≡ dΩ m P (m)C(m). The effective partition function Z involves both a trace over the Fermion degrees of freedom and an average over m: Z = dΩ m Tr f exp(−S eff (m)) . Above T C , P (m) = 1/4π is constant.
Likewise above T C , the lattice Green's function G(m, k, iν n ) αβ has the expectation value with z n = iν n + µ. Summing Eq.(5) over all k and equating the result to Eq.(4), we obtain [10,11] where W = 4t √ z c is the full bandwidth of the semi-circular DOS.
Our goal is to evaluate the general susceptibility [15] l,n where ω m = 2mπT and C involves both a Fermion trace and an average over all the Each term in this equation is a matrix in Matsubara space and matrix multiplication involves a summation over intermediate Matsubara frequencies, as shown in Fig.2(c) [15]. The local susceptibility χ βα,δκ iωm is obtained from the lattice susceptibility χ βα,δκ q,iωm by summing over q and satisfies the local BS equation: Since the vertex function is independent of momentum in infinite dimensions, it appears identically in both the lattice and local BS equations [23,24].
For the bare susceptibilities, the electrons in each Green's function are uncorrelated, which means that each Green's function is separately averaged over m. So the bare, lattice susceptibility is given by The bare, local susceptibility is obtained by summing this expression over all q: where G(iν l ) αβ = δ αβ G(iν l ) is the m-averaged, local Green's function given by Eq. (4). It is also straightforward to evaluate the full, local susceptibility: where G(m, iν l ) αβ is the m-dependent, local Green's function given by Eq.(3).
After this decoupling, both the lattice and local BS equations separate into two sets of equations for the charge and spin degrees of freedom: where a = c or s. The same charge and spin vertex functions enter the lattice and local equations. The spin BS equations with a = s were previously solved [15] in the strongcoupling limit J ≫ W and J ≫ T .
To generally solve Eq.(13) for the charge and spin lattice susceptibilities, we first obtain the charge and spin vertex functions from the local susceptibilities given by Eq. (14). The vertex function may be written as where a n = G 0 (iν n ) −2 − J 2 , b n = G 0 (iν n ) −2 − J 2 /3, and C = −(2J 2 /3) n b −1 n . In terms of A ln and B l m , the charge and spin vertex functions are then given by Γ c iωm (iν l , iν n ) = 6B l m δ ln and Γ s iωm (iν l , iν n ) = 4A ln δ m,0 − 2B l m δ ln .

III. THERMODYNAMIC CONSISTENCY
A thermodynamically consistent theory produces identical results from either the Green's function (which is on the single-particle level) or the partition function (which involves interactions on the two-particle level). Baym and Kadanoff [22] demonstrated that a sufficient condition for a theory to be thermodynamically consistent is that it be Φderivable, which means that a functional Φ({G(iν n )}), constructed from compact diagrams involving the full Green's functions and the bare vertex functions, satisfies the condition Σ(iν n ) αβ = δΦ/δG(iν n ) αβ . The partition function of a Φ-derivable theory satisfies the rela- which is stationary under variations of G(iν n ). Whereas Baym and Kadanoff considered systems of interacting Fermions and Bosons, their ideas were later extended to systems of interacting electrons and spins [25] and to disordered alloys [26]. The Φ-derivability of the DMFT of the DE model was first demonstrated in Ref. [17].
The bare vertex function Γ (0) βα,δκ iωm (iν l , iν n ) may be associated with the two-particle interaction in the purely electronic effective action [27] Because the Fermion operators anticommute, the bare vertex function must satisfy the crossing symmetries Γ There are two ways to evaluate the bare vertex function. First, we can take the J → 0 limit of the full vertex function Γ βα,δκ iωm given by Eq. (15). Alternatively, we can associate the lowestorder, J 2 contribution to the partition function Z = dΩ m Tr f exp(−S eff (m)) with the J 2 contribution to the partition function Z ′ = Tr f exp(−S ′ eff ) . Both methods yield the same result: which does indeed satisfy the required crossing symmetries.
But S ′ eff produces an inequivalent infinite-dimensional theory than produced by S eff (m), as seen in the different expansions of Z and Z ′ to order J 4 : Therefore, the DE action cannot be replaced by a purely electronic action containing only two-body interactions and it is not possible to construct a diagrammatic expansion in powers of Γ (0) .
Nevertheless, we can construct a functional Φ({G(iν n )}) satisfying the condition Σ(iν n ) αα = δΦ/δG(iν n ) αα . Starting from Eq.(3) and Dyson's equation for the self-energy, Inverting this Jacobian yields the general result where the right-hand side is the full vertex function of Eq. (15). The functional Φ must exist because the vertex function is well-defined and the curl of the self-energy vanishes: So despite the absence of a diagrammatic perturbation theory, the DMFT of the DE model is Φ-derivable and hence, thermodynamically consistent.

IV. TRANSITION TEMPERATURES
Since the electronic spin-susceptibility is given by χ el (q) = l,n χ s q,iωm=0 (iν l , iν n ), the condition that χ el (q) diverges is equivalent to the condition that Det(χ s q,iωm=0 ) −1 = 0. To obtain the FM or AF transition temperatures, we require the q = 0 or Q susceptibilities with ω m = 0. The AF wavevector Q is defined so that the Fermi surface is nested at half filling or ǫ k+Q = −ǫ k . Although the Bethe lattice sketched in Fig.1 lacks translational symmetry and nonzero wavevectors q are ill-defined, the AF wavevector Q may be associated with the bipartite nature of the Bethe lattice. To be safe, the Néel temperature will be evaluated in two different ways, only one of which relies on the existence of an AF wavevector.
For either FM or AF ordering, the electron filling p can be related to the chemical potential µ through the relation where G 0 (iν n ) −1 = z n + R n and the complex function R n satisfies the implicit relation Since a n = (z n + R n ) 2 − J 2 , R n falls off like −W 2 /(16z n ) for large n. Notice that p = 1 when µ = 0. Since all of our results are particle-hole symmetric about p = 1, we shall restrict consideration to the range 0 ≤ p ≤ 1.
As discussed in Section II, the BS equations decouple into charge and spin relations.
Since the charge vertex function Γ c iωm (iν l , iν n ) = 6B l m δ ln is diagonal in Matsubara space, the condition Det(χ c q,iωm=0 ) −1 = 0 has no solution for q = 0 or Q. In other words, there is no charge-density wave instability within the DE model.

A. Ferromagnetic ordering
When q = 0, the bare, zero-frequency lattice susceptibility is given by With the help of Eq.(26), we find that the full spin-susceptibility at q = 0 satisfies where b l = (z l + R l ) 2 − J 2 /3. Consequently, the condition that Det(χ s q=0,iωm=0 ) −1 = 0 can be written as an implicit condition for the Curie temperature: which must be supplemented by Eq.(25) for the electron filling.
If J/W > 1/4, the interacting DOS splits into two bands [16]. In the lower band centered where R n is now given explicitly by R n = z 2 n − W 2 /8 − z n /2. So in this limit, T C is proportional to W and is independent of J. Due to the strong Hund's coupling, T C vanishes at half filling (p = 1) or when the lower band is completely full because electrons are unable to hop to neighboring sites.
In the weak-coupling regime of T ≪ J ≪ W , the summation in Eq.(29) may be replaced by an integral over frequency ν, as shown in Appendix A. Carefully treating the branch cut along the real axis between ±W/2, we find that which is proportional to J 2 /W as expected in the RKKY limit. For small filling (µ close to −W/2) and a fixed J, T C initially increases with p and then begins to decrease, vanishing at it is the same for p electrons (2 − p holes) and 2 − p electrons (p holes) per site. These results are in qualitative agreement with Nolting et al. [28,29], who evaluated the Curie temperature of the s − f model with quantum local spins using both equations-of-motion [28] and modified RKKY [29] techniques. Nolting et al. obtained FM solutions for small J in a concentration range significantly narrower than found here, probably because their approach retains the non-local fluctuations produced by the RKKY interaction.

From the susceptibility
Evaluating the Néel temperature from the susceptibility requires the bare, zero-frequency lattice susceptibility for q = Q: After some algebra, we find that the full spin-susceptibility at q = Q satisfies Consequently, T N is found from the condition that Det(χ s which bares some similarity to Eq.(29) for T C .
In the strong-coupling limit, Eq.(34) reduces to with a plus sign multiplying 3W 2 /32 compared to the minus sign in Eq. (30). It is straightforward to show that T N ≤ 0 for all p in this limit. Hence, the AF phase instabilities disappear as J/W → ∞.
In the weak-coupling regime, we may once again replace the Matsubara sum by an integral, with the result For p < 0.73, T N is negative so the system becomes FM for 0 < p ≤ 0.39.
Close to half filling, T N given by Eq.(36) becomes positive and the DE model supports an AF phase. We find that T N > 0 for 0.73 ≤ p ≤ 1, in which range T C given by Eq. (31) is negative. The divergence of T N in Eq.(36) as p → 1 or µ → 0 signals a breakdown in the weak-coupling expansion and the appearance of a gap in the interacting DOS, as discussed later.

From coupled Green's functions
The previous sub-section used the nesting condition for the AF wavevector Q whereas, strictly speaking, nonzero wavevectors are not defined on the Bethe lattice. Hence, it is worthwhile to re-evaluate T N using a Green's function approach that skirts the question of whether an AF wavevector exists. This Green's function approach will also come in handy when considering the SRO state later in Section IV.
Dividing the Bethe lattice into A and B sub-lattices, we obtain coupled relations between the bare and full local Green's functions: where η =↑ or ↓ on the A or B sites andη is opposite to η. If the bare inverse Green's function is parameterized as then Eq.(37) implies that R n and Q (η) n are solved from the condition To linear order in the sublattice magnetization, R n and Q (η) n satisfy the implicit relations where m η ≡ M (η) z = ±Mz, the sign depending on whether η =↑ (A site) or η =↓ (B site).
The linear relations for Q (η) n may be readily solved, with the result After integrating exp(−S eff (m)) over the Grassman variables, we find that the probability for the local moment on sublattice η to point along m is The last relation defines the effective interaction: Earlier studies of the Heisenberg model on the Bethe lattice [30,31] proposed a spiral magnetic state with a wavevector that characterizes the twist of the spin between nearest neighbors. But it is quite easy to show that a Bethe lattice can not be partitioned into more than two sublattices. For example, if we try to partition a Bethe lattice with z c = 4 into three sublattices, then sites C and A (red and purple dots in Fig.3(a)) would be next-nearest neighbors but the two sites C (purple dots) (presumably on the same magnetic sublattice) are actually next-nearest neighbors of each other. So unless the central site is singled out, a Bethe lattice does not support a magnetic state with more than two sublattices.
More recently, Gruber et al. [32] argued that this restriction may be removed as z c → ∞ because the paths with the largest weight always move outward from the central site. This analysis uses the fact that the surface of the Bethe lattice contains a non-vanishing fraction of its sites [33]. Since the period-three state constructed in Ref. [32] can only be formed around a central site, it is not isotropic. Such states will not be considered further in this paper. For the special case of a colinear SRO state, θ i = 0 or π on every site and q is the fraction of neighbors of any site that have the opposite spin. An example of a colinear SRO state with z c = 4 and q = 3/4 is sketched in Fig.3(b).
Also working on the Bethe lattice, Chattopadhyay et al. [5] assumed that all of the angles θ i were identical and characterized the resulting state by its "incommensurate spin correlations." We avoid this description because nonzero wavevectors different are ill-defined on the Bethe lattice and the X(q) formalism of Müller-Hartmann [8] cannot be adapted from the hypercubic to the Bethe lattice. In our view, the SRO phase is more accurately characterized by the magnetic correlations around any site.
In the absence of the SRO states, a large portion of the magnetic phase diagram is NM.
This poses a problem because Nernst's theorem requires that the entropy vanishes at T = 0.
Generally, the entropy of a SRO state is difficult to evaluate. But in the z c → ∞ limit, the entropy of a colinear SRO phase is which vanishes only in the FM or AF phases. Of course, the entropy of a non-colinear SRO phase must be greater than S col . Therefore, the formal dilemma posed by Nernst's theorem is not resolved by the presence of SRO states. As discussed in the conclusion, entropic ground states frequently arise in infinite dimensions. In finite dimensions, the SRO state may evolve into a state with incommensurate SRO.
Since a SRO phase with correlation parameter q between 0 and 1 has no long-range order, the transition temperature T SRO (p, q) cannot be obtained from the condition that the magnetic susceptibility diverges. But we can easily solve for T SRO (p, q) by generalizing the Green's function technique used in sub-section IV.B.2. With the SRO state defined above, the self-consistency relation between the bare and full local Green's functions is derived in where G (η) and G (η) are spin-reversed Green's functions. The quantization axis η is defined by the spin on the central site 0. The above relation was previously obtained in Ref. [5] under the more restrictive condition that θ i is constant. Once again parameterizing the bare inverse Green's function by Eq.(38), we find that R n is independent of the magnetization M = m z on site 0 (assumed spin up) and is given by Eq.
The effective interaction J ef f is again given by the first equality in Eq.(44) so that which reduces to both the FM and AF results, Eq. (29) and (34), when q = 0 and q = 1. It also follows that T SRO (p, q) = 0 in the NM state with q = 1/2.
It is straightforward to show that T SRO (p, q) is negative for 1/2 ≤ q ≤ 1 and is positive but always smaller than T C (p) for 0 ≤ q < 1/2. So as expected, the SRO states are unstable in the strong-coupling limit.
Results for T SRO (p, q) in the weak-coupling limit are given in Appendix A. Maximizing T SRO (p, q) with respect to q, we obtain T (max) SRO (p) plotted in Fig.4. Remarkably, a SRO phase with AF correlations can always be found with a higher transition temperature than the AF state. As q → 1, the AF and SRO transition temperatures meet at p = 1 or half filling. Even in the range of fillings between about 0.26 and 0.39, where T C is nonzero, a SRO phase with FM correlations or 0 < q < 1/2 has the higher critical temperature! Also notice that the SRO transition temperature is nonzero in the range of fillings between 0.39 and 0.73, where neither the FM nor AF states was stable for small J/W . The point at which q = 1/2 and T (max) SRO (p) = 0 lies slightly below p = 0.5. Recall that q changes discontinuously at T (max) SRO (p) from 1/2 in the NM phase above to a value greater than or smaller than 1/2 in the SRO phase below.

V. MAGNETIC INSTABILITIES OF THE DOUBLE-EXCHANGE MODEL
We shall first present results for the magnetic transition temperatures in the absence of the SRO states. These results are then revised to include the SRO states. Bare in mind that just because a FM, AF, or SRO state has the highest transition temperature for a given J/W and p does not imply that it remains stable down to zero temperature or even that it is reached at all: a pure state may be bypassed by the formation of a PS mixture. The issue of PS can only be resolved by calculating the free energy as a function of filling. We shall perform such a study for T = 0 in Section VI.
In Fig.5, the Curie and Néel temperatures are plotted versus filling for a variety of couplings. When p ≥ 0.39, a critical value of J is required for T C to be nonzero. So the weak-coupling regime is restricted to electron concentrations smaller than 0.39. As J/W increases, the AF instabilities are restricted to a narrower range of concentrations near p = 1.
For large J/W at half filling, the nearest-neighbor hopping can be treated perturbatively and superexchange leads to a Néel temperature proportional to W 2 /J. So the sliver of AF phase stabilized around half filling rapidly narrows with increasing J/W . By contrast the FM phase dominates as J/W increases. In the limit J/W → ∞, the Néel temperature vanishes even at p = 1.
For a fixed filling, T C and T N are plotted versus J/W in Fig.6. As expected, the weakcoupling curves track the numeric results for small J/W but deviate for larger J/W . Whereas  Fig.7(a). The NM region shrinks as J/W increases and is absent for J/W > 0.5. Since the interacting DOS splits for J/W > 1/4 [16], the NM region survives even after the formation of sub-bands. Whereas the AF region persists at half filling for all J/W , the sliver of AF phase rapidly narrows with increasing coupling. The FM/NM phase boundary in Fig.7(a) agrees quite well with that found by Auslender et al. [14]. Now including the SRO states, we plot the correlation parameter q along with T  phase to a nonzero value 0.5 < q < 1 in the SRO phase. For J/W > 0.5, one of the ordered phases always has a higher transition temperatures than the SRO states and q jumps from 0 to 1 at the FM/AF boundary. The deviation between the numeric and weak-coupling results in Fig.8(c) as p → 1 is quite expected: in the limit p → 1, the weak-coupling result for the SRO transition temperature diverges, thereby invalidating the condition T ≪ J under which it was derived.
The magnetic instabilities of the DE model including SRO states are mapped in Fig.7(b).
Notice that the region formerly labeled NM in Fig.7(a) is now occupied by SRO states. The SRO phase also occupies the formerly AF region at very small J/W . As anticipated above, the NM curve (where T (max) SRO (p) = 0 and q = 1/2) intersects the FM phase boundary at J/W ≈ 0.33. This curve separates a SRO region with AF correlations (1 > q > 1/2) on the left from a SRO region with FM correlations (1/2 > q > 0) on the right.

VI. MAGNETIC PHASE DIAGRAM OF THE DOUBLE-EXCHANGE MODEL
The presence of a continuous range of SRO states with correlation parameter q obviously complicates the magnetic phase diagram of the DE model and the issue of PS. Because the weak-coupling behavior of the DE model has been the subject of some debate [3,4,5,14], we shall treat this limit analytically. We then provide numerical results for the interacting DOS that explain the stability of the SRO phase. Finally, we present the T = 0 phase diagram of the DE model, both with and without the SRO states.

A. Weak-coupling limit
In their numerical work at T = 0, Chattopadhyay et al. [5] obtained a very complex phase diagram, with a range of SRO states above J/W ≈ 1/16 and PS between AF and FM phases for 0.1 < p < 1 below this value. For small J/W , a pure FM phase was found to be stable for 0 < p ≤ 0.1. We now show that PS diappears in the exactly solvable limit of zero temperature and weak coupling.
The potential energy can generally be written as V /N = −2JM s 0z , where the expectation value of the electronic spin on site 0 for any temperature is For any J, the potential energy may be written as Of course, the NM potential energy is obtained by setting q = 1/2.
Second-order results for V (p, q) /N are provided in Appendix A. For small J/W , s 0z is of order J/W so the electronic spins are not frozen at T = 0. Bearing in mind that T SRO (p, q = 0) = T C (p) and T SRO (p, q = 1) = T N (p), the difference between the potential energies of a FM, AF, or SRO phase and a NM is given by where both sides are evaluated to order J 2 /W . So the potential energy is lowered when the transition temperature is positive.
A general expression for the kinetic energy of the SRO phase is proven in Appendix C.
At T = 0, Eq.(C3) becomes In contrast to alternative expressions for the kinetic energy of the FM and AF phases [10], Eq.(54) does not assume any particular wavevector for the ground state. Independent of q, the zeroth-order kinetic energy is given by E 0 (p)/N = −2W/(3π)(1 − δ 2 ) 3/2 , where δ = 2µ 0 /W and µ 0 is the chemical potential evaluated to zeroth-order in J/W . For any J, the SRO kinetic energy is given by and the NM kinetic is recovered by setting q = 1/2. It is important to realize that the second-order kinetic energy contains two contributions: one from the dependence of the integrand on J/W for a fixed µ; the other from the dependence of the chemical potential µ(p) on J/W for a fixed p.
Incorporating the change in chemical potential, the result for the second-order, qdependent correction to K(p, q) /N is given in Appendix A. To second order in J/W , the difference between the kinetic energies of a FM, AF, or SRO phase and a NM is ∆ K(p, q) /N = 3T SRO (p, q)/2. Hence, kinetic energy is lost (the electrons become more confined) when the transition temperature is positive.
Consequently, the net change in energy E(p, q) = K(p, q)+V (p, q) in the weak-coupling limit is given by ∆E(p, q)/N = −3T SRO (p, q)/2, which includes the FM (q = 0) and AF (q = 1) phases as special cases. The same relation would be obtained for a Heisenberg (and the analogy with the Heisenberg model) only holds in the weak-coupling limit.
In the limit of small J/W , the ground state energy is minimized by the same correlation parameter q that maximizes the transition temperature! This result is not surprising: for small J/W , the transition temperature is also small so the correlation parameter that appears at T SRO (p, q) also minimizes the energy at T = 0. As a consistency check, the results in Appendix A may be used to show that the derivative of the energy is related to the chemical potential by dE(p, q)/dp = Nµ, where both sides are expanded to second order in J/W .

B. Interacting Density-of-States
To better interpret the weak-coupling results, we construct the interacting DOS per spin As J/W → 0, N(ω, q) reduces to the bare DOS N 0 (ω) for any q. The AF DOS can be solved analytically for all J: which vanishes for |ω| < J and |ω| > W 2 /4 + J 2 . Hence, N(ω, q = 1) contains a squareroot singularity on either side of a gap with magnitude 2J. As J/W → ∞, the width of each side-band narrows like W 2 /8J. It is also possible to explicitly evaluate the DOS at ω = 0 for any q: N(ω = 0, q) = (8/πW 2 (1 − q))Re J 2 c − J 2 , which vanishes for J > J c ≡ W (1 − q)/2. For a NM, the critical value of J required to split the DOS is J c = W/4, as found earlier [16]. For an AF at T = 0, J c = 0 since a gap forms for any nonzero coupling constant. For a SRO phase, J c > 0.
For energies between the kinks, both up-and down-spin states appear; on either side of the kinks, the bands are fully spin-polarized. These kinks disappear in a SRO state with q > 0 due to the absence of long-range magnetic order. The width of the DOS shrinks as q increases from 0 to 1. These results agree with the second-order change to the chemical potential given above. In fact, the divergence of the AF result for µ 2 as p → 1 is caused by the appearance of a gap in the AF DOS for any nonzero J.
For any temperature, coupling constant, and correlation parameter, the total energy of the DE model may be written as an integral over the interacting DOS: where f (ω) = 1/(exp(β(ω − µ)) + 1) is the Fermi function and µ = µ(p). At T = 0, this relation may be expressed in the compact form which agrees with Eqs. (53) and (55).
From these results, it is not hard to understand why a SRO phase can have a lower energy than an AF. Due to the narrowing of the AF DOS, the AF energy may be higher than that of a SRO state with J < J c (q) and no energy gap. If the chemical potential of the AF phase lies sufficiently far from the energy gap, then the broader DOS of a SRO phase with J < J c will favor that state over an AF. As p → 1, this condition is impossible to satisfy for arbitrarily small J and the AF becomes stable.
For a given p < 1, the expansions of the kinetic and potential energies in powers of (J/W ) 2 are valid when J < J c (q) so that the DOS of a SRO state with correlation parameter q is not gapped. Conversely, for a fixed J, the expansions fail sufficiently close to p = 1 and q = 1 that a gap develops in N(ω, q).

C. Ground-state phase diagram and phase separation
When only FM, AF, and NM states are considered, PS occurs due to the discontinuity in the chemical potential at the FM/NM and NM/AF phase boundaries. Since these discontinuities become weaker as J/W decreases, PS disappears as J/W → 0.
Once the SRO states are introduced, the physical energy E (min) (p) is obtained by minimizing E(p, q) with respect to q for a fixed p. Although the chemical potential is a continuous function of filling, PS still occurs for non-zero J due to the formation of an energy gap in the AF state, so that (1/N)dE (min) /dp| p=1 − = −J. A necessary condition for PS between fillings p 1 < 1 and p 2 = 1 is that the second derivative of the energy E (min) (p) must change sign for some filling p ⋆ between p 1 and p 2 . But if E (min) (p) is expanded in powers of (J/W ) 2 , then the condition that d 2 E (min) (p)/dp 2 | p ⋆ = 0 may be written where δ = 2µ 0 /W and µ 0 is the chemical potential evaluated to zeroth-order in J/W . Since the value q ⋆ of q at p ⋆ is less than 1, the expansion in powers of (J/W ) 2 is valid for J < J c (q ⋆ ), as discussed previously. When J/W → 0, the right-hand side of Eq.(61) vanishes so the condition for PS cannot be met in the limit of vanishing J/W except as δ → 0 or p ⋆ → 1.
Near p = 1, δ ≈ (1 − p)π/4 so Eq.(61) also implies that the difference p 2 − p 1 grows linearly with J/W . These results contradict Chattopadhyay et al. [5], who seem to use a Maxwell construction involving the difference E (min) (p)−E 0 (p) rather than the total minimum energy To show this more explicitly, we have used E (min) (p) to find the PS region through Maxwell constructions between fillings p 1 < 1 and p 2 = 1. One such Maxwell construction is pictured in Fig.10, where a phase mixture along the dashed line provides a lower energy than the pure phase on the solid curve. For J/W = 0.5, the SRO states are bypassed and PS occurs between a FM with p 1 = 0.647 and an AF with p 2 = 1.
The T = 0 phase diagram of Fig.11(a) assumes that only the NM, FM, and AF phases are allowed. Without the SRO state, the ground-state phase diagram is fairly complicated. Surprisingly, the T = 0 phase diagram of Fig.11(b) becomes much simpler once the SRO states are included. For J/W < 0.33, a SRO state is stable over a range of fillings that diminishes with increasing J. Near half-filling, PS is found between a SRO state with 0 < q < 1 and an AF with p = 1. In agreement with the above discussion, the PS region grows linearly with J/W . For J/W > 0.33, the SRO phase is skipped and PS occurs between a FM with q = 0 and an AF with p = 1. The special value J/W ≈ 0.33 agrees with the coupling where the NM line intersects the FM boundary in the instability map of Fig.7(b). It also corresponds to the coupling where the AF instability covers the widest range of fillings.
The PS region in Figs.11(a) and (b) diminishes with increasing J/W . In the strongcoupling regime, our results are in qualitative agreement with other authors [3,11]. But for J/W = 2, we obtain PS over a somewhat larger range of fillings than obtained in Ref. [13].
Aside from the behavior at very small J/W , the phase diagram of Fig.11 For small J/W , the energy of the entropic SRO state is lowered below the energies of the FM and AF phases by the RKKY interaction [7], which generates competing FM and AF interactions [14]. As the dimension is lowered, the SRO state may evolve into the state with incommensurate correlations (IC) obtained in one-and two-dimensional Monte-Carlo simulations by Yunoki et al. [3]. Replacing "SRO" by "IC", the phase diagram of Fig.11(b) bears a striking resemblance to the phase diagrams of Ref. [3]. In agreement with our results, Yunoki et al. found that the short-range correlations in the IC phase change from AF to FM as p decreases. The phase diagram of Fig.11(b) also looks quite similar to the threedimensional phase diagram obtained by Yin [6], who compared the energies of the AF and FM phases with the energy of a two-dimensional spiral phase carrying momenta along the (111) direction. Hence, the infinite-dimensional phase diagram of Fig.11(b) is closely related to the results of finite-dimensional calculations.
On a translationally-invariant hypercubic lattice [10] in infinite dimensions, the SRO phase may map onto an incommensurate spin-density wave (SDW) with a definite wavevector. In that case, the transition temperature of the SDW could be obtained from either the coupled Green's functions or the magnetic susceptibility. However, there are no indications for SDW ordering in the limited studies of the DE model on an infinite-dimensional hypercubic lattice [4] and we consider such a possibility to be remote. Much more likely, the SRO phase would map onto a disordered state with short-range magnetic order and broad maxima at wavevectors q associated with the nesting of the Fermi surface, as found in Monte-Carlo simulations of one-and two-dimensional hypercubic lattices [3]. Working on a Bethe lattice has allowed us to clarify the behavior of the DE model at weak coupling. We have demonstrated both analytically and numerically that PS disappears as J/W → 0. Whereas the FM phase is stable for p below about 0.26, a SRO state is stable above. As p → 1, the SRO phase evolves into the long-range ordered AF phase.
The transition temperature of the SRO phase exceeds that of the FM and AF phases for 0.26 < p < 1.
Whether the SRO phase is a new kind of spin glass or a spin liquid can only be resolved by future studies of the static and dynamic susceptibilities. An intriguing possibility is that the local moments are frozen within a spin-glass state but that the electronic spins are subject to quantum fluctuations within a spin-liquid state. Two of the most important unresolved questions about spin glasses are whether there exists a true thermodynamic spinglass transition [34] and whether a model without quenched disorder can support a spin glass [35]. We have answered those questions for the DE model on an infinite-dimensional Bethe lattice. Above T SRO (p, q), the NM state is characterized by the absence of SRO around any site. Below T SRO (p, q), short-range magnetic order develops with a correlation length ξ(q) = −a/ log |2q − 1| and the entropy, while still extensive, is reduced. Although not marked by a divergent susceptibility, the transition at T SRO (p, q) is characterized by the development of short-range magnetic order and a reduction in the entropy. Future work may establish connections between the SRO phase and the spin-glass solution to the random Ising model on a Bethe lattice [36], where the Edwards-Anderson order parameter can be explicitly constructed.
The strength of DMFT applied to DE-like models with a Kondo coupling 2Jm i · s i between the local-moment spin S i = Sm i and the electron spin s i is that the Fermion degrees of freedom in the local action may be integrated exactly, leaving a function only of the local moment direction m i . Consequently, formally exact results are possible for a range of models where the local moments are treated classically. This stands in stark contrast to the case for the Hubbard model [10], where the action cannot be analytically integrated and quantum Monte Carlo must be used to solve the local problem. We should also mention that the classical treatment of the local spins is more questionable in the manganites, where In the regime 0 ≤ q < 1/2, the integral for T SRO (p, q) with b < 0 gives This result can be extended to the regime 1/2 < q ≤ 1 by using the identity These expressions yield Eq.(31) for T C and Eq. (36) for T N as q → 0 and q → 1, respectively.
Likewise, it is straightforward to evaluate the potential energy to order (J/W ) 2 , with the result: In the limit q → 1/2, the NM result is V (p, q = 1/2) /N = −(32J 2 /3πW )(1−δ 2 ) 3/2 . Hence, the difference between the q-dependent and NM potential energies in the weak-coupling limit is given by ∆ V (p, q) /N = −3T SRO (p, q). It follows that the change in total energy is given by ∆E(p, q)/N = −3T SRO (p, q)/2, which includes the FM (q = 0) and AF (q = 1) phases as special cases.
where Z f = Π i dΩ m i Tr f exp(−S) and the sum over i is restricted to nearest neighbors of site 0. The action S = ∆S + S 0 + S (0) is separated into terms that involve site 0 and the action S (0) of the cavity [10] with site 0 removed. The kinetic part of the action containing site 0 is Performing perturbation theory in the term involving the (0i) bond and using t = W/4 √ z c , we find where the local averages at sites 0 and i are performed using S eff of Eq. (2). Removing the single (0i) bond from ∆S does not affect those local averages in the z c → ∞ limit.