CERN-TH-2023-086 Ferromagnetic phase transitions in SU ( N ) Alexios P. Polychronakos1,2 and Konstantinos Sfetsos3,4 arXiv:2306.01051v3 [hep-th] 24 Oct 2023 1 Physics Department, the City College of the New York 160 Convent Avenue, New York, NY 10031, USA apolychronakos@ccny.cuny.edu 2 The Graduate School and University Center, City University of New York 365 Fifth Avenue, New York, NY 10016, USA apolychronakos@gc.cuny.edu 3 Department of Nuclear and Particle Physics, Faculty of Physics, National and Kapodistrian University of Athens, Athens 15784, Greece ksfetsos@phys.uoa.gr 4 Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland October 26, 2023 Abstract We study the thermodynamics of a non-abelian ferromagnet consisting of "atoms" each carrying a fundamental representation of SU ( N ), coupled with long-range twobody quadratic interactions. We uncover a rich structure of phase transitions from non-magnetized, global SU ( N )-invariant states to magnetized ones breaking global invariance to SU ( N − 1) × U (1). Phases can coexist, one being stable and the other metastable, and the transition between states involves latent heat exchange, unlike in usual SU (2) ferromagnets. Coupling the system to an external non-abelian magnetic field further enriches the phase structure, leading to additional phases. The system manifests hysteresis phenomena both in the magnetic field, as in usual ferromagnets, and in the temperature, in analogy to supercooled water. Potential applications are in fundamental situations or as a phenomenological model. Contents 1 Introduction 2 2 A system of interacting SU ( N ) "atoms" 3 2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Decomposition of n fundamentals of SU ( N ) into irreps . . . . . . . . . . 7 2.3 The thermodynamic limit of the model . . . . . . . . . . . . . . . . . . . . 9 3 4 Phase transitions with vanishing magnetic fields 3.1 Stability analysis and critical temperatures . . . . . . . . . . . . . . . . . 15 3.2 Phase transitions and metastability . . . . . . . . . . . . . . . . . . . . . . 18 Turning on magnetic fields 4.1 4.2 5 11 25 Small fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.1 The singlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1.2 The symmetric representation . . . . . . . . . . . . . . . . . . . . . 26 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2.1 One-row and conjugate one-row states . . . . . . . . . . . . . . . 30 4.2.2 Two-row states and their (N − 1)-row conjugates . . . . . . . . . 35 Conclusions 38 A Analysis of one-row states with a magnetic field 41 A.1 Resolving metastability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 B Analysis of two-row states in a magnetic field 45 B.1 The case of low temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 49 1 1 Introduction Magnetic materials are of considerable physical and technological interest, and their properties have long been the subject of theoretical research. Ferromagnets, the first type of magnetism ever observed, hold a special place among them, as they manifest nontrivial properties and symmetry breaking. All known ferromagnets consist of interacting localized magnetic dipoles and break rotational invariance below the Curie temperature. Each (quantum) dipole provides a representation of the group of rotations, that is, of SU (2). Although this is a nonabelian group, it is of a particularly simple type: it has a unique Cartan generator, and SU (2) dipoles can interact with external abelian magnetic fields that couple to their Cartan generator. Nevertheless, the exact quantitative properties of physical ferromagnets remain an active topic of research [1]. Independently, nonabelian unitary groups SU ( N ) of higher rank play a crucial role in particle physics and, indirectly through matrix models, in string theory and gravity. Ungauged and gauged SU (3) groups are the most common, representing "flavor" and "color" degrees of freedom, respectively. A collection of nucleons, or the constituents of the quark-gluon plasma, are physical systems of components carrying representations of SU (3). This raises the obvious question of the properties that large collections of such SU (3) or, more generally, SU ( N ) entities would have if they interacted with each other as well as with external nonabelian magnetic fields. In this work we investigate the properties of such systems in the ferromagnetic regime, that is, in the regime where the mutual interaction of its components would tend to "align" their SU ( N ) charges, in a way that we will make precise. The results on the decomposition of the direct product of an arbitrary number of representations of SU ( N ) into irreducible components that we derived in a recent publication [2] will be a crucial tool in our calculations. We will also study the effects of an external nonabelian magnetic field coupled to the system. As we shall demonstrate, the properties of nonabelian (N > 2) ferromagnets are qualitatively different from these of ordinary (N = 2) ferromagnets. They display a rich phase structure involving various critical temperatures, hysteresis both in the temperature and in the magnetic field, coexistence of phases, and latent heat transfer during phase transitions. In the sequel we will present the basic SU ( N ) model, consisting of distinguishable 2 quantum components in the fundamental representation, and will review the relevant group theory results of [2]. We will proceed to study the thermodynamic phases of the model in the absence, and subsequently in the presence, of external magnetic fields, and will derive its symmetry breaking patterns, critical temperatures, and magnetization. Stability issues will be crucial and will determine the pattern of SU ( N ) breaking in the various phases. We will further study the nontrivial situation of a magnetic field inducing an enhanced breaking of SU ( N ), and will conclude with some speculations about the phenomenological relevance of the model. 2 A system of interacting SU ( N ) "atoms" Magnetic systems with SU ( N ) symmetry have been considered in the context of ultracold atoms [3–7] or of interacting atoms on lattice cites [8–14] and in the presence of SU ( N ) magnetic fields [15–17]. In this section we lay out the basic structure of any model of interacting SU ( N ) atoms, specify its ferromagnetic regime, and review the group theory results necessary for its analytic treatment. 2.1 The model To motivate the basic model, consider a set of n atoms (or molecules) on a lattice, interacting with two-body interactions. Each atom is in one of N degenerate or quasidegenerate states |s⟩, s = 1, 2, . . . , N. The generic two-body interaction between atoms 1 and 2 with states |s1 ⟩ and |s2 ⟩ would be N H12 = ∑′ ′ s1 ,s1 ,s2 ,s2 =1 hs1 s2 ;s′ s′ |s1 ⟩ ⟨s1′ | ⊗ |s2 ⟩ ⟨s2′ | , 1 2 hs1 s2 ;s′ s′ = h∗s′ s′ ;s1 s2 . 1 2 1 2 (2.1) Define ja , a = 0, 1, . . . , N 2 − 1, the generators of U ( N ) in the fundamental N-dimensional representation, with j0 the identity operator (the U (1) part). Using the fact that the ja form a complete basis for the operators acting on an N-dimensional space, the above interaction can also be written as H12 = N 2 −1 ∑ a,b=0 h ab j1,a j2,b , 3 h ab = h∗ab , (2.2) where j1,a = ja ⊗ 1 , j2,a = 1 ⊗ ja , (2.3) are the fundamental U ( N ) operators acting on the states of atoms 1 and 2. We now make the physical assumption that the above interaction is invariant under a change of basis in the states |s⟩, that is, under a common unitary transformation of the states |s1 ⟩ of atom 1 and |s2 ⟩ of atom 2. This implies two equivalent facts: first, the interaction will necessarily be, up to trivial additive and multiplicative constants (proportional to the identity), the operator exchanging the states of the atoms, ′ + C12 H12 = C12 N ∑ ′ s,s =1 |s⟩ ⟨s′ | ⊗ |s′ ⟩ ⟨s| , (2.4) ′ , C being real constants. Second, the interaction will necessarily be of the with C12 12 form H12 = ′ c12 + c12 N 2 −1 ∑ a =1 j1,a j2,a . (2.5) ′ (due to the U (1) part), we obtain a unique two-body Omitting the trivial constant c12 interaction depending on a single coupling constant c12 . Note that the group U ( N ) emerges from the requirement of invariance under general changes of basis of the N states, and leads to interactions linear in the operators in each atom. Using, instead, an N-dimensional representation of a smaller group would require the inclusion of higher polynomial terms in j1,a and j2,a . The interaction Hamiltonian of the full system will be of the form n H= ∑ crs N 2 −1 ∑ jr,a js,a , (2.6) a =1 r,s=1 where crs = csr is the strength of the interaction between atoms r and s (and crr = 0). This Hamiltonian involves an isotropic quadratic coupling between the fundamental generators of the n commuting SU ( N ) groups of the atoms. Reasonable physical assumptions restrict the form of the couplings crs . We assume that the interaction is homogeneous, that is, crs is translationally invariant under the shift of both r and s by the same lattice translation (away from the boundary of the lattice). In terms of the lattice positions of the atoms ⃗r, c⃗r,⃗s = c⃗r−⃗s , c0 = 0 . 4 (2.7) Therefore, each atom couples to a fixed weighted average of the SU ( N ) generators of its neighboring atoms. We will also assume that interactions are reasonably longrange, that is, each atom couples to several of its neigboring atoms. This technical assumption justifies the mean field condition that, in the thermodynamic limit, the weighted average of the neighboring atoms is well approximated by their average over the full lattice. That is ∑ c⃗s j⃗r+⃗s,a ≃ ⃗s  ∑ c⃗s ⃗s 1 n c js,a = − Ja , ∑ n s =1 n (2.8) where we defined the total SU ( N ) generator n Ja = (2.9) ∑ js,a s =1 and the effective mean coupling1 c = − ∑ c⃗s . (2.10) ⃗s The minus sign is introduced such that ferromagnetic interactions, driving atom states to align, correspond to positive c. Altogether, the full effective interaction assumes the form c H=− n N 2 −1 ∑ a =1 Ja2 n − ∑ s =1 2 js,a  , (2.11) where the second term in the parenthesis eliminates the terms r = s. The first part of H is proportional to the quadratic Casimir of the total SU ( N ) group C (2) = ∑ a Ja2 . The second part is proportional to the sum of the quadratic Casimirs of each individual atom. Since all js,a are in the fundamental representation, their quadratic Casimir is a (common) constant, independent of their state. So the second term contributes a trivial constant and can be discarded. In addition to the atoms’ mutual interaction, we can couple the states of the atoms to a global external field, contributing an additional term n HB = N ∑ ∑ ′ r =1 s,s =1 1 The Bss′ |s⟩r ⟨s′ |r . (2.12) validity of the mean field approximation is strongest in three dimensions, since every atom has a higher number of near neighbors and the statistical fluctuations of their averaged coupling are weaker, but is expected to hold also in lower dimensions. 5 Parametrizing this one-atom operator in terms of the complete set of operators jr,a and omitting the trivial constant terms corresponding to jr,0 , it becomes HB = n N 2 −1 ∑ ∑ Ba jr,a = r =1 a =1 N 2 −1 ∑ Ba Ja . (2.13) a =1 We see that Ba acts as a global nonabelian magnetic field on the SU ( N ) "spins" jr,a . Finally, making use of the fact that the interaction Hamiltonian is invariant under global SU ( N ) transformations, we may choose a basis of states in which the sum ∑ a Ba Ja is rotated to the Cartan subspace spanned by the commuting generators Hi , i = 1, 2, . . . , N − 1. The full Hamiltonian of the model then emerges as c (2) N −1 H = − C − ∑ Bi Hi . n i =1 (2.14) We will assume that c is positive, so that the model is of the ferromagnetic type. For N = 2 the above model reduces to the ferromagnetic interaction of spin-half components. For higher N, the model has the same number of states per atom as a spin-S SU (2) model with 2S + 1 = N. The dynamics of the two models, however, are distinct: the SU (2) model is invariant only under global SU (2) transformations, which cannot mix the N states of the atoms in an arbitrary way, unlike the SU ( N ) case. The enhanced symmetry of the SU ( N ) model leads, as we shall see, to a richer structure and to qualitatively different thermodynamic properties. Finding the eigenstates of the above model and determining its thermodynamics involves decomposing the full Hilbert space of states into irreducible representations (irreps) of the total SU ( N ), evaluating the quadratic Casimir C (2) and the magnetic sum ∑i Bi Hi in each irrep, and calculating the partition function as a sum over these irreps. This requires determining the decomposition of the direct product of a large, arbitrary number n of SU ( N ) fundamentals into irreps and the multiplicity of each irrep in the decomposition, as well as calculating the Casimir and the magnetic sum for large irreps of SU ( N ). This task was performed in a recent publication [2], and the relevant results will be reviewed in the next subsection. 6 2.2 Decomposition of n fundamentals of SU ( N ) into irreps We summarize the group theory results pertaining to the decomposition of the direct product of n fundamentals of SU ( N ) into irreps, as presented in [2] (results on the simpler case of SU (2) were previously derived in [18, 19] and were applied in [18] to regular ferromagnetism). The setting and results become most tractable and intuitive in the momentum representation, in which irreps of SU ( N ) are labeled by a set of distinct integers k i , i = 1, 2, . . . , N ordered as k1 > k2 > · · · > k N . (2.15) Each irrep corresponds to a given set {k i }, for which we will use the symbol k. The corresponding Young Tableaux (YT) of the irrep may be described by its lengths ℓi , i.e., number of boxes per row, for i = 1, 2, . . . , N − 1. The correspondence with k i is ℓi = k i − k N + i − N , ℓ1 ⩾ ℓ2 ⩾ · · · ⩾ ℓ N −1 ⩾ 0 . (2.16) Note that the k i representation is redundant, since a shift of all k i by a common constant k i → k i + c leaves ℓi invariant and leads to the same irrep of SU ( N ) (the shift changes the U (1) charge of the irrep, which equals the sum of the k i ). This freedom can be used to simplify relevant formulae. In our situation, where irreps will arise from the direct product of n fundamentals, it will be convenient to choose the convention N ∑ ki = n + i =1 N ( N − 1) . 2 (2.17) For the singlet representation (n = 0) all ℓi are zero, which in the above convention corresponds to k i = N − i, i = 1, 2, . . . , N. The fundamental (n = 1) has a single box, and corresponds to k1 = N and the rest of the k i as above. In SU ( N ) there are N − 1 Casimir operators which, for the irrep k, can be expressed in terms of the k i ’s. For our purposes we need the quadratic Casimir, which is given in terms of the k i by C (2) 1 N 2 N ( N 2 − 1) 1 (k) = ∑ ki − . [n + N ( N − 1)/2]2 − 2 i =1 2N 24 7 (2.18) Note that, using (2.16), (2.17), C (2) takes the more familiar form C (2) (ℓ) = 1 2 N −1 ∑ i =1 ℓi (ℓi + N + 1 − 2i ) − N −1 1 2N ∑ ℓi i =1 !2 . (2.19) For the singlet C (2) = 0, while for the fundamental C (2) = ( N − N −1 )/2. For our purposes we also need the trace of the exponential of the magnetic term in a giver irrep k,  N  Trk exp β ∑ Bj Hj , (2.20) j =1 which will appear in the calculation of the partition function of our model. This was calculated in [2]. To express it, define the Slater determinant z1k1 z1k2 · · · ψk (z) = (z1 · · · z N 1 ) − N ∑i k i z2k1 z2k2 · · · .. .. .. . . . zkN1 zkN2 · · · k z1N −1 z1k N k z2N −1 z2k N .. .. . . , z = { zi ∈ C } , (2.21) k z NN −1 zkNN which is antisymmetric under the interchange of any two zi ’s and of any two k i ’s. Also define the Vandermonde determinant ∆ ( z ) = ( z1 · · · z N ) − N −1 2 z1N −1 z1N −2 · · · z2N −1 z2N −2 · · · .. .. .. . . . N −1 −2 zN zN ··· N z1 z2 .. . 1 1 .. . , (2.22) zN 1 which is the Slater determinant (2.21) for the singlet irrep. Then   N ψ (z) , Trk exp β ∑ Bj Hj = k ∆(z) j =1 z j = e βBj . (2.23) The prefactors involving the product z1 · · · z N in (2.21) and (2.22) eliminate the U (1) part of the irrep, which couples to the trace of the magnetic field ∑i Bi . If B is traceless, then the U (1) charge decouples and we can ignore these prefactors. As a check of (2.23), we can take the limit zi → 1 and verify that the ratio of determinants goes to Trk 1 = dim(k) = N ki − k j ∏ j−i = j > i =1 8 N ℓi − ℓ j + j − i , j−i j > i =1 ∏ (2.24) which is the standard expression for the dimension of the irrep. The last nontrivial element needed for our purposes is the multiplicity dn,k of each irrep k arising in the decomposition of n fundamental representations. This was also calculated in [2], and the result is N dn,k = δk1 +···+k N ,n+ N ( N −1)/2 Dn,k = n! ∏rN=1 kr ! ∏ j > i =1 (Si − S j ) Dn,k , (2.25) , where Si is a shift operator acting on the right by replacing k i by k i − 1. Note that Dn,k and dn,k are manifestly symmetric and antisymmetric, respectively, under exchange of the k i . In [2] the action of the operator ∏ N j>i =1 ( Si − S j ) on Dn,k was performed and an explicit combinatorial formula for dn,k was obtained, but it will not be needed for our purposes. 2.3 The thermodynamic limit of the model We now have all the ingredients to study the statistical mechanics of our SU ( N ) ferromaget. The partition function is Z= ∑ e states − βH ∑ dn;k e = βc (2) n C (k)  N  Trk exp β ∑ Bj Hj , ⟨k⟩ j =1 (2.26) where β is the inverse of the temperature T and ⟨k⟩ denotes distinct ordered integers k1 > k2 > · · · > k N satisfying the constraint (2.17). Using the results (2.18), (2.23) and (2.25), and removing the trivial (k i -independent) terms in the Casimir (2.18), the partition function becomes 1 δk +···+k ,n+ N ( N −1) Z= N 1 N! ∑ 2 k = ∑ δk k N ( N −1) 1 +···+ k N ,n + 2 = ∑ δk1 +···+k N ,n k  N ∏ j > i =1 1 ∆(z)  1 n! ∆(z) ∏rN=1 kr ! ( Si − S j ) n! ∏rN=1 kr ! N ∏ j > i =1 ( Si − S j ) N ∏ j > i =1 9  n! ∏rN=1 kr ! ψk (z) βc ∑s k2s e 2n ∆(z)  βc 2 e 2n ∑s ks + βBs ks
βc ∑ k2 + βB k 1 s s Si−1 − S− e 2n s s . j (2.27) In the first line above we made the sum unrestricted, since the summand is symmetric under permutation of the k i and vanishes for k i = k j , and introduced the constraint explicitly. The second line follows since dn,k (the expression in the parenthesis) is antisymmetric in the k i , and thus it picks the fully antisymmetric part of z1k1 · · · zkNN , reproducing ψk (z). The third line is obtained by shifting summation variables. In doing so, the term N ( N − 1)/2 in the Kronecker δ is absorbed. The above holds for arbitrary n. We now take the thermodynamic limit n ≫ 1. The typical k i is of order n, and thus the exponent in the expression is of order n, and any prefactor polynomial in n is irrelevant, as is the factor ∆(z). Similarly, the 1 action of ∏(Si−1 − S− j ) produces a subleading factor that can be ignored. (One way j >i to see this is to note that in the large n limit the shift operators act as derivatives 1 (Si−1 − S− ≃ ∂ki − ∂k j ) and bring down subleading terms). Further, we apply to kr ! j the Stirling approximation. Altogether we obtain Z= 0 ∑ δk1 +···+k N ,n e−β F(k)+O(n ) , (2.28) k where the free energy of the system is, up to a trivial overall constant,  N  c 2 k i − Bi k i . F(k) = − Tn ln n + ∑ Tk i ln k i − 2n i =1 (2.29) In the large-n limit, quantities k i and F are extensive variables of order n. We will now transition to intensive variables, that is, quantities per atom. To this end, we define rescaled variables xi as k i = nxi , i = 1, 2, . . . , N . (2.30) satisfying the constraint N ∑ xi = 1 . (2.31) i =1 In terms of the xi , the non-extensive term − Tn ln n in the free energy cancels and F becomes properly extensive,  N  NT0 2 xi − Bi xi = nF (x) , F(x) = n ∑ Txi ln xi − 2 i =1 10 (2.32) where we have defined c = NT0 , (2.33) introducing a temperature scale T0 . From now on we will work with the intensive quantities xi (magnetization per atom) and F (free energy per atom) and will omit the qualifier "per atom". In the large-n limit the sum in (2.28) can be obtained by a saddle-point approximation, as the exponent is of order n, by minimizing the free energy F (x) while respecting the constraint ∑iN=1 xi = 1. This can be done with a Lagrange multiplier. Adding the term λ(1 − ∑iN=1 xi ) to (2.32) and varying with respect to xi we obtain ∂i Fλ = T ln xi − NT0 xi − Bi − λ = 0 , i = 1, 2, . . . , N . (2.34) The Lagrange multiplier λ can be eliminated by subtracting one of the relations, say for i = N, from the rest (which is equivalent to solving the constraint and expressing one of the xi , say x N , in terms of the others). We obtain T ln xi − NT0 ( xi − x N ) − ( Bi − BN ) = 0 , xN i = 1, 2, . . . , N − 1 , (2.35) where x N is determined from the constraint (2.31). Also, from (2.34) we obtain the second derivatives ∂i ∂ j Fλ = T xi  − NT0 δij , i, j = 1, 2, . . . , N , (2.36) subject to (2.31). The above Hessian will be needed later in order to investigate the stability of the solutions. The simpler form of the equations (2.34) involving λ will also be useful in determining the nature of solutions and in the stability analysis. 3 Phase transitions with vanishing magnetic fields We now put Bi = 0 (setting all of the Bi equal is equivalent, as this would be a U (1) field and would contribute a trivial constant to the energy) and examine the phase structure of the system. We can collectively write equations (2.34) for Bi = 0 as T ln x − NT0 x = λ , 11 (3.1) dropping the index i in xi to emphasize that it is the same equation for all xi ’s, unlike the case with generic non-vanishing magnetic fields. The value of λ is fixed by the summation condition (2.31). We note that (3.1) always admits the trivial solution xi = 1/N (for an appropriate λ), corresponding to the singlet irrep and an unbroken SU ( N ) phase. Generically, however, the above equation has two solutions (see fig. 1). So, each xi can have one of two fixed values, x− or x+ > x− . This means that the dominant irreps are those with M equal rows, where M is the number of xi having the large stvalue x+ in the solution, and gives SU ( M) × SU ( N − M) × U (1) as the possible a priori spontaneous breaking of SU ( N ), the subgroup that preserves a matrix with M equal and N − M different and equal diagonal entries. We will see, however, that stability of the configuration requires that at most one xi value in the full solution be x = x+ ; that is, either M = 0, corresponding to the singlet, or M = 1, corresponding to a one-row YT, a completely symmetric representation. T ln x - N T0 x λ x- x0 x+ x T . The intersection NT0 with some constant value of the Lagrange multiplier λ occurs at x = x± , with x− < x0 < x+ . Figure 1: Plot of the LHS of (3.1), with its maximum occurring at x0 = Then equations (2.35) with Bi = 0 become T ln xi = NT0 ( xi − x N ) , xN i = 1, 2, . . . , N − 1 , (3.2) where x N = 1 − x1 − · · · − x N −1 is determined by the constraint in (2.31). As argued before from (3.1), each xi can have one of two possible values. Hence, take M of the xi to be equal, and the remaining N − M also equal and different. The integer M can take any value from 0 to N, but the values M = 0 and M = N correspond to the singlet configuration xi = 1/N that trivially satisfies (3.1). For M ̸= 0, N, taking 12 into account the summation to one condition, we set 1+x , N 1 − ax xi = , N i = 1, 2, . . . , M , xi = i = M + 1, . . . , N , M a= . N−M (3.3) Note that, according to (2.15), the xi ’s cannot strictly be equal for finite n. However, in the large n limit, differences of O(1/n) are ignored. Further, the choice of the specific xi that we set to each value is irrelevant, since the saddle point equations for Bi = 0 are invariant under permutations of the xi . With the choice (3.3), N − M of the equations are identically satisfied and the remaining M amount to T ln 1+x − (1 + a) T0 x = 0 . 1 − ax (3.4) This transcendental equation is invariant under the transformation x → − ax , a → 1/a (equivalently M → N − M) . (3.5) Thus, without loss of generality we can choose M ⩽ [ N/2] , or 0<a⩽1, x ∈ (−1, 1/a) , (3.6) where [ · ] denotes the integer part. Solutions with x > 0 specify an irrep with M equal rows of length, using (2.16) and (3.3), ℓi = x n + O(1) , N−M i = 1, . . . , M , ℓi = O(1) , i = M + 1, . . . , N − 1 . (3.7) Instead, an x < 0 specifies an irrep with N − M equal rows (corresponding to the conjugate representation), with length given by (3.7) but with x replaced by x → − x. The reason is that in this case the xi ’s in the second line of (3.3) are larger than those of the first line and therefore the roles of M and N − M are reversed. There are generically either one or three solutions to (3.4) depending on T (see fig. 2). If the temperature is higher than a critical temperature Tc , then the only solution is that with x = 0, that is, the singlet. If T < Tc , then there are two additional solutions. For T = Tc these two solutions coalesce at x = xc , implying that the x-derivate of (3.4) 13 is zero as well at xc . These conditions are summarized as 1 + xc Tc ln = (1 + a ) x c , T0 1 − axc (3.8) Tc (1 + xc )(1 − axc ) = ≡t. T0 Solving the first condition for t = Tc /T0 and substituting into the second we obtain a transcendental equation that determines xc (1 + a ) x c 1 + xc = ln (1 + xc )(1 − axc ) 1 − axc (3.9) and from that and the first of (3.8) the critical temperature Tc . Alternatively, solving the second equation in (3.8) for xc and substituting into the first one yields the transcendental equation for t = T/T0 ln  p  q  1 + a 1 1 + a + (1 + a)2 − 4at p − 1 − a + (1 + a)2 − 4at = 0 , a 1 + a − (1 + a)2 − 4at 2at (3.10) which assumes that a ⩽ 1. For a ⩾ 1 we simply replace a → 1/a. For SU (2), a = 1 is the only possibility, and for a = 1 the solution of (3.10) is t = 1. Hence, the critical temperature is just T0 , i.e. the one in (2.33). For generic a, it can be checked that the left hand side of (3.10) is monotonically increasing in t and the (1 + a )2 equation has a unique solution in the range 1 < t < for all a ⩽ 1. For a near 4a 1 we have 1 (3.11) t = 1 + (1 − a )2 + . . . , 6 whereas for a near 0
1 = a − ln a + ln(− ln a) + . . . t (3.12) and therefore t → ∞. Hence, the solution to (3.10) varies monotonically between these two limiting cases. Note that for the case M = 1, a = 1/( N − 1) (which is particularly relevant, as we shall see) M=1: Tc ≃ T0 N ≫ T0 , ln N as N ≫1. (3.13) In conclusion, Tc > T0 for any group SU ( N ) with N ⩾ 3. For T > Tc the only solution is the trivial one, x = 0, while for T < Tc there are two additional solutions x1 , x2 : two positive ones 0 < x1 < xc < x2 for T0 < T < Tc , and one positive and one negative 14 one x1 < 0 < x2 for T < T0 . A few generic cases are depicted in Fig. 1 where the left hand side of (3.4) is plotted. 1.0 0.5 0.5 -1.0 -0.5 0.5 1.0 x -1 1 2 3 4 5 6 x -0.5 -1.0 -0.5 -1.5 Figure 2: Plots of the left hand side of (3.4) for N = 7. Left: M = 3, T = 0.7. Right: M = 1, T = 1.6 (blue) and for T = Tc ≃ 1.72 (corresponding to x = xc ≃ 3.88) (orange). Temperature is in units of T0 . 3.1 Stability analysis and critical temperatures The solutions of (3.4) may be local extrema or saddle points of the action. To determine their stability we examine the second variation of the free energy. From (2.36), δ2 F is δ2 F = 1 N −1 2 Ci δxi , 2 i∑ =1 Ci−1 = T − NT0 , xi N ∑ δxi = 0 . (3.14) i =1 Note that this remains the same even in the presence of magnetic fields, so the stability argument below is fully general. If all coefficients Ci are positive at a stationary point, then clearly the solution is stable. If two or more Ci ’s are negative, on the other hand, it is unstable. Indeed, we can take, e.g., δxi1 + δxi2 = 0 for two of the negative coefficients Ci1 and Ci2 , and set the rest of the δxi to zero in order to satisfy the constraint. Then, an obvious instability arises. However, if only one Ci is negative and the rest of them positive, then the solution could still be stable due to the presence of the constraint. A standard analysis 15 shows that the condition for stability in this case is2 N ∑ Ci < 0 . (3.15) i =1 For the case of vanishing magnetic field, xi satisfy the common equation (3.1) T ln x − NT0 x = λ . (3.16) The function on the left hand side is plotted in fig. 1, repeated here as fig. 3, and has a maximum at x0 = T NT0 for all i. For x = x0 , the coefficient Ci−1 determining the perturbative stability at this value vanishes, while Ci−1 > 0 for x < x0 and Ci−1 < 0 for > x0 . Therefore, the left branch of the curve represents a priori stable points and the right branch unstable ones. T ln x - N T0 x 15 λ 10 5 5 10 15 20 x T . The intersection with some NT0 constant value of the Lagrange multiplier λ occurs at the values xi = x± , with x− < x0 < x+ . The left (green) part of the curve corresponds to stable solutions, whereas the right one (red) to unstable ones. Figure 3: Plot of (3.16), with its maximum occurring at x0 = From the above general discussion, we understand that fully stable configurations correspond either to choosing all xi ’s on the stable branch (all Ci > 0), or N − 1 of them on the stable branch and one on the unstable branch (only one Ci < 0). This last con- figuration can still be stable if it satisfies the condition (3.15). So the only potentially stable configurations for zero magnetic field are the singlet (M = 0), corresponding to a paramagnetic phase, and the fully symmetric single-row irrep (M = 1), corre2A nice way to derive this is to view δxi as covariant coordinates on a space with metric gij = Ci δij of Minkowski signature (−, +, . . . , +). Then δ2 F and the constraint become δ2 F = gij δxi δx j , ui δxi = 0 with ui = 1 . For the space spanned by the restricted δxi to be spacelike (with positive definite metric), ui must be timelike, and ui ui = gij ui u j < 0 implies (3.15). 16 sponding to a specific ferromagnetic phase, with order parameter the variable x in (3.3) determining the length of the single row. For the remainder of the zero magnetic field discussion we will focus on the nontrivial solution M = 1 and set for the constant a = M/( N − M) the corresponding value a= 1 . N−1 (3.17) For this choice, and with xi as in (3.3), the coefficients Ci become C1−1 = N A( x ) , Ci−1 = N A(− ax ) , where we defined A( x ) = i = 2, 3, . . . , N , T − T0 . 1+x (3.18) (3.19) The stability of the no magnetization solution x = 0 corresponding to the singlet is easy to find. In that case A(0) = T − T0 , so that for T < T0 the solution x = 0 is a local maximum and unstable, and for T > T0 it is a local minimum. For M = 1, it is clear from fig. 3 that we must have C1 < 0, corresponding to the single value x1 = (1 + x )/N on the unstable branch, and the remaining Ci > 0, so A( x ) < 0 and A(− ax ) > 0. Also, 1 + x > 1 − ax, so that x > 0. Condition (3.15) must also be satisfied, A( x ) + aA(− ax ) > 0 =⇒ (1 + x )(1 − ax ) < T , T0 (3.20) where we used N − 1 = a−1 . Using (3.4) to eliminate T this rewrites as (1 + a ) x 1+x > ln . (1 + x )(1 − ax ) 1 − ax (3.21) Note that this has the same form as (3.9) determining the critical xc . It can be seen that it is satisfied for x > xc and violated for x < xc . As analyzed in the previous section, the existence of solutions with M = 1 requires T < Tc . For such temperatures, equation (3.4) has two solutions, one larger and one smaller than xc . Only the solution with x > xc satisfies (3.21). Therefore, for temperatures T < Tc the solution with x > xc is stable and a local minimum and the one with x < xc unstable. Referring again to Fig. 2, the solution to the left of x = xc for T < Tc (blue) on the right plot is unstable, whereas the one to the right is stable. 17 We conclude by noting that at low temperatures T ≪ T0 , we expect the stable configuration of the system to be the fully polarized one with x ≃ xmax = N − 1, corresponding to the maximal one-row symmetric representation with ℓ1 ≃ n boxes. Indeed, the solution of equation (3.4) in that case can by well approximated by  x ≃ ( N − 1) 1 − Ne − NT0 /T  , T ≪ NT0 , (3.22) manifesting a nonperturbative behavior in T around T = 0. 3.2 Phase transitions and metastability We saw in the previous section that for T0 < T < Tc both the completely symmetric representation and the singlet are locally stable. The globally stable configuration is determined by comparing the free energies of the two solutions. The free energy per atom was found in (2.32). For zero magnetic fields, it takes the form N F (x, T ) = ∑ i =1  Txi ln xi − NT0 2  x 2 i (3.23) and for the single-row zero magnetic field solution the free energy per atom becomes Fsym ( x, T ) =
T a(1 + x ) ln(1 + x ) + (1 − ax ) ln(1 − ax ) 1+a a − T0 x2 − T ln N . 2 (3.24) Variation of this expression with respect to x leads to (3.4).3 For the singlet we have Fsinglet ( T ) = − T ln N . (3.25) For a specific temperature T1 and magnetization x1 the singlet and symmetric configurations will have the same free energy. Equating the two expressions and using (3.4) 3 Positivity of the second derivative gives the stability condition (3.20), but without any restriction on a, that is, the number M of equal rows in the YT. The reason is that this expression only captures stability under variations of the length of the YT. Taking also into account perturbations into configurations with additional rows recovers the general stability condition requiring M = 0 or M = 1. 18 we obtain T1 ln 1 + x1 = (1 + a) T0 x1 , 1 − ax1 (3.26) T0 T1 ln(1 + x1 ) = x1 (2 − ax1 ) , 2 where we used (3.4) to simplify the expression for Fsym . This system is solved by4 T1 = N ( N − 2) T0 , 2 ( N − 1) ln( N − 1) x1 = N − 2 . (3.27) In general T0 < T1 < Tc . (3.28) except for N = 2 where we have T0 = T1 = Tc , implying that for the SU (2) case there is a single critical temperature. For large N T1 ≃ T0 N ≫ T0 , 2 ln N as N ≫1. (3.29) Recalling the limiting behavior of Tc in (3.13), we note that for large N, T1 ≃ Tc /2. For T0 < T < T1 , Fsinglet > Fsym , while for T1 < T < Tc , Fsinglet < Fsym . The situation is summarized in table 1. irrep T < T0 Singlet unstable Symmetric stable T0 < T < T1 metastable stable T1 < T < Tc stable metastable Tc < T stable not a solution Table 1: Phases in various temperature ranges for N ⩾ 3 and their stability characterization. Hence, at high enough temperature the only solution is the singlet with no magnetization. At temperature Tc a magnetized state corresponding to the symmetric irrep (one-row) also emerges, and is metastable until some lower temperature T1 . Between these two temperatures the singlet is the stable solution. Below T1 and down to T0 the roles of stable and unstable solutions are interchanged. Below T0 the only stable solution is the one-row symmetric representation. Hence, we have a spontaneous symmetry breaking as SU ( N ) → SU ( N − 1) × U (1) . (3.30) Note that the free energy changes discontinuously at T = T0 and at T = Tc . The plot 4 In the next section we will present a method for solving it even in the presence of a magnetic field. 19 of the free energy is at Fig. 4. The low temperature plateaux in Fig. 4 for the one-row configuration (blue curve) is explained by the fact that due to (3.22) we have Fsym T − NT0 /T  N−1  T0 1 + 2 e , ≃− 2 T0 T ≪ NT0 . (3.31) F 0.5 -2.0 1.0 T 1.5 -2.5 Fsinglet Fsym -3.0 -3.5 Figure 4: Plot of the free energy F for N = 7, for the one-row solution (blue) up to Tc ≃ 1.72 and for the singlet one (red) from T = 1, in units of T0 . These cover the temperature range in which we have stability or metastability. The crossover behavior is at T = T1 ≃ 1.63, in agreement with table 1. To better understand the situation, we may use the thermodynamic relations between the free energy F, the internal energy U and the entropy S F = U − TS ,   F , U = −T ∂T T 2 S = −∂T F , (3.32) for Fsym given by (3.24) to obtain a Usym ( x ) = − T0 x2 , 2
1 a(1 + x ) ln(1 + x ) + (1 − ax ) ln(1 − ax ) . Ssym ( x ) = ln N − 1+a (3.33) These are readily recognized as the coupling energy and the logarithm of the number of states per atom for representations close to the dominant symmetric one (we emphasize that x = x ( T ) via (3.4)). For the singlet (x = 0) we simply have Usinglet = 0 , Ssinglet = ln N , (3.34) that is, the maximal energy and maximal entropy. Even though the free energy is discontinuous at T = Tc , we may still think of the transitions at T0 and Tc as a first order phase transitions, in the sense that the discontinuity of U implies that latent heat has to be transferred for the phase transition to occur. In 20 0.5 1.0 1.5 2.0 T Usinglet Ssinglet -0.5 1.5 -1.0 Ssym 1.0 Usym -1.5 -2.0 0.5 -2.5 -3.0 0.5 1.0 1.5 T Figure 5: Plots of U (left) and S (right) for N = 7. For the one-row solution (blue) up to Tc ≃ 1.72 and for the singlet one (red) from T = 1, in units of T0 . The sharp rise for T → Tc− is according to (3.38). detail we have T0 x2 , 2( N − 1) c T → Tc− : Usym = − T → Tc+ : Usinglet = 0 , Ssym = ln N + xc − ln(1 + xc ) , 1 + xc (3.35) Ssinglet = ln N , where we simplified Ssym by using relations (3.8). When we transition from below to above Tc , we must give energy to the system, which also increases its entropy (recall that, in the absence of volume effects, dU = TdS). The behavior of the energy and the entropy with temperature is depicted in fig. 5. At the intermediate temperature T = T1 , given in (3.27), there is the possibility of a first order phase transition from a metastable to a stable phase. Near that temperature Fsym = − T ln N + Fsinglet = − T ln N .
N−2 ln( N − 1) T − T1 + . . . , N (3.36) This transition is typical in statistical physics where latent heat transfer is involved. Hence we have that T → T1− : T→ T1+ : Usym = − 1 ( N − 2)2 T0 , 2 N−1 Usinglet = 0 , Ssym = ln N − N−2 ln( N − 1) , N (3.37) Ssinglet = ln N , The transition from metastable to stable configurations near the temperature T1 will not occur spontaneously under ideal conditions, leading to hysteresis. Only when the system is perturbed, or given an exponentially large time such that large thermal fluctuations occur, will it transition from a metastable to a stable configuration. This is reminiscent of the hysteresis in temperature exhibited in certain materials and in 21 supercooled water [20]. The discontinuity in U implies that (latent) heat has to be transferred for the phase transition to occur. When we transit T1 from above the system releases energy, which also lowers its entropy, the opposite happening when it transit above T1 (The term "pseudo phase transition" has been used in the literature for this kind of process). To understand the nature of the phase transitions that the system can undergo, imagine that we start at a temperature T > Tc with the (paramagnetic) singlet state and adiabatically lower the temperature by bringing the system into contact with a cooling agent (reservoir). If the system is not perturbed, it will stay at the paramagnetic state until T = T0 , where this state becomes unstable, the (ferromagnetic) symmetric representation solution takes over, and the system undergoes a phase transition releasing latent heat into the reservoir. Similarly, starting at a temperature below T0 with the (ferromagnetic) symmetric irrep state and raising adiabatically the temperature, the system will remain in this state if it is not perturbed and will transition to the singlet at T = Tc , absorbing latent heat from the reservoir and undergoing a phase transition. Clearly the system presents hysteresis, and no unique Curie temperature exists, since in the range T0 < T < Tc the two phases coexist. The situation is somewhat different if the system is isolated (decoupled from the reservoir) while it is in a metastable state (that is, a ferromagnetic state at T1 < T < Tc or a paramagnetic state at T0 < T < T1 ). A transition to the stable state would involve exchange of latent heat, which can only be provided by, or absorbed into, the stable phase. However, the heat capacity of the paramagnetic phase is zero (see figure 5), so no such exchange can take place and the system is "trapped" in the unstable phase. This is unlike, say, supercooled water, where perturbations nucleate the formation of a stable solid state, releasing latent heat into the unstable liquid phase and raising the temperature until the liquid phase is eliminated or the temperature reaches the point at which the two phases become equally stable. This feature of our system is somewhat unrealistic, since we have ignored all other degrees of freedom except SU ( N ) spins. A realistic system would also have vibrational degrees of freedom of the atoms, which would serve as a reservoir absorbing or receiving latent heat and thus enabling transitions from metastable to stable states. The parameter x is a measure of the spontaneous magnetization and constitutes the order parameter for the phase transition. Recalling (3.7), x is a measure of the 22 length of the single row YT corresponding to the solution. From (3.4) and (3.8) we can deduce that x ( T ) near Tc behaves as5 x − xc ≃ s ( N − 1) x c xc + 1 − N/2 s Tc − T . T0 (3.38) Hence, the deviation from xc near the critical temperature follows Bloch’s law for spontaneous magnetization of materials with an exponent of 1/2 and an N-dependent coefficient. The behavior of x ( T ) is depicted in fig. 6. 4.0 x(T) 3.5 3.0 2.5 0.6 0.8 1.0 1.2 1.4 T Figure 6: Typical plot of x ( T ) (here for N = 5) for the symmetric irrep for 0 < T < Tc ≃ 1.40. For T = 0 it reaches the maximum value x (0) = N − 1 and for T → Tc− it goes sharply to xc according to (3.38). The SU (2) case: The above results are valid for a ̸= 1 that is for N ⩾ 3. When a = 1, as in the SU (2) case, T0 = T1 = Tc and the stable-metastable range in table 1 does not exist. The transition between the singlet and the symmetric representations occurs at the unique Curie temperature T = T0 and at xc = 0. In this case we have Symm. irrep ( T < T0 ): Singlet ( T > T0 ): x≃ √ 3 x=0, s
2 T0 − T 3T0 T0 − T , , F ≃ − T ln 2 − T0 4T0 F = − T ln 2 , (3.39) which shows that at T = T0 there is a second order phase transition. Note that the naïve limit a → 1 of the result (3.38) for a < 1 would not recover the above behavior. The reason is that the range of validity of (3.38) is T0 < T ≲ Tc , and it shrinks to zero as Tc → T0 . The two first-order phase transition points at T0 and Tc fuse into a single second-order transition in the limit a → 1. Fig. 7 depicts the free 5 Positivity of xc + 1 − N/2 follows from the fact that between the unstable and the stable solutions the left hand side of (3.4) as a function of x reaches a minimum (see the right plot in fig. 2. 23 energy against the temperature for the SU (2) case. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T -0.2 -0.4 Fsym -0.6 -0.8 Fsinglet -1.0 Figure 7: Plot of the free F for N = 2. The continuity of the expression and its first derivative between the one-row solution (blue) up to T = 1 and the singlet one (red) from T = 1, in units of T0 , is manifest. The behavior of x ( T ) near T = T0 follows Bloch’s law with an exponent 1/2, as in the general SU ( N ) case, but with a different coefficient. The internal energy and entropy are continuous but their first derivatives at T = T0 are not, as depicted in fig. 8. U 0.7 T 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ssinglet 0.6 Usinglet -0.1 Ssym 0.5 Usym 0.4 -0.2 0.3 -0.3 0.2 -0.4 0.1 -0.5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T Figure 8: Plots of U (left) and S (right) for N = 2 for T in units of T0 , for the magnetized (blue) state up to T0 and the singlet (red) state above T0 . For comparison between SU (2) and higher groups, we present in table 2 the values of the critical temperature in units of T0 , the critical magnetization as a fraction of the maximal magnetization xmax = N − 1 and the coefficient of x − xc in Bloch’s law in (3.38) again as a fraction of the maximal magnetization, i.e. K = p ( N − 1) xc /( xc + 1 − N/2)/xmax for the first few values of N, as well as for N ≫ 1. N Tc /T0 xc /xmax K 2 1 0 1.7320 3 1.0926 0.3772 1.2177 4 1.2427 0.5071 0.9862 5 1.4027 0.5749 0.8480 N≫1 N/ ln N √1 2/N Table 2: Behavior near critical point for N = 2, 3, 4, 5, as well as for N ≫ 1 (the corresponding entries are leading). 24 4 Turning on magnetic fields We now turn to incorporating nonzero magnetic fields in the system. We recall the equilibrium equation (2.34) T ln xi − NT0 xi = Bi + λ , i = 1, 2, . . . , N , (4.1) where the xi sum to 1 due to the constraint in (2.31). 4.1 Small fields For small magnetic fields the solution of the coupled equations (4.1) can be found as a perturbation of the solution for vanishing fields. In fact, we can do better than that. We may consider the response of the system in the presence of generic magnetic fields Bi under small perturbations δBi . The state will change as xi → xi + δxi , i = 1, 2, . . . , N , (4.2) where δxi is a perturbation to the solution of (4.1). To linear order we have that  Since  T − NT0 δxi = Ci−1 δxi = δBi + δλ . xi (4.3) N ∑ δxi = 0 we obtain the change of λ as i =1 δλ = − ∑iN=1 Cj δBj ∑iN=1 Cj . (4.4) Then, combining with (4.3) we get δxi = Ci δBi − ∑N j=1 C j δB j ∑N j =1 C j ! , (4.5) from which the magnetizability matrix mij obtains as mij = Ci Cj ∂xi = Ci δij − N , ∂Bj ∑k=1 Ck 25 i, j = 1, 2, . . . , N . (4.6) Note that mij is symmetric and satisfies N ∑ mij = 0 as a consequence of the fact that i =1 the U (1) part decouples. We may infer the signs of mij for general Ci ’s from the stability condition on the configuration. Recall that, for stability, either all Ci are positive (including infinity), or only one of them is negative, say C1 , and the rest positive (x1 is the largest among the xi ’s). In the latter case, stability further requires that (3.15) be satisfied. Using these we can show that C1 > 0 : m11 , mii > 0 , m1i < 0 , mij < 0 , C1 < 0 : m11 , mii > 0 , m1i < 0 , mij > 0 , C1−1 = 0 : m11 , mii > 0 , m1i < 0 , mij = 0 , (4.7) where i, j = 2, 3, . . . , N and i ̸= j. According to (3.14), the last case above happens for T = NT0 x1 , which is possible only for T < NT0 , and corresponds to one of the solutions of (4.1) reaching the top of the function in the LHS of (3.1), depicted in fig. 1. 4.1.1 The singlet For the unmagnetized singlet configuration, stable for temperatures T > T0 , xi = 1/N and Ci−1 = N ( T − T0 ). The magnetizability is singlet :   1 1 mij = δ − . N ( T − T0 ) ij N (4.8) As expected, it diverges as ( T − T0 )−1 at the critical temperature T0 where the con- figuration destabilizes, and the signs of its components are in agreement with (4.7). This determines the linear response of the system to small magnetic fields, of typical magnitude B such that B ≪ T − T0 . 4.1.2 The symmetric representation For the spontaneously magnetized configuration corresponding to the symmetric representation M = 1, stable for T < Tc , the xi are as in (3.3), with x the solution of (3.4) that corresponds to a stable configuration. The components of the magnetization ma- 26 trix take the form m11 = N−1 , N∆( x ) 1 , i ̸= 1, N∆( x )   A( x ) 1 δ − , mij = N A(− ax ) ij ∆( x ) m1i = − (4.9) i, j ̸= 1 , where a = 1/( N − 1) and A( x ) = T/(1 + x ) − T0 as before, and   1 T ∆( x ) = A(− ax ) + A( x ) = N − T0 . a (1 + x )(1 − ax ) (4.10) As T → Tc− , (3.8) shows that the magnetization diverges. To compute its asymptotic behavior at T ≃ Tc we use (3.38) to obtain Tc − (1 + x )(1 − ax ) = T0 q 2xc (2axc + a − 1) which implies T0 ∆=N Tc q 2xc (2axc + a − 1) s s Tc − T +... , T0 Tc − T +... . T0 (4.11) (4.12) Hence the entries of the magnetizability matrix as T → Tc− become m11 ≃ N−1 Q √ , 2 N Tc − T m1i ≃ − mij ≃ 1 Q √ , 2 N Tc − T (4.13) 1 Q √ , N 2 ( N − 1) Tc − T where Q= p i, j = 2, . . . , N , Tc /T0 2xc (2axc + a − 1) T0 . (4.14) So the magnetizability diverges as ( Tc − T )−1/2 , and the signs of its components are in agreement with (4.7). We have depicted these in fig. 9 below. 27 0.03 m11 2.0 m22 m11 0.02 1.5 m23 0.01 1.0 0.6 0.8 1.0 1.2 1.4 T 1.6 0.5 -0.01 m12 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T Figure 9: Left: Plots of the four different types of magnetization matrix entries for the symmetric irrep. up to T = Tc for N ⩾ 3. Right: Same plot for SU (2). The SU (2) case: This is special since Tc = T0 . For the singlet solution (T > T0 ), (4.8) remains valid for N = 2, giving m11 = m22 = −m12 = 1 1 , 4 T − T0 T → T0+ . (4.15) For the magnetized solution m11 = m22 = −m12 = 1 1 − x2 , 4 T − T0 (1 − x2 ) T < T0 . (4.16) Using (3.39) we obtain the magnetizability m11 = m22 = −m12 ≃ 1 1 , 8 T0 − T T → T0− . (4.17) We note that mij diverges as | T − T0 |−1 on both sides of the critical temperature, un√ like for N ⩾ 2. The reason is that the coefficient of Tc − T0 in the expansion (4.11) vanishes for N = 2 and thus the next order in the expansion, O( T0 − T ), becomes the leading one. We have depicted these in fig. 9. 4.2 Finite fields We now consider the state of the system, given by (4.1), for general non-vanishing magnetic fields. The general qualitative picture can be obtained by the same considerations as in the case of Bi = 0. For fixed λ, each xi satisfies (4.1) for a different effective Lagrange multiplier λi = λ + Bi and can take two possible values xi± , the two solutions of (4.1) for fixed i. The stability considerations of section 3.1, which remain valid for arbitrary magnetic fields, determine that at most one of these solutions can lie on the unstable 28 branch of the curve x+ . So, fully stable configurations correspond either to choosing all xi− on the stable branch, or N − 1 of them on the stable branch and one on the unstable branch. The stability condition ∑ Ci < 0 in (3.15) must also be satisfied in i the latter case. To gain intuition on the behavior of the system, we focus on the case when only one of the magnetic fields, say B1 , is different from the rest. We can absorb the equal terms Bi , i > 1 in the Lagrange multiplier and call B = B1 − Bi . Then the set of equations (4.1) becomes just two distinct equations: one for x1 , with RHS λ + B, and one for the remaining N − 1 xi ’s, with RHS λ, as depicted in fig. 10. 18 T ln x - N T0 x λ+B 16 14 λ 12 10 8 6 5 10 15 20 x T . We consider a magnetic NT0 field B along one direction, say for i = 1. The intersection with the lines λ and λ + B occur at two values of xi on each side of x0 for each line. Figure 10: Plot of (3.16), with its maximum occurring at x0 = Referring to fig. 10, denote the four solutions of (4.1) by x± (at the intersections of the B (at the intersections with λ + B). For the choice of B > 0 in curve with λ), and by x± B < x < x B < x . For B < 0 the ordering the figure we have the ordering x− < x− + 0 + B < x < x < x < x B . Since we have a magnetic field only in would change to x− − + 0 + B while each x (i > 1) can take values x . direction 1, x1 can take values x± ± i B . Thus, we For a stable configuration we may choose at most one value at x+ or x+ have the following possible cases: B , corresponding to a one-row YT (if x B > x ) • a) N − 1 values x− and one value x− − − B < x ). This is the deformation of the singlet for B = 0. or its conjugate (if x− − B , corresponding, again, to a one-row YT. • b) N − 1 values x− and one value x+ This is the deformation of the one-row solution for B = 0. B , and one value x , corresponding to a two-row • c) N − 2 values x− , one value x− + YT. This is a deformation of the one-row YT solution at B = 0 by increasing its 29 depth and breaking the SU ( N ) symmetry further. Following a similar analysis, these would correspond either to irreps with two rows, or to irreps with N − 1 rows, N − 2 of which have equal lengths. We will encounter such cases below. Although we will not examine in detail the more general configuration of a magnetic field with M equal components and the remaining N − M equal and distinct, its qual- itative analysis is similar. Fig. 10 remains valid, but now with M values of xi at λ + B and N − M at λ. Implementing the stability criterion we have the following cases (for B > 0): B • N − M values x− and M values x−i , corresponding to a YT with M rows. This is the deformation of the singlet for B = 0. B B • N − M values x− , M − 1 values x−i , and one value x+i , corresponding to a YT with M rows. This is the deformation of the one-row solution for B = 0. B • N − M − 1 values x− , M values x−i , and one value x+ , corresponding to a YT with M + 1 rows. This is the deformation of the one-row solution for B = 0 by increasing its depth and breaking the SU ( N ) symmetry further. Overall, equality of magnetic field components results in states with equal rows in their YT. 4.2.1 One-row and conjugate one-row states We proceed to investigate quantitatively the effect of a magnetic field in one direction (say, 1) in the case where the system is in a one-row solution in the same direction, or the corresponding conjugate representation. That is, we will examine the cases (a) and (b) above (case (c) will be examined in the next subsection). Then the equation for the 30 system becomes6 T ln 1+x = T0 (1 + a) x + B , 1 − ax −1 < x < 1 = N−1. a (4.20) Solutions to this equation with x > 0 correspond to a single row YT, whereas solutions with x < 0 to its conjugate, that is, a YT with N − 1 rows of equal lengths. These are related by observing that (4.20) is invariant under x → − x/a, B → − B, and a → 1/a, which maps symmetric irreps to their conjugate. The stability conditions for the solution are determined by the general discussion in the previous subsection. As before, according to (3.18) C1−1 = N A( x ), with A( x ) defined in (3.19), and thus Ci−1 = N A(− ax ) for i > 1. By its definition, A( x ) satisfies x>0: A( x ) < A(− ax ) , x<0: A( x ) > A(− ax ) . (4.21) The general stability argument requires Ci > 0 for i > 1, that is, A(− ax ) > 0. For C1 < 0 we must also have ∑i Ci < 0, or A( x ) + aA(− ax ) > 0, as it was shown in (3.15) and (3.20). Altogether, combined with (4.21), the stability conditions that cover all cases are x>0: A( x ) + aA(− ax ) > 0 , (4.22) which guarantees positivity of A(− ax ) no matter what the sign of A( x ), and x<0: A(− ax ) > 0 . (4.23) The condition (4.22) above will be satisfied for x < x− or x > x+ , 6 Note x± = N−2± p that for a = 1 (the SU (2) case) and after setting x = y= N 2 − 4( N − 1) T/T0 , 2 B T T0 y − 2T0 T B + 0 tanh y , 2T T (4.24) equation (4.20) becomes (4.18) which is the standard expression in phenomenological investigations of ferromagnetism [21]. For a ̸= 1 2Ty− B (the SU ( N ) case with N ⩾ 3) this is modified by setting x = (1+ a)T and results to the generalization 0 y= B T 1 + 0 2T T coth y − 31 N −2 N . (4.19) while (4.23) will be satisfied for  T x0 = ( N − 1) 1 − T0 x0 < x < 0 ,  . (4.25) The existence of x± , and the condition that x0 > −1, introduce two more temperatures T+ = N2 T0 > T0 , 4( N − 1) T− = N T0 > T0 . N−1 (4.26) For T > T+ , (4.22) is satisfied for all x > 0, while for T > T− , (4.23) is satisfied for all x < 0. Note that both x± ∈ (−1, N − 1), and that x+ > 0, while x− > 0 for T > T0 and x− < 0 for T < T0 . In terms of relative ordering of temperature scales, N=3: N=4: N>4: 3T 9T0 < T− = 0 , 8 2 4T0 , T+ = T− = 3 T+ = (4.27) T+ > T− . We proceed to the analysis of the states of the system. It is most convenient to keep x and T as the free variables and consider the magnetic field B as a function of x with T as a parameter. Then (4.20) implies B( x ) = T ln 1+x − T0 (1 + a) x . 1 − ax (4.28) Note that
dB a(1 + a) T0 = ( x − x+ )( x − x− ) = T0 A( x ) + aA(− ax ) . dx (1 + x )(1 − ax ) (4.29) Hence, dB/dx is proportional to the stability condition (4.22) for x > 0. Therefore, B( x ) is an increasing function of x for T > T+ and T < T− , and a decreasing one for x− < x < x+ , with x− and x+ as local maxima and minima. The function B( x ) is plotted in figure (11) for various values of the temperature. The intersection of these graphs with the horizontal at B determines the solutions for the configuration of the system. The above allow us to determine the stability of solutions for various values of the temperature and magnetic field. We consider two cases, according to (4.22 and 4.23). Positive x: The constraint (4.22) is relevant. Therefore, when T > T+ , all solutions with 32 B(x) B(x) B(x) B B B x Figure 11: Plot of B( x ) for T > T+ (left), T0 < T < T+ (middle) and T < T0 (right). x > 0 are stable. For T0 < T < T+ , stability singles out solutions with 0 < x < x− and x > x+ . Finally, for T < T0 , since then x− < 0, stability requires that x > x+ . Negative x: The constraint (4.23) is now relevant, or equivalently x > x0 . Since x < 0, we conclude that no stable solutions exist for x0 > 0, or T < T0 . For x0 < −1, or T > T− , all x < 0 solutions are stable. Finally, for intermediate temperatures T0 < T < T− , we have stability for −1 < x0 < x < 0. The above are tabulated in table 3 below (we assume that N > 4 so that T+ > T− ): x x<0 x>0 T < T0 none x > x+ T0 < T < T− −1 < x0 < x x < x− & x > x+ T− < T < T+ all x < x− & x > x+ T+ < T all all Table 3: Stable solutions for various ranges of T and x and for N > 4. We can now investigate the existence of stable solutions for the full range of values of the temperature and the magnetic field. The complete analysis is relegated to appendix A. The results are summarized in the temperature-magnetic field phase diagram of figure 12, presented for a generic value for N > 4. The phase diagram is qualitatively the same for N = 3 and N = 4, changing only for N = 2. The only difference is that, for N = 4, T+ = T− , while for N = 3 T+ < T− . This does not affect the general features of the diagram, simply shifting the critical point vertex ( T+ , B+ ) to the left of the bottom asymptote T = T− , for N = 3, or on top of it, for N = 4. Each region in the T−B plane depicted in the figure corresponds to a discrete phase of the system. Moving within these regions without crossing any of the critical lines interpolates continuously between configurations. In the connected regions C1 , C2 , and C3 there is a unique one-row configuration at each point ( T, B). In regions A and B inside the curvilinear triangle there are two locally stable configurations at each point, one absolutely stable and the other metastable, with the line separating A and 33 B B+ C2 0 T0 T A T1 Tc T- T+ N T0 B (Tb ,Bb ) C3 C1 D Figure 12: The phase diagram of the system for generic N > 4. The (orange) curve separating regions A and C2 is B( x− ); the (blue) curve separating regions A, B, C1 and C3 , D is B( x+ ), intersecting the T-axis at the B = 0 critical temperature T = Tc ; and the (red) curve separating regions A, B, C3 and C1 , D is B( x0 ) and it asymptotes to the vertical T = T− . Crossing any of these lines precipitates a discontinuous change in the magnetization, i.e. the order parameter x. The (green) straight line separating regions A and B is the metastability frontier of the two coexisting phases inside these regions (see appendix A.1); crossing it exchanges the metastable and the absolutely stable states, and its intersection with the T-axis is the B = 0 critical temperature T = T1 . Regions C1,2,3 constitute one continuous phase with nonzero magnetization (except at B = 0 and T > Tc ), all points being accessible through continuous paths in the B − T space, while regions A, B and D are separated from C1,2,3 by discontinuous transitions in the order parameter. The (gray) dashed curve from T0 to its vertical asymptote at T = NT0 separates "broken-like" and "unbroken-like" configurations but otherwise mark no sharp phase transition. The shaded region D corresponds to a two-row (double magnetization) phase. For N = 3 the phase diagram remains qualitatively the same with the order of T+ and T− interchanged, while for N = 4, we simply have T+ = T− . B being the border of metastability where the two phases have equal free energy. In region D there are no stable one-row solutions, signifying that a two-row solution must exist there. The lines separating regions C1,2,3 and the other regions are phase boundaries, the configuration changing discontinuously as we cross a boundary. The dashed curve for T0 < T < NT0 represents configurations with C1 = ∞, that is, A( x ) = 0. This corresponds to points where x1 = (1 + x )/N reaches the top of the curve in fig. 10, transiting from the unstable to the stable branch of the curve or vice 34 versa. For such points, x = T/T0 − 1, and (4.28) gives B on this curve as B( T ) = T ln ( N − 1) T N ( T − T0 ) − . NT0 − T N−1 (4.30) Configurations to the left of this curve are in a "broken-like" SU ( N ) phase, with one of the solutions of (4.1) in the unstable branch of the curve in fig. 10, while those to the right of the curve are in an "unbroken-like" phase, with all solutions on the stable branch. For B = 0 these are the true spontaneously broken or unbroken phases of the system. A nonzero magnetic field breaks SU ( N ) explicitly, and the dashed line represents a soft phase boundary, which must be crossed to transit between the two phases as we move on the T − B plane. The physical signature of crossing this boundary is that the off-diagonal elements of the magnetizability mij , with i ̸= j ̸= 1, change sign, vanishing on the boundary (see (4.7)). The line separating regions B and C3 intersects the T axis at the critical temperature Tc found in the B = 0 section. Points ( T0 , 0) and ( T+ , B+ ), with N B+ = B( x+ ( T+ )) = 2( N − 1)  N ln( N − 1) − N + 2 2  . (4.31) are critical points, while point ( Tb , Bb ) at the lower tip of region B, satisfying the transcendental equation B( x0 ( Tb )) = B( x+ ( Tb )) ≡ Bb (4.32) is a multiple critical point, connecting several different configurations: one one-row state in C1 , two in B, and one in C3 , as well as a two-row state in D, and possible tworow states in the other neigboring regions (see next section). One of the states in B, and possibly other one-row or two-row states, are metastable. 4.2.2 Two-row states and their (N − 1)-row conjugates As we have seen, for T < T− and for sufficiently negative magnetic fields there is no stable solution to (4.20), and thus no state corresponding to the one-row YT symmetric representation. From the general analysis of subsection 4.2, we expect the solution to be the only other allowed configuration, that is, a state corresponding to a two-row YT. In this subsection we recover this solution and check its stability. We consider a configuration with two (generally unequal) lengths x1 and x2 and an 35 applied magnetic field in the x1 -direction, representing the generic breaking pattern SU ( N ) → SU ( N − 2) × U (1) × U (1) , (4.33) of the SU ( N ) symmetry. This includes a spontaneous breaking of SU ( N ) in addition to the dynamical breaking SU ( N ) → SU ( N − 1) × U (1) due to the magnetic field. We note that in the special case x1 = x2 the symmetry breaking pattern would be SU ( N ) → SU ( N − 2) × SU (2) × U (1), but as we shall demonstrate this pattern is never realized in the present case of a magnetic field in a single direction. The xi must satisfy the system of equations x1 = NT0 ( x1 − x N ) + B , xN x T ln i = NT0 ( xi − x N ) , i = 2, 3, . . . , N − 1 , xN T ln (4.34) where x N is determined from the constraint in (2.30). We write x1 = 1+x , N x2 = 1+y , N 1 − α( x + y) x3 = · · · = x N = , N 1 α= . N−2 (4.35) For y = −αx/(1 + α) = − ax = − x/( N − 1) this ansatz reduces to the one for the one-row solution. (4.34) gives rise to the system of transcendental equations T ln
1+x = T0 αy + (1 + α) x + B , 1 − α( x + y)
1+y T ln = T0 αx + (1 + α)y . 1 − α( x + y) (4.36) The free energy of the configuration is   T T0 2 2 2 (1 + x ) ln(1 + x ) ( x − y) − N ( x + y ) + F ( x, y, T ) = 2N ( N − 2) N   x+y B + (1 + y) ln(1 + y) + ( N − 2 − x − y) ln 1 − − x − T ln N . N−2 N (4.37) The transcendental equations (4.36) will be solved numerically. The full analysis of solutions and their stability is relegated to the Appendix. Here we simply state the results and present relevant plots. 36 We consider temperatures T < T0 for which spontaneous magnetization exists. As discussed earlier, for such temperatures and negative magnetic fields we expect the state to be in a two-row stable state, which includes the posssility of "antirows," that is, N − 2 equal rows plus an additional row. Further, such states may coexist with a one-row state and be either globally stable or metastable. All cases refer to plots in Fig. 13. The blue and orange curves represent the solutions of the first and second equation in (4.36), resp. The orange line y = − ax, in particular, solves the second equation in the system (4.36) while the first one reduces to the one for the one-row configuration (4.20). Intersections of blue and orange lines represent the solutions the (4.36). Only locally stable solutions are considered. • B > 0: We recover the known one-row solution on the y = − ax line for x > 0. There is also a metastable two-row solution with x < 0, y > 0. 6 B0 6 B>0 5 5 4 4 3 x+y  2 3 x+y  2 1 B0 < B < 0 5 4 1 α 6 3 α 2 1 y  -a x 0 y  -a x 0 1 2 3 4 5 -1 -1 6 y  -a x 0 -1 -1 1 α 1 0 -1 x+y  1 0 1 2 3 4 5 -1 6 0 1 2 3 4 5 6 6 6 B(x+ ) < B < B0 B < B(x+ ) 5 5 4 4 3 x+y  2 3 x+y  1 α 2 1 α 1 1 y  -a x 0 y  -a x 0 -1 -1 -1 0 1 2 3 4 5 -1 6 0 1 2 3 4 5 6 Figure 13: Typical contour plots for T < T0 in the x − y plane of the two eqs. in (4.36), in blue (1st eq.) and orange (2nd eq., for which one of the branches is a straight line). The intersection points of the blue curves with the orange curves represent solutions of (4.36). Stable and metastable solutions are indicated by black and magenta colored dots, respectively. B0 is the value of B for which the central bulges of the curves would touch (bet. 3rd and 4th plot). Plots are for N = 7 and T = 0.9, and for B = 0.2, 0, −0.4, −2, −3 (note that B0 ≃ −1.01 and B( x+ ) ≃ −2.62) in units of T0 . 37 • B = 0: The system is symmetric under x ↔ y and we recover the known onerow solution on the y = − ax line for x > 0. There is also a stable solution on the y = − x/a line for x < 0, which is equivalent to the previous one, representing spontaneous magnetization in direction x2 . • B( x+ ) < B < 0: We recover the known one-row solution on the y = − ax line for x > 0, but now it is metastable. There is a stable two-row solution with x < 0, y > 0. • B < B( x+ ): There are no stable one-row solutions, in accordance with region D of the one-row phase diagram of fig. 12. There is a unique stable two-row solution. We note that there are no solutions with x = y since this is not a consistent truncation of the system (4.36), unless B = 0 in which case we already know that the corresponding two-row solution is unstable. Thus the symmetry breaking pattern SU ( N ) → SU ( N − 2) × SU (2) × U (1) is never realized. Recalling fig. (12), the picture that emerges, at least for T < T0 , is that a tworow solution coexists with the one-row solution in both regions C1 , C2 . The two-row solution is metastable in C2 (B > 0) and becomes absolutely stable in C1 (B < 0). The one-row solution is absolutely stable in C2 , becomes metastable in C1 , and ceases to exist in region D, leaving the two-row state as the only stable solution there. The line B = 0 is a metastability frontier between one-row and two-row solutions for T < T0 . We expect this picture to extend for a range of temperatures T > T0 , with a two-row state coexisting with the one-row one outside of region D, although for high enough temperatures the two-row solution should cease to exist. 5 Conclusions The thermodynamic properties and phase structure of the SU ( N ) ferromagnet emerge as surprisingly rich and nontrivial, manifesting qualitatively new features compared to the standard SU (2) ferromagnet. The phase structure of the system, in particular, is especially rich and displays various phase transitions. Specifically, at zero magnetic field the system has three critical temperatures (vs. only one for SU (2)), one of them signaling a crossover between two metastable states. Spontaneous breaking of the 38 global SU ( N ) group in the ferromagnetic phase at zero external magnetic fields happens only in the SU ( N ) → SU ( N − 1) × U (1) channel. In the presence of a nonabelian magnetic field with M nontrivial components (M < N), the explicit symmetry break- ing (paramagnetic state) is SU ( N ) → SU ( N − M ) × U (1) M , while the spontaneous breaking (ferromagnetic state) is SU ( N ) → SU ( N − M − 1) × U (1) M+1 . Finally, due to the presence of metastable states, the system exhibits hysteresis phenomena both in the magnetic field and in the temperature. The model studied in this work, and its various generalizations described below, could be relevant in a variety of physical situations. It could serve as a phenomenological model for physical ferromagnets, in which the interaction between atoms is not purely of the dipole type and additional states participate in the dynamics. In such cases, the SU ( N ) interactions could appear as perturbations on top of the SU (2) dipole interactions, leading to modified thermodynamics. The model could also be relevant to the physics of the quark-gluon plasma [22], which can be described as a fluid of particles carrying SU (3) degrees of freedom, assuming their SU (3) states interact. Exotic applications, such as matrix models and brane models in string theory, can also be envisaged (see, e.g. [23, 24]). Various possible generalizations of the model, relevant to or motivated by potential applications, and related directions for further investigation suggest themselves. They can be organized along various distinct themes: starting with atoms carrying a higher representation of SU ( N ), generalizing the form of the two-atom interaction, or including three- and higher-atom interactions. The choice of fundamental representations for each atom was imposed by the physical requirement of invariance of their interaction under common change of basis for the atom states. Its effect on the thermodynamics is to "bias" the properties towards states with a large fundamental content. This manifests, e.g., in the qualitatively different properties of the system under positive and negative magnetic fields (with respect to the system’s spontaneous magnetization). Starting with atoms carrying a higher irrep of SU ( N ) would modify these properties. In particular, starting with atoms in the adjoint of SU ( N ) would eliminate this bias altogether. It might also eliminate phases of spontaneous magnetization, and this is worth investigating. The interaction of atoms jr,a js,a was isotropic in the group indices a, an implication of the requirement of invariance under change of basis. Anisotropic generalizations 39 of the form (2.2) can also be considered, involving an "inertia tensor" h ab in the group. Clearly this generalization contains the higher representation generalizations of the previous paragraph as special cases. E.g., SU (2) interactions with the atoms in spin-1 states can be equivalently written as SU (3) fundamental atoms with a tensor h ab equal to δab when a, b are in the SU (2) subgroup of SU (3) that admits the fundamental of SU (3) as a spin-1 irrep, and zero otherwise. The more interesting special case in which h ab deviates from δab only along the directions of the Cartan generators, in the presence of magnetic fields along these directions, seems to be the most motivated and most tractable, and is worth exploring. The phase properties of the model under generic h ab is also an interesting issue. Including higher than two-body interactions between the atoms is another avenue for generalizations. Physically, such terms would arise from higher orders in the perturbation expansion of atom interactions, and would thus be of subleading magnitude, but the possibility to include them is present. Insisting on invariance under common change of basis and a mean-field approximation would imply that such interactions appear as higher Casimirs of the global SU ( N ) and/or as higher powers of Casimirs, the most general interaction being a general function f (C (2) , · · · , C ( N −1) ) of the full set of Casimirs of the global SU ( N ). These can be readily examined using the formulation in this paper and may lead to models with richer phase structure. An interesting extension of this study is in the context of topological phases nonabelian models. Such topological phases have been proposed in one dimension [25–28] and it would be interesting to see if they exist in higher dimensions. Another independent direction of investigation is the large-N limit of the model. This could be conceivably relevant to condensed matter situations involving interacting Bose condensates, or to more exotic situations in string theory and quantum gravity. The presence of two large parameters, n and N, presents the possibility of different scaling limits. These will be explored in an upcoming publication. Finally, the nontrivial and novel features of this system offer a wide arena for experimental verification and suggest a rich set of possible experiments. The experimental realization of this model, or the demonstration of its relevance to existing systems, remain as the most interesting and physically relevant open issues. 40 Acknowledgements We would like to thank I. Bars for a very useful correspondence, D. Zoakos for help with numerics, and the anonymous Reviewer for comments and suggestions that helped improve the manuscript. A.P. would like to thank the Physics Department of the U. of Athens for hospitality during the initial stages of this work. His visit was financially supported by a Greek Diaspora Fellowship Program (GDFP) Fellowship. The research of A.P. was supported in part by the National Science Foundation under grant NSF-PHY-2112729 and by PSC-CUNY grants 65109-00 53 and 6D136-00 02. The research of K.S. was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant” (MIS 1857, Project Number: 16519). A Analysis of one-row states with a magnetic field In this appendix we present the details of the analysis that we have summarized in the main text. T > T+ (fig. 14): B( x ) is increasing and all values of x are stable, so there exsits a unique stable solution for all B, one-row for B > 0 and its conjugate for B < 0. B(x) 6 4 B 2 -1 1 2 3 4 5 6 x -2 Figure 14: Plot of B( x ) for T > T+ . T− < T < T+ (fig. 15): We have stability for −1 < x < x− and x+ < x < N − 1, the regions of increasing B( x ). Further, note that x− > 0 and B( x− ) > B( x+ ), and that B( x− ) > 0 while B( x+ ) can be positive or negative. The critical temperature Tc satisfies the condition T = Tc ⇐⇒ 41 B( x+ ) = 0 , (A.1) which is precisely the condition (3.8). Note that, for N > 4, T− < Tc < T+ . For T > Tc , B( x+ ) > 0 and for T < Tc , B( x+ ) < 0. B(x) B(x) 1.5 1.0 1.0 B B 0.5 0.5 -1 1 2 3 4 5 6 x -0.5 -1 1 2 3 4 5 6 x -1.0 -1.5 Figure 15: Plot of B( x ) for Tc < T < T+ (Left) and for T− < T < Tc (Right). So within this temperature range we distinguish two sub-cases: Tc < T < T+ (left plot in fig. 15): we have for various values of the magnetic field: • For B < 0, x varies from −1 to 0 and there is a unique stable solution corresponding to the conjugate one-row YT. • For 0 < B < B( x+ ), x varies from 0 to some x < x− and there is a unique stable solution corresponding to a one-row YT. • For B( x+ ) < B < B( x− ) we have two locally stable solutions, one for some 0 < x < x− and one for some x > x+ (a third solution in between is unstable). The first one corresponds to an unbroken phase, as it represents a continuous deformation of the singlet for B = 0, and the second one to a broken phase. One is absolutely stable and the other metastable. To decide which, we need to compare their free energies. • For B > B( x− ), x varies from some value greater than x+ to N − 1 and there is a unique stable solution. Note that for B = 0 there is only the solution x = 0, as expected. T− < T < Tc (right plot in fig. 15): we have for various values of the magnetic field: • For B < B( x+ ), x varies from −1 to some negative value x obtained from B( x+ ) = B( x ) and there is a unique stable solution corresponding to the con- jugate one-row YT. 42 • For B( x+ ) < B < B( x− ) we have two locally stable solutions, one for some x < x− and one for some x > x+ . The first one represents an unbroken phase and the second one a broken phase, as they map to the singlet and the one-row solutions for B = 0. One is absolutely stable and the other metastable. To decide, we need to compare their free energies. • For B > B( x− ), x varies from some x > x+ to N − 1 and there is a unique stable solution corresponding to a one-row YT. Note that B = 0 is in the range B( x+ ) < B < B( x− ) and we recover the expected two solutions, x = 0 (singlet) and x > 0 (one-row). T0 < T < T− (fig. 16): The situation is as in case T− < T < Tc , except now x cannot be more negative than x0 = ( N − 1)(1 − T/T0 ) defined in (4.25). The relative size of B( x0 ) and B( x+ ) will also play a role: • For B < min{ B( x0 ), B( x+ )} there is no stable one-row solution and the stable solution must necessarily have more rows. According to the general stability analysis, it must be one with N − 1 rows out of which N − 2 have equal length. • For min{ B( x0 ), B( x+ )} < B < max{ B( x0 ), B( x+ )} there is one stable solution, for x < 0 (x > x+ ) if B( x0 ) < B( x+ ) (B( x0 ) > B( x+ )). • For max{ B( x0 ), B( x+ )} < B < B( x− ) there are two stable solutions, one abso- lutely stable and the other metastable. To decide, we need to compare their free energy. • For B > B( x− ), x there is a single stable solution varying from some x > x+ to N − 1. T < T0 : Only x > x+ solutions are stable. • For B < B( x+ ) there is no stable one-row solution, and the solution must again be one with ( N − 1)-rows. • For B > B( x+ ) there is one stable solution for x > x+ 43 2 B(x) 1 B -1 1 2 3 4 5 6 x B(x0 ) -1 -2 Figure 16: Plot of B( x ) for T0 < T < T− . The dashed lines refer to the value B( x0 ) which could be higher (green) or lower (red) than B( x+ ). A.1 Resolving metastability To determine which configuration is absolutely stable and which is metastable when there are two locally stable solutions, we need to compare their free energies. The free energy of the system is given by (3.24) with the addition of the magnetic field term, Fsym ( x, T ) =
T a(1 + x ) ln(1 + x ) + (1 − ax ) ln(1 − ax ) 1+a B( x ) a x − T ln N , − T0 x2 − 2 N (A.2) where a = 1/( N − 1) and where B = B( x ) is expressed in terms of x, T via (4.28). To facilitate the comparison, define the modified free energy Φ (we use F ( x ) instead of Fsym ( x, T ) for notational convenience) Φ( x ) = F ( x ) + N−2 B( x ) . 2N (A.3) We can show that Φ( x ) and B( x ) satisfy Φ( N − 2 − x ) = Φ( x ) and B( x ) + B( N − 2 − x ) = 2T ln( N − 1) − T0 (A.4) N ( N − 2) . N−1 (A.5) For two solutions with different x, x ′ but the same B, F ( x ) − F ( x ′ ) = Φ( x ) − Φ( x ′ ), so we can compare their Φ to resolve metastability. At the transition point, when the two 44 solutions have the same free energy, (A.4) implies Φ( x ) = Φ( x′ ) =⇒ x′ = N − 2 − x (A.6) and from (A.5) with B( x ) = B( x ′ ) = B( N − 2 − x ) the magnetic field Bt at which this happens is Bt = T ln( N − 1) − T0 N ( N − 2) . 2 N−1 (A.7) For fixed B, this gives the transition temperature Tt at which the two solutions will transit from stable to metastable as 1 Tt = ln( N − 1)  N ( N − 2) B + T0 2( N − 1)  . (A.8) For B = 0 this reproduces the temperature T1 and magnetization x1 that we determined before in (3.27). B Analysis of two-row states in a magnetic field In this appendix we investigate in detail two-row solutions, including their ( N − 1)- row conjugates. We analyze the conditions for their stability, present the corresponding YT of their irreps, and derive numerical results for the case of temperatures T < T0 . To proceed, we write the coefficients Ci defined in (3.14) in terms of the variables x and y of (4.35). We obtain C1−1 = N A( x ) , Ci−1 C2−1 = N A(y) ,
= N A − α( x + y) , (B.1) i = 3, 4, . . . , N , with our usual A( x ) defined in (3.19). The variables x and y are restricted by the conditions 0 < xi < 1 to the range x, y > −1 , x+y < 1 = N−2. α (B.2) Thus, the allowed solutions are within the triangle in the ( x, y)-plane with corners at the points (1/a, −1), (−1, 1/a) and (−1, −1), depicted in fig. 17. This triangle is further subdivided by the curves y = x, y = − ax and y = − x/a into six regions repre- senting the possible ordering of x1 , x2 , xi (i ⩾ 3) and thus the various YT renditions of 45 the two-row solution. These regions are accordingly labelled by (ijk) for xi > x j > xk . Assuming N > 3, most of the Ci ’s are proportional to 1/A(−α( x + y)), so we choose solutions with A(−α( x + y)) > 0. Then at most one of the functions A( x ) and A(y) can be negative. We list below the various possibilities together with conditions for stability: y (−1, 1/a) y=-x/a 231 213 x+y=1/α y=x 123 321 x y=-ax 132 312 (−1, −1) (1/a, −1) Figure 17: The domain of x, y. Coordinate axes do not create subdivisions. Region A, or (123): x1 > x2 > x3 , with − ax < y < x =⇒ A( x ) < A(y) < A(−α( x + y)) . (B.3) The stability condition is A(y) > 0 , α[ A( x ) + A(y)] A(−α( x + y)) + A( x ) A(y) > 0 . (B.4) In this region all Ci ’s are positive, excect C1 which may have either sign. We define ℓ1 = (1 + α) x + αy , N ℓ2 = (1 + α)y + αx , N ℓ1 > ℓ2 > 0 . (B.5) The YT has the partition (ℓ1 , ℓ2 ) . (B.6) Hence, it represents a two-row YT with ℓ1 and ℓ2 boxes, respectively. This follows from the fact that x3 is the smallest among the three xi ’s and appears N − 2 times. 46 Region B, or (132): x1 > x3 > x2 , with − x/a < y < − ax , A( x ) < A(−α( x + y)) < A(y) . (B.7) The stability condition is A(−α( x + y) > 0 , α[ A( x ) + A(y)] A(−α( x + y)) + A( x ) A(y) > 0 . (B.8) In this region all Ci ’s are positive expect C1 which may have either sign. We define ℓ1 = x−y , N ℓ2 = − (1 + α)y + αx , N ℓ1 > ℓ2 > 0 . (B.9) The ( N − 1)-row YT has the partition (ℓ1 , ℓ2 , ℓ2 , . . . , ℓ2 ) . | {z } (B.10) N −2 Hence, it represents a YT with ℓ1 boxes in the first line and ℓ2 boxes in the following N − 2 lines. This follows from the fact that the smallest among the three xi ’s, x2 appears only once. Region C, or (213): x2 > x1 > x3 , with − ay < x < y , A(y) < A( x ) < A(−α( x + y)) . (B.11) The stability condition is A( x ) > 0 , α[ A( x ) + A(y)] A(−α( x + y)) + A( x ) A(y) > 0 . (B.12) In this region all Ci ’s are positive expect C2 which could be either positive or negative. Defining ℓ1 = (1 + α)y + αx , N ℓ2 = (1 + α) x + αy , N ℓ1 > ℓ2 > 0 , (B.13) this case represents a two-row YT with the partition (B.6). Region D, or (231): x2 > x3 > x1 , with − ax < y < − x/a , A(y) < A(−α( x + y)) < A( x ) . 47 (B.14) The stability condition is A(−α( x + y) > 0 , α[ A( x ) + A(y)] A(−α( x + y)) + A( x ) A(y) > 0 . (B.15) In this region all Ci ’s are positive expect C2 which could be either positive or negative. Defining ℓ1 = y−x , N ℓ2 = − (1 + α) x + αy , N ℓ1 > ℓ2 > 0 . (B.16) this case represents a YT with the partition (B.10). Region E, or (312): with x3 > x1 > x2 , with y < x < − ay, equivalently x3 > x1 > x2 , y < x < − ay , A(−α( x + y)) < A( x ) < A(y) . (B.17) The stability condition is A(−α( x + y)) > 0 . (B.18) In these two regions all Ci ’s are positive. Defining ℓ1 = − (1 + α)y + αx , N ℓ2 = x−y , N ℓ1 > ℓ2 > 0 , (B.19) this case represents a ( N − 1)-row YT with the partition (ℓ1 , ℓ1 , . . . , ℓ1 , ℓ2 ) , {z } | (B.20) N −2 Region F, or (321): x3 > x2 > x1 , with x < y < − ax A(−α( x + y)) < A(y) < A( x ) . (B.21) The stability condition is A(−α( x + y)) > 0 . (B.22) In these two regions all Ci ’s are positive. Defining ℓ1 = − (1 + α) x + αy , N ℓ2 = y−x , N ℓ1 > ℓ2 > 0 . this case represents a ( N − 1)-row YT with the partition (B.20). 48 (B.23) B.1 The case of low temperatures We consider temperatures T < T0 for which spontaneous magnetization exists. Various cases arise depending on the sign of B and, if negative, on the value B( x+ ) < 0 with x+ defined in (4.24) (recall that B( x+ ) is negative for T < T0 ) and some other intermediate value B0 to be defined shortly. In the plots below all points on the red line y = − ax solve identically the second equation in the system (4.36), whereas the first one reduces to the equation for the one-row configuration (4.20). This line can be approached for x > 0 from the regions A and B and for x < 0 from the regions D and F. In addition, the stability conditions for these regions reduce to those in (4.22). All cases below map to one of the plots in Fig. 13. • B > 0: We know that the one-row configuration has a stable solution which in this two-parameter plot is on the y = − ax red line for x > 0. The intersection point in the middle (region C in Fig. 17) is unstable, whereas that on the upper left corner is, having a higher value for the free energy, metastable (region C). • B = 0: We have included the case with B = 0, which is symmetric with respect to the y = x line. We know that the one-row configuration has a stable solution which in this two-parameter plot is on the y = − ax red line for x > 0. Due to the above symmetry there is stable solution also on the y = − x/a line for x < 0, which however is equivalent to the above. The other three intersecting points are unstable. • B0 < B < 0: The value of the magnetic field B0 arises when the two curves in the plot meet tangentially. The one-row configuration has a stable solution which in this two-parameter plot is on the y = − ax red line for x > 0. However, this becomes now metastable, as the stable intersection point is on the upper left corner (region D in fig. 17), corresponding to an ( N − 1)-row YT with partition (B.10). The other intersection points are unstable. • B( x+ ) < B < B0 : There are typically three intersection points along the y = − ax line corresponding to the one-row configuration, the far right is now metastable and the other two unstable. However, there are two additional intersecting points in the far left of that plot, the lower one being unstable and the upper one stable (region D in Fig. 17) corresponding to an ( N − 1)-row YT with parti- tion (B.10). 49 • B < B( x+ ): There is one intersection points along the y = − ax line corresponding to the one-row configuration. This has x < 0 and is, as we have shown un- stable. 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