The Hamiltonian of the transverse field Ising model ofN fully connected spins reads H=− 1
2N XN i,j=1
sziszj −Γ XN i=1
sxi, (2.1)
where sx,y,zi = σix,y,z/2 are spin-1/2 operators of the ith elementary spin in terms of the Pauli matrices σx,y,z, and Γ is the homogeneous transverse magnetic field. The ferromagnetic coupling between the spins is all-to-all, and its uniform strength is scaled by a factor of 1/N such that both sums in (2.1) scale extensively asO(N).
The model (2.1) is also known as the Lipkin Meshkov Glick model [62]. It has a quantum critical point at Γc= 1/2, which separates the ferromagnetic (ordered) phase Γ <Γc from the paramag-netic (disordered) phase Γ>Γc[28, 81].
The Hamiltonian (2.1) is defined on the 2N-dimensional Hilbert space HN = ⊗NC2 of N ele-mentary 1/2 spins. The subspace of invariant states w.r.t. permutations of the N spins is denoted by DN, and referred to as the symmetric Dicke space. As H is invariant under permutations of the spins, the Dicke subspace is invariant underH and under the family of unitary time evolution U =e−iHt.
In Sec. 2.2 we will encounter another Hilbert space, the Hilbert space of square integrable func-tions on the unit intervalHeff:=L2(0,1). In the limit of N → ∞, states in DN are approximated by functions inHeff. The three Hilbert spaces, HN,DN, andHeff, occur frequently in this treatise.
Pictorially, their relation is summarized as
HN ⊃ DN N→∞
−−−−→ Heff.
In the following subsection, we discuss the symmetric Dicke space and its relation to the full Hilbert space ofN spins in more detail. Thereafter, in Sec. 2.2 a mapping of the symmetric spin model onto an effective model with Hilbert spaceHeff is explained. This effective description has a semiclassical limit forN → ∞.
2.1.1 Permutation invariance and Dicke subspace
We spend some time on the theory of permutation invariant systems. First, we review some gen-eral consequences of permutation invariant Hamiltonians on the structure of the Hilbert space by following [82]. Although, we will mostly confine to the totally symmetric subspace, we sketch the theory in more generality. Second, we apply the general findings to a fully symmetric spin Hamiltonian ofN spin-1/2 particles.
Generic permutation invariant Hamiltonian of N particles. The following discussion can be found in chapter IX. of [82]. Let H be a generic N particle Hamiltonian1 that commutes with all particle permutation operatorsP, i.e. [H, P] = 0. Moreover, any permutation operator is unitary, P† = P, and can be viewed as a conserved (complex-valued) quantity. There are N! conserved quantities resulting from permutation invariance in this way. However, as two permutation op-erators P1 and P2 do not commute in general, one cannot give a value to all these conserved quantities simultaneously. In other words, the permutation operators are complementary, and there is no common eigenbasis of all P and H. The best one can do is to give a maximal set of pairwise commuting conserved quantities χ1,· · ·χm, i.e. [χi, χj] = 0 and [χi, H] = 0. Every χi
must be a linear combination of the permutation operators, because any polynomial of permutation operators can be written as a linear combination of permutation operators. It turns out that the choice of χi = χ(Pi) = (N!)−1P
P P PiP†, where the summation is over all permutations, fulfills the requirements. Evidently, χi commutes with all permutations. Two similar2 permutations P and Q give identical χ(P) = χ(Q). On the contrary, any two permutations P and Q that are not similar yield different operators, χ(P) 6= χ(Q). Hence, to each equivalence class of similar permutations one associates a conserved quantity χi. As χi is the average over all permutations in the respective equivalence class, and because any permutation P is similar to its inverse P†, χi is Hermitian, i.e. a real-valued conserved quantity. The number of equivalence classes modulo similarity ism, the number of ways thatN can be decomposes into a sum of integers. A trivial conserved quantity is given by the identity, χ1 = 1. We denote the collection of all m operators (χ1,· · ·χm) by the bold face letterχ. The eigenvalues of χare not independent, since they have to fulfill certain relations among them. It turns out that there are m different valid solutions (each solution is related to a character of the group of permutations). One obvious solution is χ1 = 1,· · ·χm = 1, corresponding to permutation invariant states. Another solution is χi = ±1, depending on the parity of the equivalence class, corresponding to antisymmetric states. These two irreducible representations of the permutation group, the totally symmetric and the totally antisymmetric, are the most prominent ones, as they are the mathematical basis of the theory bosons and fermions, respectively. We can now construct common eigenstates of H and χ, and denote the orthonormal eigenbasis by |E,χi. The permutation invariant energy eigenstates are
|E,{χi = 1}i. These states are special and they obey P|E,{χi = 1}i = |E,{χi = 1}i for any permutationP. In general, acting with a permutation on|E,χichanges the state, but it remains an eigenstate of H and χ with the same eigenvalues. In other words, |E,χi are usually highly degenerate. (The fact that many energy eigenstates are degenerate already follows from the fact that H has many non-commuting conserved quantities.) The degeneracy3 of |E,χi is a function
1For the sake of concreteness, one may think ofH as the Hamiltonian in (2.1), but the discussion applies to allN particle Hamiltonians of any particle type.
2Two permutations P1 and P2 are similar if there exists another permutation P such that P1 = P P2P−1. In other words, two permutationsP1 andP2 are similar if they are identical modulo relabeling of the labels by a permutationP.
3Here, we only mean the degeneracy as a consequence of the indistinguishability of the particles. Of course, there may be additional degeneracies, unrelated to the permutation invariance.
2.1 The fully connected transverse field Ising model n(χ) ofχ alone. It is impossible to distinguish these degenerate eigenstates by physically mean-ingful, i.e. permutation invariant, operators. Only eigenstates ofχ with different eigenvalues can be distinguished by physically meaningful observables. Examples of such meaningful observables are the Hermitian operators χ.
Permutation invariant Hamiltonian of N spin-1/2 particles. Now, we turn to the special situa-tion of a permutasitua-tion invariant Hamiltonian ofN spin one half particles, i.e. when the local Hilbert space of a single particle is two dimensional. Prior to discussing the concrete spin Hamiltonian (2.1), we make some general remarks about the structure of the Hilbert space of a system of N identical spin one-half particles. A system of N elementary spins is built up by starting with a single spin-1/2 and successively adding elementary spins one by one until the N-th spin is added.
Adding two spin one half spins produces an antisymmetric singlet (j = 0) state and three sym-metric triplet states (j = 1). The procedure of subsequently adding more spins is schematically sketched in Fig. 2.1. From one row to the next a spin one half is added. The numbers in every
1
2(2) N = 1
0(1) 1(3) N = 2
1
2(2) 12(2) 3
2(4) N = 3
0(1) 1(3) 0(1) 1(3) 1(3) 2(5) N = 4
1
2(2) 12(2) 32(4) 12(2) 12(2) 12(2) 12(2) 32(4) 32(4) 5
2(6) N = 5
Figure 2.1: Successive buildup of the Hilbert space of N spin-1/2 particles, starting with a single spin (top row), and adding one additional spin according to the spin coupling rules by going to the next row below. Each cell represents subspace of the Hilbert space with fixed number of particles (given by the row index N in the last column), and a fixed total spin j given by the first number in each cell. The number in brackets is (2j+ 1), the dimension of the subspace. All of these subspaces are mutually orthogonal. The total dimension of all subspaces in the N-th row sums to 2N, the dimension of the Hilbert space of N spin-1/2 particles. The last cell in each row (boxed) denotes the totally symmetric, permutation invariant, Dicke subspace of N spins.
cell indicates the spin-j representation, the number in brackets is the dimension (2j+ 1) of this representation. The sum of all dimensions in the Nth row add up to 2N, i.e. the Hilbert space dimension ofN elementary spins. The arrows indicate how the irreducible spin-j representations splits into the two irreducible spin|j−1/2|and (j+ 1/2) representations upon adding a new spin.
Thereby, the dimension doubles from (2j+ 1) to [2(j−1/2) + 1] + [2(j+ 1/2) + 1] = 2(2j+ 1).
The representation with the largest spin in each row (marked with a box) is totally symmetric,
i.e. states of this representation are eigenvalue one eigenstates of χ1,· · ·χm acting on the N el-ementary spins. Those totally symmetric states span the (N + 1) dimensional Dicke space. In other words, the Dicke space is the spin-N/2 irreducible representation obtained by adding N elementary spin-1/2’s. In general, the representations of the remaining cells in the above diagram are not spanned byχ eigenstates. However, in the direct product space of representations of the same spin j, one can choose an eigenbasis of the χ1,· · ·χm operators. This is because the op-erators χ commute with the total spin operator S = PN
j Sj, and hence, eigenstates of χ must lie in an eigenvalue j(j+ 1) eigenspace of S2. We do not proceed further with the discussion of the non-symmetric χeigenspaces, other than noting that these eigenspaces need to be taken into account when constructing thermal density matrices of permutation invariant spin Hamiltonians onHN = (C2)⊗N.
It is interesting that the spin-N/2 representation obtained by addingN elementary spin-1/2’s is invariant under permutations of the spins. We confirm this fact by a direct computation. To this end, let be the normalized permutation invariant state of N elementary spins with N+ up spins, where P = N!1 P
p∈SNp is the orthogonal projection operator onto the permutation invariant subspace.
There are (N + 1) independent states, parametrized by N+ ∈ {0,· · ·N}. We show that these states are N2(N2 + 1) eigenvalue eigenstates of S2. The off-diagonal terms in the double sum S2 =PN parallel and antiparallel spins, the number of up and down spins is not changed under the action ofSi·Sj. Using this result and the fact that|sii is an 12(12 + 1) eigenstate ofS2i in
shows that the symmetric Dicke states are indeed states of a spin-N/2 representation.