Dynamical Symmetries in

Supersymmetric Matrix^{1}

V. Bach, J. Hoppe, D. Lundholm

Received: January 29, 2008 Revised: April 16, 2008

Communicated by Heinz Siedentop

Abstract. We reveal a dynamical SU(2) symmetry in the asymptotic description of supersymmetric matrix models. We also consider a recursive approach for determining the ground state, and point out some additional properties of the model(s).

2000 Mathematics Subject Classification:

Keywords and Phrases:

1 Introduction

Supersymmetric matrix models derive from superstring theory which ultimately aims at a quantum field theoretic model of all known forces, including gravity. Some of the basic mathematical properties of supersymmetric matrix models are still open and pose a challenge to mathematics. One of the key properties of supersymmetric matrix models - often assumed for granted in physics, but difficult to prove mathematically - is the existence of a ground state. I.e., the self-adjoint and nonnegative Hamiltonian operator H = H^{∗} ≥ 0 specifying the supersymmetric matrix model under consideration is assumed to have an eigenvalue at 0, the bottom of its spectrum. Since its spectrum is purely essential and covers the entire positive half axis, σ(H) = [0, ∞) (see [3, 11]), the existence of zero-energy eigenstates, i.e., the non-triviality Ker(H) ≠ 0 of the zero-energy subspace, is not a consequence of standard methods of regular perturbation theory. The Hamiltonian H acts on (an appropriate dense domain in) the Hilbert space H := L^{2}(ℝ^{d(N^{2}−1)}) ⊗ F of square-integrable functions of coordinate variables x ≡ (x_{tA})_{t∈D,A∈N} with values in the fermion Fock space F spanned by vectors of the form λ_{α_{1},A_{1}}^{†} ⋯ λ_{α_{k},A_{k}}^{†}|0⟩, where {λ_{αA}^{†}}_{α∈S,A∈N} are standard fermion creation operators, and |0⟩ is the vacuum vector. The index ranges are denoted by D := {1, 2, …, d}, N := {1, 2, …, N^{2} − 1}, and S := {1, 2, …, d − 1}. Note that dim_{ℂ} F = 2^{(d−1)(N^{2}−1)} < ∞. On H, the Hamiltonian assumes the form

1Supported by the Swedish Research Council

Documenta Mathematica 13 (2008) 103–116

H = −Δ_{x} ⊗ 1 + V (x) ⊗ 1 + H_{F} ,  (1)

where Δ_{x} is the Laplacian on ℝ^{d(N^{2}−1)}. The potential V is a homogenous, quartic polynomial in the coordinates x,

V (x) = \frac{1}{2} f_{ABC} f_{AB′C′} x_{sB} x_{tC} x_{sB′} x_{tC′},  (2)

with (f_{ABC})_{A,B,C∈N} being real, antisymmetric structure constants of SU(N) and using Einstein’s summation convention (i.e., repeated indices are summed over). The operator H_{F}, the fermionic part of the Hamiltonian, is linear in x, but quadratic in the fermion creation operators λ_{αA}^{†} and their adjoints λ_{αA} = (λ_{αA}^{†})^{∗}, the fermion annihilation operators,

H_{F} = f_{ABC} { − 2i( ∑_{t=1}^{d−2} x_{t,C} Γ_{α,β}^{t}) λ_{αA}^{†} λ_{βB}  (3)

+ (x_{d−1,C} + ix_{d,C}) λ_{αA}^{†} λ_{αB}^{†} − (x_{d−1,C} − ix_{d,C}) λ_{αA} λ_{αB}},

where (Γ^{t})_{t=1}^{d−2} are purely imaginary, antisymmetric (d − 1) × (d − 1) matrices that represent the Clifford algebra {Γ^{s}, Γ^{t}} = 2δ_{st} · 1_{(d−1)×(d−1)}, with s, t ∈ {1, 2, …, d − 2} and d ∈ {2, 3, 5, 9}. The Hamiltonian H commutes with the generators {J_{A}}_{A∈N} of su(N), where J_{A} = \frac{−i}{2} f_{ABC}(2x_{sB}∂_{sC} +λ_{αB}^{†}λ_{αC} +λ_{αC}λ_{αB}^{†}) and ∂_{sC} := \frac{∂}{∂x_{sC}}, and the ground state sought for is required to be SU(N)-invariant. That is, the spectral analysis of H is carried out on the subspace H_{0} = ⋂_{A∈N} Ker(J_{A}) ⊆ H. On H_{0}, the Hamiltonian H arises as the restriction of the square of supercharges (Q_{β})_{β∈S}. These supercharges are selfadjoint (matrix-valued, first-order partial differential) operators on H, which we don’t describe here in detail, but we note that the Hamiltonian H ↿_{H_{0}}= Q_{β}^{2} ↿_{H_{0}}≥ 0, is manifestly nonnegative on H_{0}. Two main difficulties arise in the analysis of H:

(a) The quartic potential V has many vanishing directions. E.g., given e⃗ = (e_{A})_{A∈N} ∈ ℝ^{N^{2}−1} and denoting x⃗_{t} = (x_{tA})_{A∈N}, the potential V (x) vanishes on all hyperplanes M(e⃗) = {x | ∀t ∈ D : x⃗_{t} ∈ ℝe⃗}. So, even though the potential V grows to lim_{η→∞} V (ηx) = ∞, for almost all x ∈ ℝ^{d(N^{2}−1)}, this growth at infinity is not confining enough for H to have purely discrete spectrum, as shown in [3].

(b) The fermionic part H_{F} of the Hamiltonian is indefinite, so it doesn’t contribute an obviously confining term to −Δ + V (x). Yet, their sum H restricted to H_{0} is nonnegative and is expected to yield a zero eigenvalue at the bottom of its spectrum, for d = 9. In contrast, if d = 2 and N = 2 then zero is not an eigenvalue of H, as was shown in [5]. A lot of effort was put into the question of existence of zero-energy states in these SU(N)-invariant supersymmetric matrix models given by H ↿_{H_{0}}. The original formulation uses Clifford variables {Θ_{α̂,A}}_{α̂∈Ŝ,A∈N}, {Θ_{α̂,A}, Θ_{β̂,B}} = δ_{α̂β̂}δ_{AB} rather than fermion creation and annihilation operators employed here, where Ŝ:= {1, 2, …, 2(d − 1)} and the relation between Clifford variables and the fermion creation and annihilation operators is the standard one, Θ_{α,A} := \frac{1}{\sqrt{2}}(λ_{αA}^{†} +λ_{αA}) and Θ_{α+d−1,A} := \frac{−i}{\sqrt{2}}(λ_{αA}^{†} −λ_{αA}), for all α ∈ S and A ∈ N. In

terms of these Clifford variables, the Hamiltonian reads H̃ = [−Δ_{x} + V (x)] ⊗ 1 + H̃_{F}, where

H̃_{F} = i f_{ABC} x_{tC} γ_{α̂,β̂}^{t} Θ_{α̂,A} Θ_{β̂,B},  (4)

and (γ^{t})_{t∈D} are real, symmetric 2(d − 1) × 2(d − 1) matrices given by

γ^{t} := ( \begin{matrix}0 & iΓ^{t} \\ −iΓ^{t} & 0 \end{matrix} ) ,  γ^{8} := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,  γ^{9} := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , (5)

with t = 1, 2, …, d − 2 [6]. The matrices (γ^{t})_{t∈D} form a real representation of the Clifford algebra {γ^{s}, γ^{t}} = 2δ_{st} · 1, with s, t ∈ D, of minimal dimension, provided d = 2, 3, 5, 9. The reason for recalling the form H̃ of the Hamiltonian is that H̃ is manifestly Spin(d)-invariant. The fermion creation and annihilation operators leading to (3) correspond to the particular choice (5) of (γ^{t})_{t∈D}. In attempts to construct a ground state explicitly [9], fermion creation and annihilation operators are used rather than Clifford variables. This is so, mainly because they provide the Hilbert space on which the Hamiltonian acts irreducibly from the very beginning. Namely, the creation and annihilation operators, λ_{αA}^{†}, λ_{αA}, with α ∈ S and a ∈ N, form a representation of the canonical anticommutation relations (CAR): {λ_{αA}^{†}, λ_{βB}^{†}} = {λ_{αA}, λ_{βB}} = 0 and {λ_{αA}, λ_{βB}^{†}} = δ_{αβ}δ_{AB}, where the anticommutator is {a, b} := ab + ba. The CAR have an explicit representation as linear operators on the fermion Fock space

F = ⨁_{k=0}^{(d−1)(N^{2}−1)} span {λ_{α_{1},A_{1}}^{†} ⋯ λ_{α_{k},A_{k}}^{†}|0⟩ | α_{j} ∈ S, A_{j} ∈ N }, (6)

which is a complex Hilbert space of dimension 2^{(d−1)(N^{2}−1)}. The vectors {λ_{α_{1},A_{1}}^{†} ⋯ λ_{α_{k},A_{k}}^{†}|0⟩ | α_{j} ∈ S, A_{j} ∈ N } ⊆ F form an orthonormal basis; |0⟩ is called vacuum vector. The Hilbert space H can be viewed as a direct integral

H = ∫_{ℝ^{d(N^{2}−1)}}^{⊕} F dx = L^{2}(ℝ^{d(N^{2}−1)}; F),  (7)

whose elements are linear combinations of the form

Ψ(x) = ∑_{k=0}^{(d−1)(N^{2}−1)} Ψ_{k}(x),  (8)
Ψ_{k}(x) = ∑_{α_{1},...,α_{k}∈S} ∑_{A_{1},...,A_{k}∈N} ψ_{α_{1},A_{1}...,α_{k},A_{k}}^{(k)}(x) λ_{α_{1},A_{1}}^{†} ⋯ λ_{α_{k},A_{k}}^{†}|0⟩. (9)

While the fermion creation and annihilation operators above (and in attempts to explicitly construct a ground state as in [9]) are chosen independently of x, the asymptotic form of the ground state wave function was determined with the help of space-dependent fermions [7, 6, 5]. The analysis of the asymptotic form H^{∞}Ψ = 0 of the solutions of HΨ = 0 leads for N = 2 and d = 2, 3, 5 to absence of a zero-energy states, as proved in [5], since these solutions are not square-integrable at r → ∞, where r > 0 is introduced in (12), below. On the other hand, for d = 9, this asymptotic form of the wave function is squareintegrable at infinity, and it is believed that for d = 9, the supersymmetric matrix models do possess zero enery eigenstates, for all N ∈ ℕ. This belief is supported by the recent existence proof for a related model [4]. Main Results: In this paper we study the asymptotic Hamiltonian H^{∞} described in detail in (16)–(18). The asympotic Hamiltonian H^{∞} = H_{B}^{∞} + H_{F}^{∞} splits into a bosonic part H_{B}^{∞} and a fermionic part H_{F}^{∞}, similar to the full Hamiltonian H. The bosonic Hamiltonian H_{B}^{∞} is a sum of harmonic oscillators and we first focus our attention on the ground state subspace of H_{B}^{∞} with corresponding ground state energy 2(d − 1). This leads us to study the spectral properties of 2(d − 1) + H_{F}^{∞}. We derive a dynamical SU(2) symmetry in (39) and observe the formation of ‘Cooper pairs’ [e.g., in the ground state of 2(d − 1) + H_{F}^{∞} computed in (46) and (48)] that arise in the SO(d)-breaking formulation when diagonalizing certain ingredients of the fermionic part of the Hamiltonian. Thereafter, we transform the zero energy equation on Fock space into a system of graded equations (52) obtained by its natural grading derived from the fermion number. We show that this system of equations can be solved by a recursive insertion (58) of solutions, provided a certain invertibility condition on the graded Hamiltonians hold, which is known to hold true for the first recursion step (54). We finally observe a sum rule for the graded equations and apply this to the asymptotic ground state of s_{d} + H_{F}^{∞} (62)–(64). To ease the reading, we carry out our analysis first in the case N = 2, i.e., for the asymptotic SU(2) theory. In the last section we note that several features extend to the non-asymptotic case and/or to general N ≥ 2. We mostly restrict the dimension d to the most interesting case d = 9.

2 Asymptotic form of the Hamiltonian

The bosonic configuration space is a set of d = 2, 3, 5, or 9 traceless hermitian matrices {X_{s}}_{s=1}^{d} corresponding to the Lie algebra su(N) of the gauge group SU(N). Given selfadjoint generators T⃗ ≡ (T_{A})_{A∈N} of su(N) with [T_{A}, T_{B}] = if_{ABC}T_{C}, the coordinates x derive from expanding X_{t} = x_{tA}T_{A} = x⃗_{t} · T⃗ in these generators. For simplicity, we start by taking N = 2 and 2T⃗ to be the Pauli matrices σ⃗ ≡ (σ_{1}, σ_{2}, σ_{3}), so f_{ABC} = ϵ_{ABC}. The potential, which for general N ≥ 2 is given by

V (x) := −\frac{1}{2} ∑_{s,t∈D} Tr ([X_{s}, X_{t}]^{2}),  (10)

assumes the form

V (x) = \frac{1}{2} ∑_{s,t∈D} (x⃗_{s} ∧ x⃗_{t})^{2},  (11)

where (x⃗ ∧ y⃗)_{A} = ϵ_{ABC} x_{B} y_{C}, as usual. Observe from (11) that V (x) vanishes if x ∈ M := ⋃_{e⃗∈ℝ^{3}} M(e⃗), recalling that M(e⃗) are the hyperplanes M(e⃗) = {x | ∀t ∈ D : x⃗_{t} ∈ ℝe⃗}. We remark that, for N = 2, the condition x ∈ M is even equivalent to V (x) = 0, while a necessary condition for the vanishing of the potential is more complicated for N > 2. We coordinatize x ∈ ℝ^{3d} by (see, e.g., [6, 5])

x⃗_{t} = r E_{t} e⃗ + r^{−1/2} y⃗_{t},  (12)

for t = 1, …, d, where e⃗ ∈ ℝ^{3} and E = (E_{1}, …, E_{d}) ∈ ℝ^{d} are unit vectors, r > 0, and y⃗_{t} ∈ ℝ^{3} are perpendicular to both E and e⃗ in the sense that E_{s}y⃗_{s} = e⃗ · y⃗_{t} = 0, for all t = 1, …, d. They derive from x ∈ ℝ^{3d} \ {0} by the requirement that the euclidean length of the projection x^{∥} of x along ℝ · E ⊗ e⃗ be maximal. Indeed, e⃗ and E are normalized eigenvectors of (x_{tA}x_{tB})_{A,B=1}^{3} and (x_{sA}x_{tA})_{s,t=1}^{d}, respectively, corresponding to the largest eigenvalue r^{2} > 0 which, in turn, is the square of the length r = |x^{∥}| = ⟨E ⊗ e⃗, x⟩ = E_{t}x_{tA}e_{A} of x^{∥}. The component x^{⊥} = x−x^{∥} perpendicular to E⊗e⃗ then yields y⃗_{t} = r^{1/2}x_{t}^{⊥}. Writing E as E(Ẽ, θ, φ) = ( cos θ Ẽ, sin θ cos φ, sin θ sin φ) in spherical coordinates, the coordinate vectors x⃗_{t} assume the form

x⃗_{t} = r cos θ Ẽ_{t} e⃗ + r^{−1/2} y⃗_{t},  (13)

for t = 1, …, d − 2, and

x⃗_{d−1} + ix⃗_{d} = r sin θ e^{iφ} e⃗ + r^{−1/2} (y⃗_{d−1} + iy⃗_{d}),  (14)

where e⃗ ∈ ℝ^{3} and Ẽ = (Ẽ_{1}, Ẽ_{2}, …, Ẽ_{d−2}) ∈ ℝ^{d−2} are unit vectors, θ ∈ (−π/2, π/2), φ ∈ (0, 2π), and r > 0. To derive the asymptotic form of the Hamiltonian (cf. [5, 9]), we substitute (13)–(14) (and the corresponding differentials) into H, divide by r, and obtain

\frac{1}{r} H → H^{∞}  (15)

as the resulting limit, as r → ∞. Note that, while the difference of H/r and H^{∞} is of lower order in r, the limit r → ∞ is formal, as this difference is an unbounded (differential) operator. Moreover, it ignores the question whether the coordinate change (13)–(14) defines a global diffeomorphism. The asymptotic Hamiltonian H^{∞} in (15) is of the form

H^{∞} = H_{B}^{∞} + H_{F}^{∞},  (16)

where the bosonic part

H_{B}^{∞} := −(1 − E_{s}E_{t}) (1 − e_{A}e_{B}) \frac{∂^{2}}{∂y_{sA} ∂y_{tB}} + y_{sA} y_{sA}  (17)

is a sum of harmonic oscillators in the variables y⃗_{t} in all s_{d} = 2(d − 1) spatial directions perpendicular to both E and e⃗_{t}, with ground state energy equal to s_{d} = 2(d − 1) = 2, 4, 8, 16, for d = 2, 3, 5, 9, respectively, and ground state eigenvector Φ_{0}(y⃗_{1}, …, y⃗_{d}) = exp[−(1 − E_{s}E_{t}) (1 − e_{A}e_{B}) y_{sA} y_{tB}]. The fermionic part H_{F}^{∞} of the asymptotic Hamiltonian H^{∞} in (16) results from (3) by inserting (13)–(14), with y⃗_{t} = 0 and r = 1,

H_{F}^{∞} = 2 cos θ (−ie_{C}ϵ_{ABC}) Γ_{αβ} λ_{αA}^{†} λ_{βB}  (18)

+ sin θ e^{iφ} (e_{C}ϵ_{ABC}) λ_{αA}^{†}λ_{αB}^{†} + sin θ e^{−iφ} (e_{C}ϵ_{ABC}) λ_{αB}λ_{αA},

with Γ_{αβ} := ∑_{t=1}^{d−2} Ẽ_{t}Γ_{αβ}^{t}. We henceforth assume the unit vectors Ẽ ∈ ℝ^{d−2}, e⃗ ∈ ℝ^{3} and θ ∈ (−π/2, π/2), φ ∈ (0, 2π) to be fixed. Our goal is to find an explicit unitary transformation on Fock space F which brings H_{F}^{∞} into normal (i.e., particle number preserving) form. It is a well-known fact (see, e.g., [2, 1]) that, since H_{F}^{∞} is quadratic in the creation and annihilation operators, such a unitary exists and is of the form λ_{k}^{†} ↦ u_{k,ℓ}λ_{ℓ}^{†} + v_{k,ℓ}λ_{ℓ}^{†}, i.e., linear. While the existence of such a unitary follows from the diagonalizability of self-adjoint matrices, the determination of that unitary in an explicit and managable form can be difficult and depends on the special properties of the model (here: H_{F}^{∞}) under consideration. Note that both (M_{AB} := −ie_{C}ϵ_{ABC})_{A,B=1,2,3} and (Γ_{αβ})_{α,β=1}^{d−1} are imaginary, antisymmetric, and thus self-adjoint matrices. Since Mv⃗ = ie⃗ ∧ v⃗, an orthonormal basis {e⃗, n⃗_{+}, n⃗_{−}} ⊆ ℂ^{3} of eigenvectors, Me⃗ = 0,

Mn⃗_{±} = ie⃗ ∧ n⃗_{±} = ±n⃗_{±}  (19)

is given by the usual orthonormal dreibein: e⃗ ⊥ n⃗_{+} ⊥ n⃗_{−}. We choose n⃗_{±} = n⃗_{±}(e⃗) to depend continuously on e⃗ and to obey \overline{n⃗_{±}} = n⃗_{∓}. Hence

−ie_{C}ϵ_{ABC} = (n⃗_{+})_{A} \overline{(n⃗_{+})_{B}} − (n⃗_{−})_{A} \overline{(n⃗_{−})_{B}} = (n⃗_{+})_{A} (n⃗_{−})_{B} − (n⃗_{−})_{A} (n⃗_{+})_{B}.

(20)

Similary, for d = 3, 5, 9, we observe that, due to Γ^{2} = Ẽ^{2} · 1 = 1 and Tr Γ = 0, there is an orthonormal basis {ẽ_{σj}| σ = ±, j = 1, …, (d − 1)/2} ⊆ ℂ^{d−1} of eigenvectors of Γ such that

Γẽ_{±j} = ±ẽ_{±j},  (21)

for all j = 1, …, (d − 1)/2. Again we choose Ẽ ↦ ẽ_{±j} continuous and \overline{ẽ_{±j}} = ẽ_{∓j}. So,

Γ_{αβ} = ∑_{j} [(ẽ_{+j})_{α} (ẽ_{−j})_{β} − (ẽ_{−j})_{α} (ẽ_{+j})_{β}],  (22)

where the summation ranges over j = 1, …, (d − 1)/2. Using the orthonormal (eigen)vectors ẽ_{±j}, n⃗_{±}, and n⃗_{0} := e⃗, we define spacedependent fermion creation operators (for d = 3, 5, 9)

λ_{σjτ}^{†} := (ẽ_{σj})_{α} (n⃗_{τ})_{A} λ_{αA}^{†},  (23)

where σ ∈ {+, −}, j = 1, …, (d−1)/2, and τ ∈ {+, −, 0}. Note that the matrix U defined by U_{σjτ,αA} := (ẽ_{σj})_{α} (n⃗_{τ})_{A} is unitary and, hence, λ_{αA}^{†} ↦ λ_{σjτ}^{†} is implemented by a unitary conjugation on (the operators on) Fock space F. I.e., λ_{σjτ} |0⟩ = 0 and λ_{σjτ}^{†} , λ_{σjτ} fulfill the CAR. Using the new creation operators λ_{σjτ}^{†}, we introduce

A_{j}^{†} := ie^{iφ} λ_{+j+}^{†} λ_{−j−}^{†},  (24)
B_{j}^{†} := ie^{−iφ} λ_{−j+}^{†} λ_{+j−}^{†},  (25)

and A_{j} := (A_{j}^{†})^{∗} and B_{j} := (B_{j}^{†})^{∗}, for j = 1, …, (d − 1)/2, which may be considered (Cooper) pair creation and annihilation operators. Note that these operators obey commutation relations somewhat reminiscent to the canonical ones, namely

[A_{j}^{†} , A_{k}^{†}] = [B_{j}^{†} , B_{k}^{†}] = [A_{j}^{†} , B_{k}^{†}] = [A_{j} , B_{k}^{†}] = 0,  (26)
[A_{j} , A_{k}^{†}] = δ_{kj}(N_{j}^{(A)} − 1) := δ_{kj}(λ_{+j+}^{†} λ_{+j+} + λ_{−j−}^{†} λ_{−j−} − 1),(27)
[B_{j} , B_{k}^{†}] = δ_{kj}(N_{j}^{(B)} − 1) := δ_{kj}(λ_{−j+}^{†} λ_{−j+} + λ_{+j−}^{†} λ_{+j−} − 1).(28)

The asymptotic Hamiltonian H^{∞}, when acting on the ground state of H_{B}^{∞}, can be written as

s_{d} + H_{F}^{∞} = H_{0}^{∞} + H_{+}^{∞} + H_{−}^{∞},  (29)

where

H_{0}^{∞} := s_{d} + 2 cos θ ∑_{j}(N_{j}^{(A)} − N_{j}^{(B)}),  (30)
H_{+}^{∞} := 2 sin θ ∑_{j}(A_{j}^{†} + B_{j}^{†}),  (31)
H_{−}^{∞} := 2 sin θ ∑_{j}(A_{j} + B_{j}).  (32)

We remark that the degrees of freedom defined by the parallel fermions λ_{±j0}^{†} = (ẽ_{±j})_{α}e_{A}λ_{αA} do not appear in H_{F}^{∞} and can be dropped, henceforth.

For the d=2 case we instead of (23) define λ_{±}^{†} := (n⃗_{±})_{A}λ_{A}^{†}, and the corresponding expressions for the asymptotic Hamiltonian in (29) are simply

H_{0}^{∞} = 2, H_{+}^{∞} = 2C^{†}, H_{−}^{∞} = 2C, C^{†} := ie^{iφ} λ_{+}^{†} λ_{−}^{†}.  (33)

3 Dynamical Symmetry

For definiteness, we restrict our attention in this and the following sections to the most interesting case: d = 9. Denoting

J_{+} ⊗ 1 := A^{†} := ∑_{j} A_{j}^{†}, 1 ⊗ J_{+} := B^{†} := ∑_{j} B_{j}^{†},  (34)
J_{−} ⊗ 1 := A := ∑_{j} A_{j}, 1 ⊗ J_{−} := B := ∑_{j} B_{j} ,  (35)
J_{3} ⊗ 1 := \frac{1}{2}(N^{(A)} − 4) := \frac{1}{2}( ∑_{j} N_{j}^{(A)} − 4),  (36)
1 ⊗ J_{3} := \frac{1}{2}(N^{(B)} − 4) := \frac{1}{2}( ∑_{j} N_{j}^{(B)} − 4),  (37)

with

[J_{+}, J_{−}] = 2J_{3}, [J_{3}, J_{±}] = ±J_{±}, J_{±} = J_{1} ± iJ_{2},  (38)

Eqs. (29)–(32) can be written as

4 + \frac{1}{4}H_{F}^{∞} = (2 + cos θ J_{3} + sin θ J_{1}) ⊗ 1 + 1 ⊗ (2 − cos θ J_{3} + sin θ J_{1}), (39)

thus exhibiting the dynamical symmetry mentioned above. (We recall that a dynamical symmetry refers to the situation that the Hamiltonian, being one of the generators of a symmetry Lie group, has nontrivial commutation relations with the other symmetry generators rather than commuting with them.)

The relevant SU(2) representations are the tensor product of four spin \frac{1}{2} representations, i.e., direct sums of two singlets [note that both (A_{1}A_{3} + A_{2}A_{4} − A_{1}A_{4} − A_{2}A_{3})|0⟩ and (A_{1}A_{2} + A_{3}A_{4} − A_{1}A_{4} − A_{2}A_{3})|0⟩ are annihilated by A^{†}, A , and \frac{1}{2}(N^{(A)} − 4)], three spin 1 representations, and (most importantly, as providing the zero-energy state of H) one spin 2 representation acting irreducibly on the space spanned by the orthonormal states

|0⟩, \frac{1}{2} A^{†}|0⟩, \frac{1}{\sqrt{24}} (A^{†})^{2}|0⟩, \frac{1}{12} (A^{†})^{3}|0⟩, \frac{1}{4!} (A^{†})^{4}|0⟩.  (40)

Restricting to that space (correspondingly for the B^{†}’s), we can write

J_{+} = [ \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 \\ 0 & \sqrt{6} & 0 & 0 & 0 \\ 0 & 0 & \sqrt{6} & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \end{matrix} ] , J_{−} = [ \begin{matrix}0 & 2 & 0 & 0 & 0 \\ 0 & 0 & \sqrt{6} & 0 & 0 \\ 0 & 0 & 0 & \sqrt{6} & 0 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ] , (41)

J_{3} = [ \begin{matrix}−2 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix} ] .  (42)

Since the spectrum of sin θJ_{1} ± cos θJ_{3} is the same as that of J_{3}, the spectrum of 4 + \frac{1}{4}H_{F}^{∞} clearly consists of all integers between zero and eight,

σ(4 + \frac{1}{4}H_{F}^{∞}) = \{0, 1, 2, …, 8\}.  (43)

Its unique zero-energy state Ψ is most easily obtained by solving individually, for each A_{j}^{†} resp. B_{j}^{†} degree of freedom,

(1 ± cos θ σ_{3}^{(j)} + sin θ σ_{1}^{(j)}) Ψ = e^{∓ \frac{1}{2} θiσ_{2}^{(j)}}(1 ± σ_{3}^{(j)})e^{± \frac{1}{2} θiσ_{2}^{(j)}}Ψ =^{!} 0, (44)

where we identify

2 J_{k} = σ_{k} ⊗ 1 ⊗ 1 ⊗ 1 + … + 1 ⊗ 1 ⊗ 1 ⊗ σ_{k} ≡ ∑_{j=1}^{4} σ_{k}^{(j)}.  (45)

In our notation σ_{3}^{(j)}|0⟩ = −|0⟩ and σ_{3}^{(j)}A_{j}^{†}|0⟩ = +A_{j}^{†}|0⟩, and we easily find the solution to (44) as

Ψ = ( ∏_{j} e^{− \frac{θ}{2} iσ_{2}^{(j)}})( ∏_{j} e^{\frac{θ}{2} iσ_{2}^{(j)}}B_{j}^{†})|0⟩ = \frac{1}{4!} e^{−θi(J_{2}⊗1−1⊗J_{2})} (B^{†})^{4}|0⟩. (46)

Using the nilpotency of A_{j}^{†} and B_{j}^{†} for

e^{α(A_{j}^{†}−A_{j})}|0⟩ = cos α e^{tan α A_{j}^{†}}|0⟩ and e^{−α(B_{j}^{†}−B_{j})}B_{j}^{†}|0⟩ = sin α e^{cot α B_{j}^{†}}|0⟩,

(47)

the ground state can also be written as

Ψ =  (48)
= \frac{1}{16} e^{−4iφ} (sin θ)^{−4} ∏_{j} \{( sin θ − (1 − cos θ)A_{j}^{†})( sin θ − (1 + cos θ)B_{j}^{†})\}|0⟩

= \frac{1}{16} e^{−4iφ} (sin θ)^{4} exp [ − \frac{1 − cos θ}{sin θ} A^{†} − \frac{1 + cos θ}{sin θ} B]|0⟩ ∼ e^{−C_{θ}}|0⟩,

with C_{θ} := \frac{1−cos θ}{sin θ} (J_{+} ⊗ 1) + \frac{1+cos θ}{sin θ} (1 ⊗ J_{+}). Alternatively, one can solve the 2 × 2 matrix eigenvector equations resulting from (44),

(1 + cos θ (N_{j}^{(A)} − 1) + sin θ (A_{j}^{†} + A_{j}))Ψ = 0,  (49)
(1 − cos θ (N_{j}^{(B)} − 1) + sin θ (B_{j}^{†} + B_{j}))Ψ = 0,  (50)

to obtain (48). For d=2 the asymptotic ground state is easily found from (33),

Ψ = \frac{1}{\sqrt{2}}e^{−C^{†}}|0⟩ = \frac{1}{\sqrt{2}}(1 − C^{†})|0⟩.  (51)

An interesting feature of the form (46) for the ground state is that it expresses it as a spin-rotation by an angle θ applied to some reference state (B^{†})^{4}|0⟩ (which itself also varies in the first d − 2 directions in space according to (13), (23), (21)).

4 Graded chain of Hamiltonians

We henceforth drop the superscript “∞” and write H_{0} = H_{0}^{∞}, H_{+} = H_{+}^{∞}, and H_{−} = H_{−}^{∞}. Consider the grade- resp. fermion number-ordered equations

H_{0}Ψ_{0} + H_{−}Ψ_{2} = 0,
H_{+}Ψ_{0} + H_{0}Ψ_{2} + H_{−}Ψ_{4} = 0,
H_{+}Ψ_{2} + H_{0}Ψ_{4} + H_{−}Ψ_{8} = 0,
⋮
H_{+}Ψ_{12} + H_{0}Ψ_{14} + H_{−}Ψ_{16} = 0,
H_{+}Ψ_{14} + H_{0}Ψ_{16}  = 0,

(52)

which we obtain by writing Ψ = ∑_{n=0}^{16} Ψ_{n}, requiring (N^{(A)}+N^{(B)})Ψ_{n} = n Ψ_{n} and the ground state equation

(16 + H_{F}^{∞})Ψ = (H_{0} + H_{+} + H_{−})(Ψ_{0} + Ψ_{2} + … + Ψ_{16}) =^{!} 0.  (53)

(Recall that we have dropped the eight non-dynamical parallel fermions λ_{±j0}^{†} = (ẽ_{±j})_{α}e_{A}λ_{αA}.) The following method to construct the ground state we believe to be relevant also for the fully interacting, non-asymptotic theory. We use the first equation in (52) to express Ψ_{0} in terms of Ψ_{2},

Ψ_{0} = −H_{0}^{−1} H_{−} Ψ_{2}.  (54)

H_{0} is certainly invertible on the zero-fermion subspace, even in the full theory (cf. [9]). Inserting (54), the second equation in (52) can be written as

H_{2}Ψ_{2} + H_{−}Ψ_{4} = 0, with H_{2} := H_{0} − H_{+} H_{0}^{−1} H_{−},  (55)

which yields

Ψ_{2} = −H_{2}^{−1} H_{−} Ψ_{4},  (56)

provided H_{2} is invertible on H_{−}Ψ_{4}, resp. the two-fermion sector of Fock space. Continuing in this manner, denoting

ℋ̂_{2k} := span {(A^{†})^{m}(B^{†})^{n}|0⟩}_{m,n=0,1,2,3,4, m+n=k}  (57)

for the considered 2k-fermion subspace, we find that if we assume invertibility of H_{2k} on ℋ̂_{2k} we can form

H_{2(k+1)} := H_{0} − H_{+}H_{2k}^{−1}H_{−}  (58)

on ℋ̂_{2(k+1)} and solve for Ψ_{2k} in terms of Ψ_{2(k+1)}. The final equation for Ψ_{16} is H_{16}Ψ_{16} = 0. For concreteness, denote an orthonormal basis of ℋ̂ = ⊕_{k}ℋ̂_{2k} by |k, l⟩ := |k⟩ ⊗ |l⟩, where, as in (40),

|k⟩ := \frac{1}{k!\sqrt{\binom{4}{k}}}J_{+}^{k}|0⟩.  (59)

Then, e.g., H_{+}H_{0}^{−1}H_{−} acts on ℋ̂‘tridiagonally’ according to

\frac{1}{sin^{2} θ} H_{+}H_{0}^{−1}H_{−}|k, l⟩ =

(\frac{k(5 − k)}{4 + (k − l − 1) cos θ} + \frac{l(5 − l)}{4 + (k − l + 1) cos θ} )|k, l⟩

+ \frac{\sqrt{l(5 − l)(k + 1)(4 − k)}}{4 + (k − l + 1) cos θ}|k + 1, l − 1⟩

+ \frac{\sqrt{k(5 − k)(l + 1)(4 − l)}}{4 + (k − l − 1) cos θ}|k − 1, l + 1⟩.  (60)

Calculating the spectra of H_{2k} on ℋ̂_{2k} (e.g., with the help of a computer) one can verify the invertibility of all H_{2k} on ℋ̂_{2k} for k < 8, while H_{16} is identically zero on ℋ̂_{16} . Hence, one can also start with the state Ψ_{16} ∼ A^{4}B^{4}|0⟩ (with correct normalization in θ) and generate the lower grade parts of the full asymptotic ground state Ψ using the relations (54), (56), etc. We finish this section by noting a simple consequence of the graded form (52) of the ground state equation HΨ = 0 (for general d and N). Taking the inner product of the grade 2k-equation with Ψ_{2k} yields

⟨Ψ_{2k}, H_{−}Ψ_{2(k+1)}⟩ = −⟨H_{0}⟩_{2k} − ⟨Ψ_{2k}, H_{+}Ψ_{2(k−1)}⟩
= −⟨H_{0}⟩_{2k} − \overline{⟨Ψ_{2(k−1)}, H_{−}Ψ_{2k}⟩},  (61)

Documenta Mathematica 13 (2008) 103–116

where ⟨H_{0}⟩_{2k} := ⟨Ψ_{2k}, H_{0}Ψ_{2k}⟩. The first equation reads ⟨Ψ_{0}, H_{−}Ψ_{2}⟩ = −⟨H_{0}⟩_{0} which is real. The second then becomes ⟨Ψ_{2}, H_{−}Ψ_{4}⟩ = −⟨H_{0}⟩_{2} + ⟨H_{0}⟩_{0}, and so on, so that in the last step one obtains

∑_{k=0}^{Λ}(−1)^{k}⟨H_{0}⟩_{2k} = 0,  (62)

where Λ is the total number of fermions in the relevant Fock space. It is instructive to verify (62) for the asymptotic N = 2 case studied above, since there all relevant terms can be calculated explicitly. Using the basis (59) and the notation α := 1 − cos θ, β := 1 + cos θ, we find

Ψ ∼ e^{−C_{θ}}|0⟩ = ∑_{k} \frac{(−1)^{k}\sqrt{\binom{4}{k}}}{(sin θ)^{k}} α^{k} |k⟩ ⊗ ∑_{l} \frac{(−1)^{l}\sqrt{\binom{4}{l}}}{(sin θ)^{l}} β^{l} |l⟩.  (63)

Hence,

⟨Ψ_{2n}, H_{0}Ψ_{2n}⟩ = \frac{1}{64}(sin θ)^{8−2n} ∑_{k+l=n} \binom{4}{k}\binom{4}{l}(4+(k −l) cos θ) α^{2k} β^{2l}. (64)

5 General SU(N)

We now compute the ground state energy of

H̃_{F} = i f_{ABC} x_{tC} γ_{α̂,β̂}^{t} Θ_{α̂,A} Θ_{β̂,B},  (65)

for general N ≥ 2 and in regions of the configuration space where the potential V is zero (see eqs. (2), (3), and recalling that f_{ABC} denote the structure constants of SU(N) in an orthonormal basis). By (10), the vanishing of the potential V means that all X_{s} are commuting, hence can be written X_{s} = UD_{s}U^{†} where U is unitary and independent of s and the D_{s} are diagonal. If we look into a particular direction (which corresponds to fixing e⃗ in the SU(2) case) and choose a basis {T_{A}} accordingly, we may write X_{s} = D_{s} = x_{sA}T_{A} = x_{sk̃}Tk̃ and x_{sa} = 0, where k̃ = 1, …, N − 1 are indices in the Cartan subalgebra and a, b = N, …, N^{2} − 1 denote the remaining indices. Denoting the eigenvalues of X_{t} by μ_{k}^{t}, i.e., X_{t} = diag(μ_{1}^{t}, …, μ_{N}^{t}), the eigenvectors {e_{kl}}_{k≠l} of M_{ab}^{t} := −if_{abC}x_{tC} = −if_{abk̃}x_{tk̃} satisfy (cf. e.g. [8])

M^{t}e_{kl} = (μ_{k}^{t} − μ_{l}^{t})e_{kl} =: μ_{kl}^{t}e_{kl}, (e_{kl}^{a})^{∗} = e_{lk}^{a}.  (66)

The crucial observation is that these eigenvectors are independent of t. Now,

H̃_{F} = −γ_{α̂,β̂}^{t} M_{ab}^{t} Θ_{α̂,A} Θ_{β̂,B}, = W_{α̂a,β̂b} Θ_{α̂,a} Θ_{β̂,b},  (67)

where W := − ∑_{t} γ^{t} ⊗ M^{t}. From the above observations we have the ansatz E_{μkl} := v_{μ} ⊗ e_{kl} for the eigenvectors of W, yielding

W E_{μkl} = − ∑_{t} γ^{t} v ⊗ M^{t} e_{kl} = γ(k, l) v_{μ} ⊗ e_{kl},  (68)

where γ(k, l) := − ∑_{t} μ_{kl}^{t}γ^{t} squares to ∑_{t}(μ_{kl}^{t})^{2}. Denoting by v_{μ} = v_{±jkl} the corresponding 16 eigenvectors of γ(k, l), we find

W E_{±jkl} = ±\sqrt{∑_{t} (μ_{kl}^{t})^{2}} E_{±jkl},  (69)

and H̃_{F} therefore has

E_{0} := −16 ∑_{k<l} \sqrt{∑_{t=1}^{9}(μ_{k}^{t} − μ_{l}^{t})^{2}}  (70)

as its lowest eigenvalue. This agrees with the following two previously known cases: [8], where only X_{9} is assumed to have large eigenvalues so that E_{0} → −16 ∑_{k<l} |μ_{k}^{9} − μ_{l}^{9}|; as well as the SU(2)-case studied above and in [7], where (13) with, e.g., e_{A} = δ_{A3} gives E_{0} = −16r. Note also [10], where the eigenvalues of H̃_{F} are stated, with the SU(N) symmetry fixed.

Acknowledgements

One of us (J.H.) would like to thank Choonkyu Lee for his hospitality, and Ki-Myeong Lee for useful discussions.

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V. Bach Department of Mathematics Mainz University DE-55099 Mainz Germany vbach@mathematik.uni-mainz.de J. Hoppe Department of Mathematics Royal Institute of Technology SE-10044 Stockholm Sweden hoppe@math.kth.se

D. Lundholm

Department of Mathematics Royal Institute of Technology SE-10044 Stockholm Sweden dogge@math.kth.se