Unresolved problems in the quantization of generalized (higher-rank) spin systems and their G/H sigma-model continuum limits

Randjbar-Daemi, Salam and Strathdee [Phys. Rev. B 48, 3190 (1993)] generalized SU(2) lattice spin systems to site variables generating an arbitrary compact Lie group G and derived their continuum limits: non-relativistic sigma models on coset spaces G/H (H containing the maximal torus of G) with two independent G-invariant tensors, a Haldane-type topological term in one dimension, and a Holstein-Primakoff-type operator realization for SU(N+1) on CP^N. The concluding section of that work posed five unresolved issues: (i) the operator realization covers only irreducible representations labeled by the single quantum number n_1-n_2, not the general case (n_1,n_2), and other groups; (ii) the significance of factor-ordering ambiguities in the continuum limit is unknown; (iii) it is unknown whether the two tensors k_{μν} and g_{μν} flow under renormalization to the relativistic form k_{μν}=c^2 g_{μν} at fixed points; (iv) the topological total-derivative term has no known interpretation for more than one spatial dimension; (v) rigorous SU(2) results on long-range order, ground states and critical behavior have not been generalized to higher-rank systems.

Background

A conventional quantum spin system places generators of SU(2) — the angular-momentum operators — on every site of a lattice and couples them by exchange interactions. Its low-energy theory is organized by two approximations. In the large-spin or correspondence limit the spin operators become classical vectors on the sphere $S^2=SU(2)/U(1)$, and combined with the long-wavelength limit this yields the O(3) nonlinear $\sigma$-model (a field theory whose fields are constrained to a curved target manifold). For one-dimensional antiferromagnets the Euclidean action further acquires Haldane's topological term — a total-derivative contribution, invisible to the classical equations of motion, whose quantized coefficient distinguishes integer-spin from half-integer-spin chains. At the operator level the bridge between lattice and semiclassical expansion is the Holstein-Primakoff transformation, the standard spin-wave representation of the SU(2) generators in terms of a single bosonic oscillator mode; the ordering of its noncommuting operator factors controls the first quantum corrections and is fixed by requiring the exact spin algebra.

Randjbar-Daemi, Salam and Strathdee extended this framework to 'spin' systems whose site variables generate an arbitrary compact Lie group $G$. In the correspondence limit the classical dynamics lives on a homogeneous space $G/H$ whose stability group $H$ contains the maximal torus of $G$ (a maximal set of mutually commuting generators); for groups of rank greater than one the generic targets are flag manifolds, iterated bundles such as $SU(N)/U(1)\times\cdots\times U(1)$, rather than symmetric spaces — a symmetric space being a coset $G/H$ whose denominator is the fixed-point set of an involutive automorphism, such as the sphere or complex projective space $\mathbb{CP}^N$. Two structural results of that work frame the questions below. First, the antiferromagnetic continuum limit is a $\sigma$-model in which spatial gradients couple to one $G$-invariant rank-2 tensor $k_{\mu\nu}$ while time derivatives couple to another, $g_{\mu\nu}$; a symmetric space admits only one invariant tensor up to scale, so its model is automatically relativistic ($k_{\mu\nu}=c^2g_{\mu\nu}$), whereas a flag manifold admits several, so the generic generalized model is non-relativistic. Second, in one spatial dimension a total-derivative topological term survives and generalizes Haldane's: it distinguishes representations whose contravariant weight components $\Lambda^j$ (integer labels generalizing the spin $s$) are odd, which inherit the half-integer-spin alternating sign in the sum over configurations.

What the program had not achieved, per its own concluding section: an operator realization covering all irreducible representations — the one constructed for $SU(N+1)$ on $\mathbb{CP}^N$ works only for representations with non-degenerate weights, labeled by the single quantum number $n_1-n_2$ rather than the full pair $(n_1,n_2)$; a resolution of the factor-ordering ambiguities, whose operator-valued contributions define what the authors call the sub-correspondence regime, the layer of $1/\Lambda$ corrections between the quantum lattice problem and the classical limit; the renormalization-group behavior of the two-tensor models; the status of the topological term when the number of spatial dimensions $D$ exceeds one, where it has no known topological interpretation and singular configurations — hedgehogs, point defects where the continuum field is undefined (monopoles of the $\mathbb{CP}^1$ field in the SU(2) case) — had been argued to enter only through possible Berry-phase weights; and, finally, generalizations of the rigorous SU(2) results: the Lieb-Schultz-Mattis theorem (a constraint forbidding a trivially gapped, featureless ground state for half-integer-spin chains), theorems on long-range antiferromagnetic order, and the renormalization-group phase diagram of the two-dimensional Heisenberg antiferromagnet by Chakravarty, Halperin and Nelson. For SU(N), the square-lattice antiferromagnet had been studied at large $N$ by Read and Sachdev; for SU(2) in two dimensions, Dombre and Read showed that the Hopf-type invariant is absent from the long-wavelength action. Beyond these specific cases the $D>1$ question, and the general higher-rank questions, stood open.

Problem Statement

The problem is to complete the quantization program for generalized (higher-rank) spin systems and their $G/H$ $\sigma$-model continuum limits, as posed in the concluding section of Randjbar-Daemi, Salam and Strathdee, Phys. Rev. B 48, 3190 (1993). Five linked issues constitute one program — turning the classical continuum formalism into a controlled quantum theory:

1. Operator realization: construct a Holstein-Primakoff-type realization of the generators of $SU(N+1)$ — and of other compact groups — on the coset coordinates of $\mathbb{CP}^N$ that covers all irreducible representations, including the degenerate-weight cases labeled by the full pair $(n_1,n_2)$ and not only by $n_1-n_2$, so that physically interesting Hamiltonians can be expressed as well-defined, self-adjoint, $G$-invariant operators.

2. Factor ordering: determine whether the factor-ordering contributions — the operator-valued terms that distinguish the sub-correspondence regime on the lattice — are significant in the continuum limit, or whether they can be discarded as in dimensional-regularization treatments of $\sigma$-models; equivalently, whether the quantized continuum theory needs them (and the related total-derivative terms) for the long-wavelength description of the lattice system.

3. Renormalization group: determine the long-wavelength behavior of the quantized two-tensor $\sigma$-models, in particular whether the independent tensors $k_{\mu\nu}$ and $g_{\mu\nu}$ flow to the relativistic form $k_{\mu\nu}=c^2g_{\mu\nu}$ at fixed points. The one-loop, $(2+\epsilon)$-dimensional, flag-manifold analysis already given by the authors settles this question only perturbatively near two dimensions and for flag manifolds; the open question is the corresponding fixed-point behavior in the physical $2+1$ dimensions, nonperturbatively, for other cosets, and from generic non-relativistic initial conditions. These four respects define the general case the authors' analysis leaves open.

4. Topological term in $D>1$: establish whether the total-derivative term (Eq. (4.23) of the source) has any significance — topological or physical, including through singular hedgehog-type configurations — for generalized antiferromagnets in more than one spatial dimension, beyond the settled SU(2) case (Dombre-Read) and the large-$N$ SU(N) case (Read-Sachdev).

5. Rigorous results: generalize the rigorous SU(2) results on long-range order, ground-state properties and critical behavior to these higher-rank spin systems (one-dimensional Lieb-Schultz-Mattis-type constraints now exist; the two- and three-dimensional questions remain).

These are one problem because they are the successive layers of a single objective: the operator realization and the ordering resolution are what make the continuum quantum theory well defined, the renormalization group is its long-wavelength solution, the topological analysis fixes its configuration sum, and rigorous results anchor its physical content. The parts that later literature has already settled perturbatively or in one dimension — the authors' one-loop $(2+\epsilon)$-dimensional flag-manifold analysis for the renormalization-group issue, the $\mathrm{SU}(2)$ and large-$N$ $\mathrm{SU}(N)$ cases for the topological-term issue, and the one-dimensional Lieb-Schultz-Mattis-type constraints for the rigorous-results issue — narrow the surviving core without dissolving it. Relevant references cited below do not settle the remaining issues: the oscillator realizations on coadjoint orbits (ref. 5), within the general boson-realization framework (ref. 18), cover only the minimal and maximal orbits, and neither they nor the quaternionic-unitary construction (ref. 6) or the coherent-state ordering analysis (ref. 19) supply the explicit all-representation realization of issue 1 or resolve the lattice $G/H$ ordering question of issue 2.

References

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