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arXiv:2302.13126 · Journal of Physics A Mathematical and Theoretical

Goldstone modes and the golden spiral in the ferromagnetic spin-1 biquadratic model

Huan-Qiang Zhou, Qian‐Qian Shi, Ian P. McCulloch, Murray T. Batchelor

Published Sep 10, 2025• 0 citations
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Abstract

Abstract Ferromagnetic ground states have often been overlooked in comparison to seemingly more interesting antiferromagnetic ground states. However, both the physical and mathematical structure of ferromagnetic ground states are particularly rich. We show that the highly degenerate and highly entangled ground states of the ferromagnetic spin-1 biquadratic model are scale invariant, originating from spontaneous symmetry breaking from SU ( 3 ) to U ( 1 ) × U ( 1 ) with two type-B Goldstone modes (GMs) if the system size is even or from SU ( 2 ) to U ( 1 ) with one type-B GM if the system size is odd, when periodic boundary conditions (PBCs) are adopted. The ground state degeneracies are characterized as Fibonacci–Lucas sequences, under open and PBCs, with nonzero residual entropy per site. This implies that the ground state degeneracies for this model are asymptotically the golden spiral. In addition, sequences of atypical (periodic) degenerate ground states generated from highest and generalized highest weight states are constructed to establish that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit. The prefactor is half the number of type-B GMs, which is identified to be the fractal dimension, if one is restricted to atypical degenerate ground states. We also argue that the same conclusion is valid for typical (non-periodic) degenerate ground states, as long as the block size is sufficiently large.

DOI
10.1088/1751-8121/ae05d9

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Abstract Ferromagnetic ground states have often been overlooked in comparison to seemingly more interesting antiferromagnetic ground states. However, both the physical and mathematical structure of ferromagnetic ground states are particularly rich. We show that the highly degenerate and highly entangled ground states of the ferromagnetic spin-1 biquadratic model are scale invariant, originating from spontaneous symmetry breaking from SU ( 3 ) to U ( 1 ) × U ( 1 ) with two type-B Goldstone modes (GMs) if the system size is even or from SU ( 2 ) to U ( 1 ) with one type-B GM if the system size is odd, when periodic boundary conditions (PBCs) are adopted. The ground state degeneracies are characterized as Fibonacci–Lucas sequences, under open and PBCs, with nonzero residual entropy per site. This implies that the ground state degeneracies for this model are asymptotically the golden spiral. In addition, sequences of atypical (periodic) degenerate ground states generated from highest and generalized highest weight states are constructed to establish that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit. The prefactor is half the number of type-B GMs, which is identified to be the fractal dimension, if one is restricted to atypical degenerate ground states. We also argue that the same conclusion is valid for typical (non-periodic) degenerate ground states, as long as the block size is sufficiently large.
Type-B Goldstone modes and a logarithmic spiral in the staggered SU(4)\rm SU(4)SU(4) ferromagnetic spin-orbital model
It is found that the staggered SU(4)\rm SU(4)SU(4) ferromagnetic spin-orbital model accommodates highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. The spontaneous symmetry breaking patterns are SU(4)→U(1)×U(1)×U(1){\rm SU(4)} \rightarrow {\rm U(1)} \times {\rm U(1)} \times {\rm U(1)}SU(4)→U(1)×U(1)×U(1), with three type-B Goldstone modes or SO(4)∼SU(2)×SU(2)→U(1)×U(1){\rm SO(4)} \sim {\rm SU(2)} \times {\rm SU(2)} \rightarrow {\rm U(1)} \times {\rm U(1)}SO(4)∼SU(2)×SU(2)→U(1)×U(1), with two type-B Goldstone modes, depending on the system size being even or odd. An abstract fractal constitutes the underlying structure of the ground-state subspace. For a sequence of atypical degenerate ground states the fractal dimension is identified with the number of type-B Goldstone modes. This connection is established by evaluating the entanglement entropy for these atypical degenerate ground states. The observed universal finite system-size scaling behavior of the entanglement entropy follows a logarithmic scaling relation with the block size in the thermodynamic limit. In addition, the ground state degeneracies, depending on the boundary conditions adopted, constitute the two Fibonacci-Lucas sequences. In the limit of large system size their asymptotic forms become a self-similar logarithmic spiral. As a result, the model has a non-zero residual entropy $S_{r} = -2 \ln R $, where R=( ⁣6− ⁣2)/2R=(\! \sqrt{6}-\!\sqrt{2})/2R=(6​−2​)/2.
The one-dimensional flat-band ferromagnetic Tasaki model exhibits spontaneous symmetry breaking from SU(2){\rm SU}(2)SU(2) to U(1){\rm U}(1)U(1) with one type-B Goldstone mode, featuring that the highest weight state is entangled at quarter filling, but there is always a choice to keep the highest weight state unentangled away from quarter filling. It is found that the ground-state degeneracies under both periodic and open boundary conditions constitute essentially the Fibonacci sequences, behaving asymptotically as the golden spiral - a self-similar geometric object. A set of orthonormal basis states are generated from the repeated action of the lowering operator of the symmetry group SU(2){\rm SU}(2)SU(2) on the highest weight state at a specific filling. In particular, it is possible to construct the orthonormal basis states reflecting an abstract fractal underlying the ground-state subspace, which are permutation-invariant away from quarter filling, but not at quarter filling. As a consequence, there exists a singularity that accounts for the emergence of the saturated flat-band ferromagnetism at quarter filling. We perform a systematic finite system-size scaling analysis of the entanglement entropy, thus confirming that it scales logarithmically with the block size in the thermodynamic limit, with the prefactor being half the number of type-B Goldstone modes, for the orthonormal basis states at and away from quarter filling.
The one-dimensional spin-sSU(2)s\phantom{\rule{4pt}{0ex}}\mathrm{SU}(2)sSU(2) ferromagnetic Heisenberg model, as a paradigmatic example for spontaneous symmetry breaking (SSB) with type-B Goldstone modes (GMs), is expected to exhibit an abstract fractal underlying the ground-state subspace. This intrinsic abstract fractal is here revealed from a systematic investigation into the entanglement entropy for a linear combination of factorized (unentangled) ground states on a fractal decomposable into a set of the Cantor sets. The entanglement entropy scales logarithmically with the block size, with the prefactor being half the fractal dimension of a fractal, as first noted by Castro-Alvaredo and Doyon for the spin-1/21/21/2 ferromagnetic Heisenberg model [Phys. Rev. Lett. 108 (2012) 120401], as long as the norm for the linear combination scales as the square root of the number of the self-similar building blocks kept at each step kkk for a fractal, under an assumption that the maximum absolute value of the coefficients in the linear combination is chosen to be around one, and the coefficients in the linear combination are almost constants within the building blocks. Actually, the set of the fractal dimensions for all the Cantor sets forms a dense subset in the interval [0,1]. As a consequence, the ground-state subspace is separated into a disjoint union of countably infinitely many regions, each of which is labeled by a decomposable fractal. Hence, the interpretation of the prefactor as half the fractal dimension is valid for any support beyond a fractal, which in turn leads to the identification of the fractal dimension with the number of type-B GMs for the orthonormal basis states. Our argument may be extended to any quantum many-body systems undergoing SSB with type-B GMs.
For a quantum many-body spin system undergoing spontaneous symmetry breaking with type-B Goldstone modes, a high degree of degeneracy arises in the ground state manifold. Generically, if this degeneracy is polynomial in system size, then it does not depend on the type of boundary conditions used. However, if there exists an emergent (local) symmetry operation tailored to a specific degenerate ground state, then we show that the degeneracies are exponential in system size and are different under periodic boundary conditions (PBCs) and open boundary conditions (OBCs). We further show that the exponential ground state degeneracies in turn imply the emergence of Goldstone flat bands -- single-mode excitations generated by a multi-site operator and its images under the repeated action of the translation operation under PBCs or the cyclic permutation symmetry operation under OBCs. Conversely, we also show that the presence of emergent Goldstone flat bands implies that there exists an emergent (local) symmetry operation tailored to a specific degenerate ground state. In addition, we propose an extrinsic characterization of emergent Goldstone flat bands, revealing a connection to quantum many-body scars, which violate the eigenstate thermalization hypothesis. We illustrate this by presenting examples from the staggered SU(4){\rm SU}(4)SU(4) spin-1 ferromagnetic biquadratic model and the staggered SU(4){\rm SU}(4)SU(4) ferromagnetic spin-orbital model. We also perform extensive numerical simulations for the more general SO(3){\rm SO}(3)SO(3) spin-1 bilinear-biquadratic and SO(4){\rm SO(4)}SO(4) ferromagnetic spin-orbital models, containing the two aforementioned models as the endpoints in the ferromagnetic regimes respectively, and confirm the emergence of Goldstone flat bands, as we approach these endpoints from deep inside the ferromagnetic regimes.
Spontaneous symmetry breaking (SSB) with type-B Goldstone modes is investigated in the macroscopically degenerate phase for a quantum spin-1 many-body system with competing dimer and trimer interactions. The SSB involves three distinct patterns. The first occurs at the dimer point, with the pattern from staggered SU(3){\rm SU}(3)SU(3) to U(1)×U(1){\rm U}(1)\times{\rm U}(1)U(1)×U(1). The second occurs at the trimer point, with the pattern from uniform SU(3){\rm SU}(3)SU(3) to U(1)×U(1){\rm U}(1)\times{\rm U}(1)U(1)×U(1). The third occurs in the dimer-trimer regime, with the pattern from uniform SU(2){\rm SU}(2)SU(2) to U(1){\rm U}(1)U(1). The number of type-B Goldstone modes is thus two, two and one for the three patterns, respectively. The ground state degeneracies arising from the three patterns are exponential with the system size, which may be recognized as sequences of integers relevant to self-similar logarithmic spirals. This in turn is attributed to the presence of an emergent symmetry operation tailored to a specific degenerate ground state. As a consequence, the residual entropy is non-zero, which measures the disorder present in a unit cell of highly degenerate ground state generated from a generalized highest weight state. An exact Schmidt decomposition exists for the highly degenerate ground states, thus exposing the self-similarities underlying an abstract fractal, described by the fractal dimension. The latter is extracted from performing a universal finite system-size scaling analysis of the entanglement entropy, which is identical to the number of type-B Goldstone modes. The model under investigation thus accommodates an exotic scale invariant quantum state of matter.
Spontaneous symmetry breaking with type-B Goldstone modes in the SO(2s+12s+12s+1) ferromagnetic model: an entanglement perspective
Spontaneous symmetry breaking with type-B Goldstone modes is investigated in the SO(2s+12s+12s+1) ferromagnetic model. A set of orthonormal basis states in the ground state subspace are constructed, which admit an exact Schmidt decomposition, exposing self-similarities in real space of an abstract fractal underlying the ground state subspace. Focusing on the SO(5) and the SO(6) ferromagnetic spin chains as illustrative examples, finite system-size scaling analysis of the entanglement entropy for this set of orthonormal basis states confirms that the entanglement entropy scales logarithmically with block size in the thermodynamic limit. The prefactor in front of the logarithm is half the number of type-B Goldstone modes NBN_BNB​, which is identified as the fractal dimension dfd_fdf​ for these orthonormal basis states. For the SO(2s+12s+12s+1) ferromagnetic model NB=df=sN_B = d_f =sNB​=df​=s for integer sss and NB=df=s+1/2N_B = d_f =s+1/2NB​=df​=s+1/2 for half-odd-integer sss.