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arXiv:2309.04953

Extracting the number of type-B Goldstone modes and the dynamical critical exponent for a type of scale-invariant states

Huan-Qiang Zhou, Yan-Wei Dai, Qianqian Shi, Ian P. McCulloch, Murray T. Batchelor

Published Sep 10, 2023• 0 citations
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Abstract

A generic scheme is proposed to perform a finite-entanglement scaling analysis for scale-invariant states, which appear to be highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. This allows us to extract the number of type-B Goldstone modes and the dynamical critical exponent, in combination with a finite block-size scaling analysis, from numerical simulations of quantum many-body systems in the context of tensor network representations. The number of type-B Goldstone modes is identical to the fractal dimension, thus reflecting an abstract fractal underlying the ground state subspace. As illustrative examples, we investigate the spin-$s$ Heisenberg ferromagnetic model, the $\rm{SU}(3)$ ferromagnetic model and the $\rm{SO}(4)$ spin-orbital model.

DOI
10.48550/arxiv.2309.04953

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A generic scheme is proposed to perform a finite-entanglement scaling analysis for scale-invariant states, which appear to be highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. This allows us to extract the number of type-B Goldstone modes and the dynamical critical exponent, in combination with a finite block-size scaling analysis, from numerical simulations of quantum many-body systems in the context of tensor network representations. The number of type-B Goldstone modes is identical to the fractal dimension, thus reflecting an abstract fractal underlying the ground state subspace. As illustrative examples, we investigate the spin-sss Heisenberg ferromagnetic model, the SU(3)\rm{SU}(3)SU(3) ferromagnetic model and the SO(4)\rm{SO}(4)SO(4) spin-orbital model.
A generic scheme is proposed to investigate the entanglement entropy for a type of scale-invariant states, valid for orthonormal basis states in the ground state subspace of quantum many-body systems undergoing spontaneous symmetry breaking with type-B Goldstone modes in two spatial dimensions and beyond. It is argued that a contribution from the area law to the entanglement entropy is absent, since the closeness to the boundary between a subsystem and its environment is not well-defined, given that a permutation symmetry group with respect to the unit cells of degenerate ground state wave functions emerges. Three physical constraints imposed lead to a universal finite-system size scaling function in the dominant logarithmic contribution to the entanglement entropy. As a result, an abstract fractal underlying the ground state subspace is revealed, characterized by the fractal dimension. The latter in turn is identical to the number of type-B Goldstone modes for the orthonormal basis states. The prediction is numerically confirmed for the SU(2){\rm SU}(2)SU(2) spin-sss ferromagnetic Heisenberg model, the SU(2s+1){\rm SU}(2s+1)SU(2s+1) ferromagnetic model, and the staggered SU(3){\rm SU}(3)SU(3) spin-1 ferromagnetic biquadratic model.
Abstract It is argued that there are a set of orthonormal basis states, which appear as highly degenerate ground states arising from spontaneous symmetry breaking with a type-B Goldstone mode, and they are scale-invariant, with a salient feature that the entanglement entropy S(n) scales logarithmically with the block size n in the thermodynamic limit. As it turns out, the prefactor is half the number of type-B Goldstone modes N B . This is achieved by performing an exact Schmidt decomposition of the orthonormal basis states, thus unveiling their self-similarities in the real space—the essence of a fractal. Combining with a field-theoretic prediction (Castro-Alvaredo and Doyon 2012 Phys. Rev. Lett. 108 120401), we are led to the identification of the fractal dimension d f with the number of type-B Goldstone modes N B for the orthonormal basis states in quantum many-body systems undergoing spontaneous symmetry breaking.
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 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.
A universal finite system-size scaling analysis of the entanglement entropy is presented for highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes in exactly solvable one-dimensional quantum many-body systems. These states appear to be scale-invariant, but not conformally invariant. Our findings are based on a physical argument, imposing three constraints on the entanglement entropy, in addition to further confirmation from an asymptotic analysis of the entanglement entropy for the SU(2){\rm SU}(2)SU(2) spin-1/21/21/2 ferromagnetic states. The resulting universal scaling form is demonstrated for three fundamental models -- the SU(2){\rm SU}(2)SU(2) spin-sss Heisenberg ferromagnetic model, the SU(N+1){\rm SU}(N+1)SU(N+1) ferromagnetic model, and the staggered SU(3){\rm SU}(3)SU(3) spin-1 ferromagnetic biquadratic model. The results point towards a classification for distinct types of scale-invariant states, relevant to a complete classification of quantum states 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.