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arXiv:2304.11339

Entanglement entropy for scale-invariant states: universal finite-size scaling

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

Published Apr 22, 2023• 2 citations
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Abstract

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 ${\rm SU}(2)$ spin-$1/2$ ferromagnetic states. The resulting universal scaling form is demonstrated for three fundamental models -- the ${\rm SU}(2)$ spin-$s$ Heisenberg ferromagnetic model, the ${\rm SU}(N+1)$ ferromagnetic model, and the staggered ${\rm SU}(3)$ 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.

DOI
10.48550/arxiv.2304.11339

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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.
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.
Exact matrix product state representations for a type of scale-invariant states are presented, which describe highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes in one-dimensional quantum many-body systems. As a possible application, such a representation offers a convenient but powerful means for evaluating the norms of highly degenerate ground states. This in turn allows us to perform a universal finite system-size scaling analysis of the entanglement entropy. Moreover, this approach vividly explains why the entanglement entropy does not depend on what types of the boundary conditions are adopted, either periodic boundary conditions or open boundary conditions. Illustrative examples include the SU(2){\rm SU}(2)SU(2) spin-sss Heisenberg ferromagnetic 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.
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.
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.
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.
We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge ccc in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of ccc. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.