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Huan-Qiang Zhou

Publications and source records attributed to Huan-Qiang Zhou.

At least 19 recordsLinked to original sources
arXiv2026

Type-B Goldstone modes and a logarithmic spiral in the staggered $\rm SU(4)$ ferromagnetic spin-orbital model

It is found that the staggered $\rm SU(4)$ 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 ${\rm SU(4)} \rightarrow {\rm U(1)} \times {\rm U(1)} \times {\rm U(1)}$, with three type-B Goldstone modes or ${\rm SO(4)} \sim {\rm SU(2)} \times {\rm SU(2)} \rightarrow {\rm U(1)} \times {\rm U(1)}$, 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 $R=(\! \sqrt{6}-\!\sqrt{2})/2$.

cond-mat.stat-mech↗
arXiv2026

Green parafermions as emergent flat-band excitations in condensed matter

Green parafermions, originally introduced by Green and extended by Greenberg and Messiah through trilinear and relative trilinear commutation relations beyond Bose-Fermi statistics, are generally regarded as mathematical curiosities without physical realization. We show that these paraparticles can in fact emerge as composite excitations in a broad class of condensed-matter systems undergoing spontaneous symmetry breaking with type-B Goldstone modes. The key ingredient is the introduction of auxiliary Majorana fermions defined on emergent unit cells produced by partial translational-symmetry breaking. When the auxiliary Majoranas are treated as physical degrees of freedom, the resulting Green parafermion states (up to a projection operator) correspond to flat-band excitations, whose creation and annihilation operators satisfy the trilinear algebra. When they are regarded as fictitious, the same construction explains the appearance of exponentially many degenerate ground states and reveals a surprising correspondence between Green parafermions and self-similar geometric objects, such as the golden spiral. Explicit realizations are demonstrated for the ferromagnetic spin-1 biquadratic model and the ferromagnetic $\rm {SU}(2)$ flat-band Tasaki model, showing that condensed-matter systems with type-B Goldstone modes provide a natural setting for Green parafermions as emergent, possibly observable quasiparticles.

cond-mat.str-el↗
arXiv2026

Symmetry group factorization and unitary equivalence among Temperley-Lieb integrable models

It is shown that there is a hidden connection between the two well-studied sequences of the Temperley-Lieb (TL) integrable models -- the $q$-state quantum Potts (QP) models at the self-dual points and the staggered ${\rm SU}(n)$ spin-$s$ chains with $n=2s+1$ ($s \ge 1$), in addition to the uniform ${\rm SU}(2)$ spin-$1/2$ Heisenberg model. For each sequence, symmetry group factorization arises, in the sense that if $q$ is factorized into $q_1$ and $q_2$, then the $q$-state QP model is unitarily equivalent to a combined QP model with the symmetry group ${\rm S}_{q_1} \times {\rm S}_{q_2}$ or if $n$ is factorized into $n_1$ and $n_2$, then the staggered ${\rm SU}(n)$ spin-$s$ chain with the symmetry group ${\rm SU}(n)$ is unitarily equivalent to a combined staggered ${\rm SU}(n_1) \times {\rm SU}(n_2)$ spin chain with the symmetry group ${\rm SU}(n_1) \times {\rm SU}(n_2)$, valid for both ferromagnetic (FM) and antiferromagnetic (AF) cases. Moreover, the FM (AF) staggered ${\rm SU}(n)$ spin-$s$ chain is unitarily equivalent to the AF (FM) $q$-state QP model with $q=n^2$, as long as the size of the AF (FM) staggered ${\rm SU}(n)$ spin-$s$ chain is doubled. A combination of the two distinct types of unitary equivalences yields a family of models such that they are essentially identical, but appear in different guises. Some physical implications for unitary equivalence among different TL integrable models are clarified.

cond-mat.str-el↗
arXiv2026

Non-invertible symmetries and boundary conditions for the transverse-field Ising model

Non-invertible Kramers-Wannier (KW) duality symmetries are constructed for the transverse-field Ising model (TFIM) at the self-dual point under various boundary conditions (BCs), as long as the resultant Hamiltonian commutes with the ${\rm Z}_2$ symmetry operator. This is achieved by introducing extra degrees of freedom into the Hilbert space, in order to turn a non-translation-invariant Hamiltonian in the original Hilbert space into a translation-invariant Hamiltonian in the augmented Hilbert space. One may lift the trivial identity operator, the ${\rm Z}_2$ symmetry operator and the non-invertible KW duality symmetry operator to their counterparts in the augmented Hilbert space, valid for each of four types of toroidal BCs. As it turns out, they yield a lattice version of fusion rules, which bears a resemblance to the Tambara-Yamagami ${\rm Z}_2$ fusion category. Our construction is thus consistent with the basic physical requirement that all possible BCs should yield a converging result in the thermodynamic limit. In particular, the lattice versions of fusion rules, constructed by Seiberg, Seifnashri and Shao [SciPost Phys. \textbf{16}, 154 (2024)], are reproduced for periodic and anti-periodic BCs, but a discrepancy is revealed for duality-twisted BCs.

cond-mat.str-el↗
arXiv2025

Fractal dimension and the counting rule of the Goldstone modes

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 [O. A. Castro-Alvaredo and B. Doyon, Phys. Rev. Lett. \textbf{108}, 120401 (2012)], 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.

cond-mat.str-el↗
arXiv2025

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

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 ${\rm SU}(3)$ to ${\rm U}(1)\times {\rm U}(1)$ with two type-B Goldstone modes if the system size is even or from ${\rm SU}(2)$ to ${\rm U}(1)$ with one type-B Goldstone mode if the system size is odd, when periodic boundary conditions are adopted. The ground state degeneracies are characterized as Fibonacci-Lucas sequences, under open and periodic boundary conditions, 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 Goldstone modes, 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.

cond-mat.str-el↗
arXiv2025

Fractals and spontaneous symmetry breaking with type-B Goldstone modes: a perspective from entanglement

The one-dimensional spin-$s$ ${\rm SU}(2)$ 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 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 $k$ 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 {\it 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.

quant-ph↗
arXiv2025

Spontaneous symmetry breaking with type-B Goldstone modes in the SO($2s+1$) ferromagnetic model: an entanglement perspective

Spontaneous symmetry breaking with type-B Goldstone modes is investigated in the SO($2s+1$) 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 $N_B$, which is identified as the fractal dimension $d_f$ for these orthonormal basis states. For the SO($2s+1$) ferromagnetic model $N_B = d_f =s$ for integer $s$ and $N_B = d_f =s+1/2$ for half-odd-integer $s$.

cond-mat.str-el↗
arXiv2025

Emergent Goldstone flat bands and spontaneous symmetry breaking with type-B Goldstone modes

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 ${\rm SU}(4)$ spin-1 ferromagnetic biquadratic model and the staggered ${\rm SU}(4)$ ferromagnetic spin-orbital model. We also perform extensive numerical simulations for the more general ${\rm SO}(3)$ spin-1 bilinear-biquadratic and ${\rm SO(4)}$ 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.

cond-mat.str-el↗
arXiv2024

Entanglement entropy for the one-dimensional flat-band ferromagnetic Tasaki model: spontaneous symmetry breaking with one type-B Goldstone mode

The one-dimensional flat-band ferromagnetic Tasaki model exhibits spontaneous symmetry breaking from ${\rm SU}(2)$ to ${\rm U}(1)$ 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 ${\rm SU}(2)$ 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.

cond-mat.str-el↗
arXiv2024

Entanglement entropy for a type of scale-invariant states in two spatial dimensions and beyond: universal finite-size scaling

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 ${\rm SU}(2)$ spin-$s$ ferromagnetic Heisenberg model, the ${\rm SU}(2s+1)$ ferromagnetic model, and the staggered ${\rm SU}(3)$ spin-1 ferromagnetic biquadratic model.

cond-mat.stat-mech↗
arXiv2024

Quantum entanglement entropy and Tomonaga-Luttinger liquid to liquid transition in biquadratic spin-1 XY chain with rhombic single-ion anisotropy

Quantum phase transitions (QPTs) are investigated in biquadratic spin-$1$ XY chain with rhombic single-ion anisotropy by using the ground state energy (GE), the bipartite entanglement entropy (BEE), and the mutual information (MI). It turns out that there are three spin nematic phases and two Tomonaga-Luttinger (TL) liquid phases with the central charge $c = 1$. The TL Liquid phases emerge roughly for biquadratic interaction strength two times stronger than the absolute value of the single-ion anisotropy. The GE and the derivatives up to the second order reveal a first-order QPT between spin nematic ferroquarupole (FQ) phases but cannot capture an evident signal of QPTs between the spin nematic phases and the TL Liquid phases as well as QPT between the two TL liquid phases. The TL liquid-to-liquid transition point features a highly degenerate state and the spin-block entanglement entropy increases logarithmically with block size. The BEE exhibits a divergent or convergent behavior identifying the TL Liquid or spin nematic FQ phases, respectively. Similarly, the MI and the spin-spin correlation are shown to decay algebraically or exponentially with increasing the lattice distance in the TL Liquid or spin nematic FQ phases, respectively. In the TL liquid phase, the exponents $η_I$ and $η_z$ of the MI and the spin-spin correlation vary with the interaction parameter of the biquadratic interaction strength and the rhombic single-ion anisotropy and satisfy the relationship of $η_z <η_I$. Such changes of characteristic behavior of the BEE, the MI and the spin-spin correlation indicate an occurrence of the Berezinskii-Kosterlitz-Thouless (BKT)-type QPT between the TL Liquid phase and the spin nematic FQ phase. The staggered spin fluctuation $\langle S^x S^y \rangle$ is shown to play a significant role for the emergence of the TL liquid phase and thus give rise to the BKT-type QPT.

cond-mat.str-el↗
arXiv2024

Wigner-Yanase skew information, quantum entanglement and spin nematic quantum phase transitions in biquadratic spin-1 and spin-2 XY chains with single-ion anisotropies

Quantum phase transitions (QPTs) between uniaxial or biaxial spin nematic (SN) phases are investigated in biquadratic spin-1 and spin-2 XY infinite chains with the rhombic- and uniaxial-type single-ion anisotropies. Systematic discussions of distinctive singular behaviors are made to classify various types of QPT from one SN state to the other SN state in using the Wigner-Yanase skew information (WYSI), the bipartite entanglement entropy (BEE), and the quadrupole moments (QMs). For the spin-1 system with the three uniaxial SN phases, we find that a discontinuous QPT, signaled by discontinuous behaviors of all the considered WYSI, BEE, and QMs, occurs from the z-ferroquadrupole phase (FQP) to the x- or y-FQPs, while a continuous QPT occurs between the x- and y-FQPs. The central charge in the continuous QPT line is estimated as $c \simeq 1$ from the BEE. Compared to the spin-1 system, depending on a given strength of the uniaxial-type single-ion anisotropy, the spin-2 system undergoes four different types of QPTs between the two biaxial SN phases as the rhombic-type anisotropy varies: the quantum crossovers, connecting the two orthogonal biaxial SN states adiabatically without an explicit phase transition, the continuous and the discontinuous QPTs, and the SN to magnetic transitions via the antiferromagnetic phase (AFP). In a sharp contrast to the spin-1 system, for the transitions between the two biaxial SN phases, the discontinuous transition line is classified as a topological phase characterized by a doubly degenerate entanglement spectrum and a string order parameter defined by the Cartan generator of the $\mathrm{SO}(5)$ symmetry group in spin-2 systems, while the continuous QPT is advocated by the central charge $c \simeq 1$. Whereas the QPT lines with $c \simeq 1/2$ indicate that the transition between the biaxial SN phase and the AFP belongs to the Ising universality class.

cond-mat.str-el↗
arXiv2024

Exact matrix product state representations for a type of scale-invariant states

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 ${\rm SU}(2)$ spin-$s$ Heisenberg ferromagnetic model, the ${\rm SU}(2s+1)$ ferromagnetic model, and the staggered ${\rm SU}(3)$ spin-1 ferromagnetic biquadratic model.

cond-mat.str-el↗
arXiv2024

Entanglement and logarithmic spirals in a quantum spin-1 many-body system with competing dimer and trimer interactions

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 ${\rm SU}(3)$ to ${\rm U}(1)\times{\rm U}(1)$. The second occurs at the trimer point, with the pattern from uniform ${\rm SU}(3)$ to ${\rm U}(1)\times{\rm U}(1)$. The third occurs in the dimer-trimer regime, with the pattern from uniform ${\rm SU}(2)$ to ${\rm U}(1)$. 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.

cond-mat.str-el↗
arXiv2023

Absence of a critical nematic phase in the vicinity of the $\rm {SU}(3)$ ferromagnetic point for the one-dimensional spin-1 bilinear-biquadratic model

The absence of a critical nematic phase in the vicinity of the $\rm {SU}(3)$ ferromagnetic point for the one-dimensional spin-1 bilinear-biquadratic model is demonstrated by means of the tensor network algorithms. As it turns out, the phase transition from the ferromagnetic phase to the dimerized phase at the $\rm {SU}(3)$ ferromagnetic point is direct, but not of the first-order. The transition point features highly degenerate ground states, which are scale but not conformally invariant, with the fractal dimension being equal to 2. The conceptual developments in effective field theories - the fractal dimension and the counting rule of the Goldstone modes - play a pivotal role in clarifying the numerical artifacts arising from the finiteness of the bond dimension in the tensor network simulations, which are attributed to a proximity effect to a highly entangled scale or conformally invariant ground state.

cond-mat.str-el↗
arXiv2023

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

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.

cond-mat.stat-mech↗
arXiv2023

Instability of the Luttinger liquids towards an exotic quantum state of matter with highly degenerate ground states: an anisotropic extension of the ferromagnetic spin-1 biquadratic model

An extensive investigation, both numerical and analytical, is performed for an anisotropic extension of the ferromagnetic spin-1 biquadratic model. The ground state phase diagram accommodates three symmetry-protected trivial phases, three coexisting fractal phases and six Luttinger liquid phases. A novel universality class arises from an instability of a Luttinger liquid towards an exotic quantum state of matter with infinitely degenerate ground states. The latter in turn is a scale-invariant quantum state of matter, which may be attributed to the coexistence of ${\rm SU}(2)$ spontaneous symmetry breaking with one type-B Goldstone mode on the characteristic line: $J_y=J_z$, and ${\rm U}(1)$ spontaneous symmetry breaking without any gapless Goldstone mode on the characteristic line $J_x/J_z=0$, together with their cyclic permutations with respect to $x$, $y$ and $z$.

cond-mat.str-el↗