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Murray T. Batchelor

Publications and source records attributed to Murray T. Batchelor.

At least 19 recordsLinked to original sources
arXiv2026

Exact Matching-Polynomial Solution of the Periodic Baxter-Fendley $Z_N$ Clock Chain

The periodic non-Hermitian Baxter-Fendley $Z_N$ clock chain has lacked a complete finite-size spectral solution, whereas its open-chain counterpart admits a solution in terms of independent quasienergies. For the periodic model we show that the operator-valued matching polynomial associated with its cyclic Weyl algebra simultaneously generates a set of conserved quantities, including the Hamiltonian, and realizes a cyclic $\tau^{(2)}$ Yang-Baxter transfer matrix. Root-of-unity closure yields a finite system of polynomial spectral equations in each charge sector, which reproduces the complete finite-size energy spectrum counted with algebraic multiplicity. As a first application of this result, we show that Newton continuation of these equations provides a practical numerical route to the periodic ground-state energy without enumerating the full spectrum. For homogeneous chains the thermodynamic seam response yields a criterion for boundary-induced criticality; for $N=3$ it predicts two reciprocal critical couplings with singular ground-state curvature, in contrast to the single self-dual open boundary critical point.

quant-ph↗
arXiv2026

Exact Solution for Non-Hermitian Free Fermions: A Case Study of the XY Chain

We consider the non-Hermitian XY spin chain with open boundary conditions when the anisotropy parameter is extended to complex values. By analyzing the quasi-Hamiltonian matrix, we demonstrate that the free-fermion structure of the quasi-energy spectrum coincides with that of the Hermitian model and construct the corresponding biorthogonal fermionic basis away from exceptional points (EPs). We make use of an explicit Chebyshev-polynomial representation of the open-boundary eigenvectors in which the quasi-energy $\varepsilon$ is the natural spectral variable. This quasi-energy polynomial form is particularly useful at EPs, because EPs correspond to repeated roots of the same boundary polynomial, making the construction of generalized eigenvectors by $\varepsilon$-differentiation transparent. At EPs, where the quasi-Hamiltonian becomes defective, we derive the Jordan normal form and construct the associated generalized eigenvectors, which yields the correct counting of independent many-body eigenstates. We further show that EPs act as branch points in the complex anisotropy plane, leading to the characteristic permutation of eigenenergies and eigenstates upon encirclement. The branch-cut structure of the biorthogonal eigenstates provides direct evidence for the exchange of eigenstates when an EP is encircled. These results provide an analytically controlled many-body platform for studying EP physics and non-Hermitian topology beyond momentum-space descriptions.

quant-ph↗
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↗
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

Exceptional point rings and $PT$-symmetry in the non-Hermitian XY model

The XY spin chain is a paradigmatic example of a model solved by free fermions, in which the energy eigenspectrum is built from combinations of quasi-energies. In this article we show that by extending the XY model's anisotropy parameter $λ$ to complex values, it is possible for two of the quasi-energies to become degenerate. In the non-Hermitian XY model these quasi-energy degeneracies give rise to exceptional points (EPs) where two of the eigenvalues and their corresponding eigenvectors coalesce. The distinct $λ$ values at which EPs appear form concentric rings in the complex plane which are shown in the infinite system size limit to converge to the unit circle coinciding with the boundary between distinct topological phases. The non-Hermitian model is also seen to possess a line of broken $PT$ symmetry along the pure imaginary $λ$-axis. For finite systems, there are four EP values on this broken $PT$-symmetric line if the system size is a multiple of 4.

quant-ph↗
arXiv2025

The planar parafermion algebra: The $\mathbb{Z}_{N}$ clock model and the coupled Temperley-Lieb algebra

The Hamiltonian of the $N$-state clock model is written in terms of a coupled Temperley-Lieb (TL) algebra defined by $N-1$ types of TL generators. This generalizes a previous result for $N=3$ obtained by J. F. Fjelstad and T. M\r{a}nsson [J. Phys. A {\bf 45} (2012) 155208]. The $\mathbb{Z}_{N}$-symmetric clock chain Hamiltonian expressed in terms of the coupled TL algebra generalizes the well known correspondence between the $N$-state Potts model and the TL algebra. The algebra admits a pictorial description in terms of a planar algebra involving parafermionic operators attached to $n$ strands. A key ingredient in the resolution of diagrams is the string Fourier transform. The pictorial presentation also allows a description of the Hilbert space. We also give a pictorial description of the representation related to the staggered XX spin chain. Just as the pictorial representation of the TL algebra has proven to be particularly useful in providing a visual and intuitive way to understand and manipulate algebraic expressions, it is anticipated that the pictorial representation of the coupled TL algebra may lead to further progress in understanding various aspects of the $\mathbb{Z}_{N}$ clock model, including the superintegrable chiral Potts model.

cond-mat.stat-mech↗
arXiv2025

Characterizing phase transitions and criticality in non-Hermitian extensions of the XY model

In this work we study non-Hermitian extensions of the paradigmatic spin-1/2 XY chain in a magnetic field. Using the mapping of the model to free fermion form, we provide analytical insights into the energy spectrum of the non-Hermitian model and establish an intrinsic connection between the quasienergies and topological invariants. We also use exact diagonalization as a supplementary method to examine the performance of biorthogonal-based expectation values. Our results confirm that the theoretical analysis is consistent with the numerical results, with the extended phase diagram determined via the analytical solution and the critical behavior of the fidelity and entanglement. The entanglement transition goes hand in hand with the non-Hermitian topological phase transition. Like the Hermitian case, we analyze the critical behavior using finite-size scaling. Our results show that non-Hermiticity can induce the system into a new universality class with unusual critical exponent. We also emphasize the ability of the Loschmidt echo to characterize potential phase transitions and introduce the average of the Loschmidt echo to describe phase transitions in non-Hermitian systems.

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↗
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

Topological analysis of the complex SSH model using the quantum geometric tensor

This paper presents two methods for topological analysis of the complex Hermitian Su-Schrieffer-Heeger (SSH) model using the quantum geometric tensor: Berry phase and topological data analysis. We demonstrate how both methods can effectively generate topological phase diagrams for the model, revealing two distinct regions based on the relative magnitudes of the parameters $|v|$ and $|w|$. Specifically, when $|v| > |w|$, the system is found to be topologically trivial, whereas for $|v| < |w|$, it exhibits topologically non-trivial behavior. Our results contribute to building the groundwork for topological analysis of more complicated SSH-type models.

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

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

PT-symmetric quantum Rabi model

In this work, we explore the PT-symmetric quantum Rabi model, which describes a PT-symmetric qubit coupled to a quantized light field. By employing the adiabatic approximation (AA), we are able to solve this model analytically in the parameter regime of interest and analyze various physical aspects. We investigate the static and dynamic properties of the model, using both the AA and numerical diagonalization. Our analysis reveals a multitude of exceptional points (EPs) that are closely connected with the exactly solvable points in the Hermitian counterpart of the model. Intriguingly, these EPs vanish and revive depending on the light-matter coupling strength. Furthermore, we discuss the time evolution of physical observables under the non-Hermitian Hamiltonian. Rich and exotic behaviors are observed in both strong and ultra-strong coupling regimes. Our work extends the theory of PT symmetry into the full quantum light-matter interaction regime and provides insights that can be readily enlarged to a broad class of quantum optical systems.

quant-ph↗
arXiv2023

A Brief History of Free Parafermions

In this article we outline the historical development and key results obtained to date for free parafermionic spin chains. The concept of free parafermions provides a natural N-state generalization of free fermions, which have long underpinned the exact solution and application of widely studied quantum spin chains and their classical counterparts. In particular, we discuss the Baxter-Fendley free parafermionic Z(N) spin chain, which is a relatively simple non-Hermitian generalization of the Ising model.

cond-mat.stat-mech↗
arXiv2023

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↗
arXiv2023

Exceptional Points in the Baxter-Fendley Free Parafermion Model

Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a simple generalisation of this idea to $Z(N)$-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but $PT$-symmetric model directly generalising the Ising chain, which was much later recognised by Fendley to be a free parafermion spectrum. By extending the model's magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general $PT$-symmetric at these exceptional points, their proximity can have a profound impact on the model on the $PT$-symmetric real line. Furthermore, in certain cases of the model an exceptional point may appear on the real line (with negative field).

cond-mat.stat-mech↗
arXiv2023

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

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.

cond-mat.str-el↗