We study the S=1/2 spin chain with the Hamiltonian H=${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}\mathrm{\ensuremath{-}}1}$ {${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{x}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{x}}$+${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{y}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{y}}$ +\ensuremath{\lambda}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{z}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}$} +\ensuremath{\beta} ${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}$ (1-${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}$\ensuremath{\cdot}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}$), where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$=(${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{x}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{y}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{z}}$) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden ${\mathit{Z}}_{2}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{Z}}_{2}$ symmetry of the system. It has been argued that a similar hidden ${\mathit{Z}}_{2}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{Z}}_{2}$ symmetry of the S=1 chain is fully broken when and only when the system exhibits the Haldane gap. We prove that the present system exhibits both an excitation gap and a full breaking of the hidden ${\mathit{Z}}_{2}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{Z}}_{2}$ symmetry in a range of the parameter space including the line \ensuremath{\beta}=0, \ensuremath{\lambda}>-1. We argue that the range with such properties indeed extends to the limit \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty} in which the present model reduces to the S=1 spin chain. This observation provides support of Hida's conclusion that the Haldane gap in the S=1 chain is continuously connected to the gap in the decoupled S=1/2 system with \ensuremath{\beta}=0.We study the S=1/2 spin chain with the Hamiltonian H=${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}\mathrm{\ensuremath{-}}1}$ {${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{x}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{x}}$+${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{y}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{y}}$ +\ensuremath{\lambda}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{z}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}$} +\ensuremath{\beta}${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}$ (1-${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}$\ensuremath{\cdot}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}$), where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$=(${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{x}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{y}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{z}}$) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry of the system. It has been argued that a similar hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry of the S=1 chain is fully broken when and only when the system exhibits the Haldane gap. We prove that the present system exhibits both an excitation gap and a full breaking of the hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry in a range of the parameter space including the line \ensuremath{\beta}=0, \ensuremath{\lambda}>-1. We argue that the range with such properties indeed extends to the limit \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty} in which the present model reduces to the S=1 spin chain. This observation provides support of Hida's conclusion that the Haldane gap in the S=1 chain is continuously connected to the gap in the decoupled S=1/2 system with \ensuremath{\beta}=0.

Physical review. B, Condensed matterAug 1, 199285 citations

Spin order and spin excitation spectra of spin-1/2 tetramer chains

Junli Li, Jun-Qing Cheng, Trinanjan Datta +1 more

Abstract We investigate the spin dynamics of a 1D spin-1/2 Heisenberg tetramer chain. Employing a combination of density matrix renormalization group, quantum renormalization group, and perturbation theory techniques, we compute the energy levels and the quantum phase diagram, analyze the phase transitions, and evaluate the dynamical structure factor (DSF), which is related to the inelastic neutron scattering (INS) spectrum and the resonant inelastic x-ray scattering (RIXS) spectrum of fractionalized and collective excitations. Our calculations suggest that the chain can transition between a hidden Z 2 × Z 2 discrete symmetry preserving tetramer phase and a Haldane phase with non-vanishing string order that breaks the hidden symmetry. These two gapped phases are intervened by an intermediate deconfined quantum critical state comprising of free spins and three-site doublets, which is a gapless critical phase with deconfined spinons that can be detected either by INS or L -edge RIXS. We find that the tetramer chain can support fractionalized (spinon) and collective (triplon and quinton) excitations. In the ferromagnetic intra-tetramer limit, the chain can support a quinton excitation which has a five-fold degenerate excited state. String order parameter calculations suggest CuInVO 5 should be in a Haldane-like phase whose INS or L -edge RIXS spectrum can support observable single-spin excitations including triplon and quinton excitations. We also identify possible two-particle excitations (two-singlon, two-triplon, triplon-quinton, and two-quinton excitations) resulting from the double spin-flip effect in the K -edge RIXS spectrum.Abstract We investigate the spin dynamics of a 1D spin-1/2 Heisenberg tetramer chain. Employing a combination of density matrix renormalization group, quantum renormalization group, and perturbation theory techniques, we compute the energy levels and the quantum phase diagram, analyze the phase transitions, and evaluate the dynamical structure factor (DSF), which is related to the inelastic neutron scattering (INS) spectrum and the resonant inelastic x-ray scattering (RIXS) spectrum of fractionalized and collective excitations. Our calculations suggest that the chain can transition between a hidden Z 2 × Z 2 discrete symmetry preserving tetramer phase and a Haldane phase with non-vanishing string order that breaks the hidden symmetry. These two gapped phases are intervened by an intermediate deconfined quantum critical state comprising of free spins and three-site doublets, which is a gapless critical phase with deconfined spinons that can be detected either by INS or L -edge RIXS. We find that the tetramer chain can support fractionalized (spinon) and collective (triplon and quinton) excitations. In the ferromagnetic intra-tetramer limit, the chain can support a quinton excitation which has a five-fold degenerate excited state. String order parameter calculations suggest CuInVO 5 should be in a Haldane-like phase whose INS or L -edge RIXS spectrum can support observable single-spin excitations including triplon and quinton excitations. We also identify possible two-particle excitations (two-singlon, two-triplon, triplon-quinton, and two-quinton excitations) resulting from the double spin-flip effect in the K -edge RIXS spectrum.

arXiv:2504.00095npj Quantum MaterialsMar 18, 20260 citations

Hidden Magnetic Order in Quantum Spin Chains

Changfeng Chen

Bulletin of the American Physical SocietyMar 23, 20050 citations

Hidden Z2 × Z2 symmetry breaking in Haldane-gap antiferromagnets

Tom Kennedy, Hal Tasaki

We show that the Haldane phase of the S=1 antiferromagnetic chain is closely related to the breaking of a hidden ${\mathit{Z}}_{2}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{Z}}_{2}$ symmetry. When the chain is in the Haldane phase, this ${\mathit{Z}}_{2}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{Z}}_{2}$ symmetry is fully broken, but when the chain is in a massive phase other than the Haldane phase, e.g., the Ising phase or the dimerized phase, this symmetry is broken only partially or not at all. The hidden ${\mathit{Z}}_{2}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{Z}}_{2}$ symmetry is revealed by introducing a nonlocal unitary transformation of the chain. This unitary transformation also leads to a simple variational calculation which qualitatively reproduces the phase diagram of the S=1 chain.We show that the Haldane phase of the S=1 antiferromagnetic chain is closely related to the breaking of a hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry. When the chain is in the Haldane phase, this ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry is fully broken, but when the chain is in a massive phase other than the Haldane phase, e.g., the Ising phase or the dimerized phase, this symmetry is broken only partially or not at all. The hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry is revealed by introducing a nonlocal unitary transformation of the chain. This unitary transformation also leads to a simple variational calculation which qualitatively reproduces the phase diagram of the S=1 chain.

Physical review. B, Condensed matterJan 1, 1992476 citations

{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}

(1-${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}$\ensuremath{\cdot}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}$), where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$=(${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{x}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{y}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{z}}$) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden

{\mathit{Z}}_{2}