We study symmetric gapped ground states of one-dimensional spin $S$ chains with a continuous exponential symmetry $U(1)\rtimes\mathbb Z_2$ with integer base $q$ , where the U(1) charge has a modulated unit of charge $q^{L-j}$ decaying exponentially with site $j$ . The simplest model is a quantum breakdown model that can be viewed as a $q$ -deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on $S$ and $q$ . For integer $S$ , odd $q$ allows a nontrivial $\mathbb Z_2$ Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even $q$ allows no nontrivial SPT phase. For half-odd integer $S$ , odd $q$ prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even $q$ enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.We study symmetric gapped ground states of one-dimensional spin $S$ chains with a continuous exponential symmetry $U(1)\rtimes\mathbb Z_2$ with integer base $q$, where the U(1) charge has a modulated unit of charge $q^{L-j}$ decaying exponentially with site $j$. The simplest model is a quantum breakdown model that can be viewed as a $q$-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on $S$ and $q$. For integer $S$, odd $q$ allows a nontrivial $\mathbb Z_2$ Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even $q$ allows no nontrivial SPT phase. For half-odd integer $S$, odd $q$ prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even $q$ enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.

cond-mat.str-elCross-listedarXiv:2610.10682Oct 9, 20260 citations

Quantum phase transitions in integer spin chains

A. Zheludev

Bulletin of the American Physical SocietyMar 16, 20060 citations

Stacking-induced symmetry-protected topological phase transitions

Sang‐Jun Choi, Björn Trauzettel

We study symmetry-protected topological (SPT) phase transitions induced by stacking two gapped one-dimensional subsystems in the BDI symmetry class. The topological invariant of the entire system is a sum of three topological invariants: two from each subsystem and an emerging topological invariant from the stacking. We find that any symmetry-preserving stacking of topologically trivial subsystems can drive the entire system into a topologically nontrivial phase for a certain coupling strength. We explain this intriguing SPT phase transition by conditions set by orbital degrees of freedom and time-reversal symmetry. To exemplify the SPT transition, we provide a concrete model which consists of an atomic chain and a spinful nanowire with spin-orbit interaction and $s$ -wave superconducting order. The stacking-induced SPT transition drives this heterostructure into a zero-field topological superconducting phase.We study symmetry-protected topological (SPT) phase transitions induced by stacking two gapped one-dimensional subsystems in the BDI symmetry class. The topological invariant of the entire system is a sum of three topological invariants: two from each subsystem and an emerging topological invariant from the stacking. We find that any symmetry-preserving stacking of topologically trivial subsystems can drive the entire system into a topologically nontrivial phase for a certain coupling strength. We explain this intriguing SPT phase transition by conditions set by orbital degrees of freedom and time-reversal symmetry. To exemplify the SPT transition, we provide a concrete model which consists of an atomic chain and a spinful nanowire with spin-orbit interaction and $s$-wave superconducting order. The stacking-induced SPT transition drives this heterostructure into a zero-field topological superconducting phase.

arXiv:2302.07633Physical review. B./Physical review. BJun 7, 20237 citations

Sequential Topological Phase Transitions of J 1 - J 2 Integer Spin Chain

Shota Fubasami, Yasuhiro Hatsugai

Bulletin of the American Physical SocietyMar 8, 20180 citations

Symmetry-protected quantum phase transition in topological insulators

L. -F. Liu, X. L. Zhang, Su-Peng Kou

The European Physical Journal BAug 1, 20121 citation

Symmetry-protected topological orders of one-dimensional spin systems with D2+T symmetry

Zheng-Xin Liu, Xie Chen, Xiao-Gang Wen

In Liu et al. [Phys. Rev. B 84, 075135 (2011)] we studied eight gapped symmetric quantum phases in $S=1$ spin chains/ladders which respect a discrete spin rotation ${D}_{2}\ensuremath{\subset}SO(3)$ and time reversal $T$ symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of one-dimensional (1D) integer spin systems with ${D}_{2}+T$ symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in $S=1$ spin chain or spin ladder models. The gapped symmetric phases protected by subgroups of ${D}_{2}+T$ are also studied. Again, all these phases can be distinguished by physically measuring their end ``spins.''In Liu et al. [Phys. Rev. B 84, 075135 (2011)] we studied eight gapped symmetric quantum phases in $S=1$ spin chains/ladders which respect a discrete spin rotation ${D}_{2}\ensuremath{\subset}SO(3)$ and time reversal $T$ symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of one-dimensional (1D) integer spin systems with ${D}_{2}+T$ symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in $S=1$ spin chain or spin ladder models. The gapped symmetric phases protected by subgroups of ${D}_{2}+T$ are also studied. Again, all these phases can be distinguished by physically measuring their end ``spins.''

arXiv:1105.6021Physical Review BNov 29, 201122 citations

We study symmetric gapped ground states of one-dimensional spin

S

chains with a continuous exponential symmetry

U(1)\rtimes\mathbb Z_2

with integer base

q

, where the U(1) charge has a modulated unit of charge

q^{L-j}

decaying exponentially with site

j

. The simplest model is a quantum breakdown model that can be viewed as a

q

-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on

S

and

q

. For integer

S

, odd

q

allows a nontrivial

\mathbb Z_2

Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even

q

allows no nontrivial SPT phase. For half-odd integer

S

, odd

q

prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even

q

enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.

s

-wave superconducting order. The stacking-induced SPT transition drives this heterostructure into a zero-field topological superconducting phase.

In Liu et al. [Phys. Rev. B 84, 075135 (2011)] we studied eight gapped symmetric quantum phases in

S=1

spin chains/ladders which respect a discrete spin rotation ${D}_{2}\ensuremath{\subset}SO(3)$ and time reversal

T

symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of one-dimensional (1D) integer spin systems with

{D}_{2}+T

symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in

S=1

spin chain or spin ladder models. The gapped symmetric phases protected by subgroups of

{D}_{2}+T

are also studied. Again, all these phases can be distinguished by physically measuring their end ``spins.''