arXiv:2406.19372 · Physical review. B./Physical review. B

Refined phase diagram for a spin-1 system exhibiting a Haldane phase

Mohamad Mousa, Birgit Wehefritz–Kaufmann, Sabre Kais, Shawn Xingshan Cui, Ralph M. Kaufmann

Published Feb 18, 2025• 2 citations

Abstract

We provide the phase diagram of a two-parameter spin-1 chain that has a symmetry-protected topological (SPT) Haldane phase using computational algorithms along with tensor-network tools. We improve previous results, showing the existence of a new phase and new triple points. New striking features are the triple end of the Haldane phase and the complexity of phases bordering the Haldane phase in proximity---allowing moving to nearby non-SPT phases via small perturbations. These characteristics make the system, which appears in Rydberg excitons, e.g., in ${\mathrm{Cu}}_{2}\mathrm{O}$, a prime candidate for applications.

DOI: 10.1103/physrevb.111.085303

Related papers

Spin-1 Haldane Phase in a Chain of Rydberg Atoms

Johannes Mögerle, Katharina Brechtelsbauer, A. T. Gea-Caballero +7 more

arXiv:2410.21424PRX QuantumMay 19, 202516 citations

Symmetry-protected topological phases beyond groups: The q -deformed bilinear-biquadratic spin chain

Thomas Quella

arXiv:2011.12679Physical review. B./Physical review. BFeb 2, 20219 citations

Symmetry-protected topological phases in spin ladders with two-body interactions

Zheng-Xin Liu, Zhen‐Biao Yang, Yong‐Jian Han +2 more

arXiv:1204.5162Physical Review BNov 15, 201235 citations

Symmetry-protected topological phases in a two-leg SU(N) spin ladder with unequal spins

Sylvain Capponi, Pierre Fromholz, P. Lecheminant +1 more

arXiv:1909.13553Physical review. B./Physical review. BMay 12, 202013 citations

Three-dimensional symmetry-protected topological phase close to antiferromagnetic Néel order

Cenke Xu

arXiv:1209.4399Physical Review BApr 25, 201339 citations

Characterizing the Haldane phase in quasi-one-dimensional spin-1 Heisenberg antiferromagnets

Keola Wierschem, Pinaki Sengupta

arXiv:1501.00422Modern Physics Letters BDec 23, 201455 citations

arXiv:2406.19372 · Physical review. B./Physical review. B

Refined phase diagram for a spin-1 system exhibiting a Haldane phase

Mohamad Mousa, Birgit Wehefritz–Kaufmann, Sabre Kais, Shawn Xingshan Cui, Ralph M. Kaufmann

Published Feb 18, 2025• 2 citations

arXiv HTML PDF

Abstract

{\mathrm{Cu}}_{2}\mathrm{O}

, a prime candidate for applications.

DOI: 10.1103/physrevb.111.085303

Related papers

Spin-1 Haldane Phase in a Chain of Rydberg Atoms

Johannes Mögerle, Katharina Brechtelsbauer, A. T. Gea-Caballero +7 more

We present a protocol to implement a spin-1 chain in Rydberg systems using three Rydberg states close to a Förster resonance. In addition to dipole-dipole interactions, strong van der Waals interactions naturally appear due to the presence of the Förster resonance and give rise to a highly tunable Hamiltonian. The resulting phase diagram is studied using the infinite density-matrix renormalization group and reveals a highly robust Haldane phase—a prime example of a symmetry-protected topological phase. We find experimentally accessible parameters to probe the Haldane phase in current Rydberg systems, and demonstrate an efficient adiabatic preparation scheme. This paves the way to probe the remarkable properties of spin fractionalization in the Haldane phase.We present a protocol to implement a spin-1 chain in Rydberg systems using three Rydberg states close to a Förster resonance. In addition to dipole-dipole interactions, strong van der Waals interactions naturally appear due to the presence of the Förster resonance and give rise to a highly tunable Hamiltonian. The resulting phase diagram is studied using the infinite density-matrix renormalization group and reveals a highly robust Haldane phase—a prime example of a symmetry-protected topological phase. We find experimentally accessible parameters to probe the Haldane phase in current Rydberg systems, and demonstrate an efficient adiabatic preparation scheme. This paves the way to probe the remarkable properties of spin fractionalization in the Haldane phase.

arXiv:2410.21424PRX QuantumMay 19, 202516 citations

Symmetry-protected topological phases beyond groups: The q -deformed bilinear-biquadratic spin chain

Thomas Quella

We study the phase diagram of the ${\text{SO}}_{q}(3)$ quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of $q>1$ . Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for $q=1$ remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable $q$ deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form ${\mathcal{U}}_{q}\left[su(2)\right]$ for distinct choices of $q$ should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a $q$ -SPT phase.We study the phase diagram of the ${\text{SO}}_{q}(3)$ quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of $q>1$. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for $q=1$ remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable $q$ deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form ${\mathcal{U}}_{q}\left[su(2)\right]$ for distinct choices of $q$ should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a $q$-SPT phase.

arXiv:2011.12679Physical review. B./Physical review. BFeb 2, 20219 citations

Symmetry-protected topological phases in spin ladders with two-body interactions

Zheng-Xin Liu, Zhen‐Biao Yang, Yong‐Jian Han +2 more

Spin-1/2 two-legged ladders respecting interleg exchange symmetry $\ensuremath{\sigma}$ and spin rotation symmetry ${D}_{2}$ have new symmetry-protected topological (SPT) phases which are different from the Haldane phase. Three of the new SPT phases are ${t}_{x},{t}_{y},{t}_{z}$ , which all have symmetry-protected twofold degenerate edge states on each end of an open chain. However, the edge states in different phases have different responses to magnetic field. For example, the edge states in the ${t}_{z}$ phase will be split by the magnetic field along the $z$ direction, but not by the fields in the $x$ and $y$ directions. We give the Hamiltonian that realizes each SPT phase and demonstrate a proof-of-principle quantum simulation scheme for Hamiltonians of the ${t}_{0}$ and ${t}_{z}$ phases based on the coupled-QED-cavity ladder.Spin-1/2 two-legged ladders respecting interleg exchange symmetry $\ensuremath{\sigma}$ and spin rotation symmetry ${D}_{2}$ have new symmetry-protected topological (SPT) phases which are different from the Haldane phase. Three of the new SPT phases are ${t}_{x},{t}_{y},{t}_{z}$, which all have symmetry-protected twofold degenerate edge states on each end of an open chain. However, the edge states in different phases have different responses to magnetic field. For example, the edge states in the ${t}_{z}$ phase will be split by the magnetic field along the $z$ direction, but not by the fields in the $x$ and $y$ directions. We give the Hamiltonian that realizes each SPT phase and demonstrate a proof-of-principle quantum simulation scheme for Hamiltonians of the ${t}_{0}$ and ${t}_{z}$ phases based on the coupled-QED-cavity ladder.

arXiv:1204.5162Physical Review BNov 15, 201235 citations

Symmetry-protected topological phases in a two-leg SU(N) spin ladder with unequal spins

Sylvain Capponi, Pierre Fromholz, P. Lecheminant +1 more

Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective $\mathrm{SU}(N)$ group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with $N>2$ . The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of $\mathrm{SU}(N)$ . We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg $\mathrm{SU}(N)$ spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for $N=3$ and 4 to show that all the possible symmetry-protected topological phases with projective $\mathrm{SU}(N)$ symmetry appear in this simple ladder model.Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective $\mathrm{SU}(N)$ group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with $N>2$. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of $\mathrm{SU}(N)$. We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg $\mathrm{SU}(N)$ spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for $N=3$ and 4 to show that all the possible symmetry-protected topological phases with projective $\mathrm{SU}(N)$ symmetry appear in this simple ladder model.

arXiv:1909.13553Physical review. B./Physical review. BMay 12, 202013 citations

Three-dimensional symmetry-protected topological phase close to antiferromagnetic Néel order

Cenke Xu

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2 $N$ ) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold $\mathcal{M}$ . Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for $N>1$ , this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2 $N$ ) spin symmetry, or its subgroup SU( $N$ )$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2 $N$ ) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2$N$) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold $\mathcal{M}$. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for $N>1$, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2$N$) spin symmetry, or its subgroup SU($N$)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2$N$) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.

arXiv:1209.4399Physical Review BApr 25, 201339 citations

Characterizing the Haldane phase in quasi-one-dimensional spin-1 Heisenberg antiferromagnets

Keola Wierschem, Pinaki Sengupta

We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional (Q1D) geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean-field universality of the critical coupling times the coordination number of the lattice. Inspired by recent developments in the characterization of symmetry protected topological (SPT) states, of which the Haldane phase of the spin-1 Heisenberg antiferromagnetic chain is a preeminent example, we provide direct evidence that the Q1D Haldane phase is indeed a nontrivial SPT state.We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional (Q1D) geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean-field universality of the critical coupling times the coordination number of the lattice. Inspired by recent developments in the characterization of symmetry protected topological (SPT) states, of which the Haldane phase of the spin-1 Heisenberg antiferromagnetic chain is a preeminent example, we provide direct evidence that the Q1D Haldane phase is indeed a nontrivial SPT state.

arXiv:1501.00422Modern Physics Letters BDec 23, 201455 citations

{\mathrm{Cu}}_{2}\mathrm{O}

, a prime candidate for applications.

We study the phase diagram of the

{\text{SO}}_{q}(3)

quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of

q>1

. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for

q=1

remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable

q

deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form

{\mathcal{U}}_{q}\left[su(2)\right]

for distinct choices of

q

should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a

q

-SPT phase.

Spin-1/2 two-legged ladders respecting interleg exchange symmetry $\ensuremath{\sigma}$ and spin rotation symmetry

{D}_{2}

have new symmetry-protected topological (SPT) phases which are different from the Haldane phase. Three of the new SPT phases are

{t}_{x},{t}_{y},{t}_{z}

, which all have symmetry-protected twofold degenerate edge states on each end of an open chain. However, the edge states in different phases have different responses to magnetic field. For example, the edge states in the

{t}_{z}

phase will be split by the magnetic field along the

z

direction, but not by the fields in the

x

and

y

directions. We give the Hamiltonian that realizes each SPT phase and demonstrate a proof-of-principle quantum simulation scheme for Hamiltonians of the

{t}_{0}

and

{t}_{z}

phases based on the coupled-QED-cavity ladder.

Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective

\mathrm{SU}(N)

group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with

N>2

. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of

\mathrm{SU}(N)

. We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg

\mathrm{SU}(N)

spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for

N=3

and 4 to show that all the possible symmetry-protected topological phases with projective

\mathrm{SU}(N)

symmetry appear in this simple ladder model.

N

) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold

\mathcal{M}

. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for

N>1

, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2

N

) spin symmetry, or its subgroup SU(

N

)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2

N

) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.