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

Symmetry breaking in Heisenberg antiferromagnets

Tohru Koma, Hal Tasaki

Communications in Mathematical PhysicsNov 1, 199338 citations

Quantum Number Fractionalization in Antiferromagnets

R. B. Laughlin, Domenico Giuliano, Raffaele Caracciolo +1 more

Springer series in solid-state sciencesJan 1, 20003 citations

ZN Berry phase and symmetry-protected topological phases of the SU( N ) antiferromagnetic Heisenberg chain

Yuichi Motoyama, Synge Todo

The local ${Z}_{N}$ quantized Berry phase for the $\mathrm{SU}(N)$ antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local ${Z}_{2}$ Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional $\mathrm{SU}(N)$ spin systems. We extend the path-integral quantum Monte Carlo method for the ${Z}_{2}$ Berry phase in order to calculate the ${Z}_{N}$ Berry phase numerically. We demonstrate our method by calculating the ${Z}_{4}$ Berry phase for the bond-alternating $\mathrm{SU}(4)$ antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.The local ${Z}_{N}$ quantized Berry phase for the $\mathrm{SU}(N)$ antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local ${Z}_{2}$ Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional $\mathrm{SU}(N)$ spin systems. We extend the path-integral quantum Monte Carlo method for the ${Z}_{2}$ Berry phase in order to calculate the ${Z}_{N}$ Berry phase numerically. We demonstrate our method by calculating the ${Z}_{4}$ Berry phase for the bond-alternating $\mathrm{SU}(4)$ antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.

arXiv:1508.00960Physical review. B./Physical review. BNov 20, 201810 citations

Ground States of the Antiferromagnetic Heisenberg Chains

Hal Tasaki

Graduate texts in physicsJan 1, 20200 citations

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 $\ensuremath{\Theta}=2\ensuremath{\pi}$.

The local

{Z}_{N}

quantized Berry phase for the

\mathrm{SU}(N)

antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local

{Z}_{2}

Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional

\mathrm{SU}(N)

spin systems. We extend the path-integral quantum Monte Carlo method for the

{Z}_{2}

Berry phase in order to calculate the

{Z}_{N}

Berry phase numerically. We demonstrate our method by calculating the

{Z}_{4}

Berry phase for the bond-alternating

\mathrm{SU}(4)

antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.