Chapter 6 ties invertible topological phases to extensions of the original Lieb–Schultz–Mattis theorem. A review is made of the original Lieb–Schultz–Mattis theorem and how it has been refined under the assumption that a continuous symmetry holds. Two extensions of the Lieb–Schultz–Mattis theorem are given that apply to the Majorana chains from Chapter 5 when protected by discrete symmetries. To this end, it is necessary to introduce the notion of projective representations of symmetries and their classifications in terms of the second cohomology group. A precise definition is given of fermionic invertible topological phases and how they can be classified by the second cohomology group in one-dimensional space. Stacking rules of fermionic invertible topological phases in one-dimensional space are explained and shown to deliver the degeneracies of the boundary states that are protected by the symmetries.Chapter 6 ties invertible topological phases to extensions of the original Lieb–Schultz–Mattis theorem. A review is made of the original Lieb–Schultz–Mattis theorem and how it has been refined under the assumption that a continuous symmetry holds. Two extensions of the Lieb–Schultz–Mattis theorem are given that apply to the Majorana chains from Chapter 5 when protected by discrete symmetries. To this end, it is necessary to introduce the notion of projective representations of symmetries and their classifications in terms of the second cohomology group. A precise definition is given of fermionic invertible topological phases and how they can be classified by the second cohomology group in one-dimensional space. Stacking rules of fermionic invertible topological phases in one-dimensional space are explained and shown to deliver the degeneracies of the boundary states that are protected by the symmetries.

Cambridge University Press eBooksJan 9, 20250 citations

A linear Solution of the Four Dimensionality Problem

Vladimir Trifonov

Jan 1, 20260 citations

Discrete symmetries and the Lieb–Schultz–Mattis theorem

Takaichi Isoyama, Kiyohide Nomura

In this study, we consider one-dimensional (1D) quantum spin systems with translation and discrete symmetries (spin reversal, space inversion, and time reversal symmetries). By combining the continuous U(1) symmetry with the discrete symmetries and using the extended Lieb–Schultz–Mattis (LSM) theorem [E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961); K. Nomura, J. Morishige, and T. Isoyama, J. Phys. A 48, 375001 (2015)], we investigate the relation between the ground states, energy spectra, and symmetries. For half-integer spin cases, we generalize the dimer and Néel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or Néel states, since there was a subtle dilemma. Furthermore, a part of discrete symmetries is enough to classify possible phases. Thus we can deepen our understanding of the relation between the LSM theorem and discrete symmetries.In this study, we consider one-dimensional (1D) quantum spin systems with translation and discrete symmetries (spin reversal, space inversion, and time reversal symmetries). By combining the continuous U(1) symmetry with the discrete symmetries and using the extended Lieb–Schultz–Mattis (LSM) theorem [E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961); K. Nomura, J. Morishige, and T. Isoyama, J. Phys. A 48, 375001 (2015)], we investigate the relation between the ground states, energy spectra, and symmetries. For half-integer spin cases, we generalize the dimer and Néel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or Néel states, since there was a subtle dilemma. Furthermore, a part of discrete symmetries is enough to classify possible phases. Thus we can deepen our understanding of the relation between the LSM theorem and discrete symmetries.

arXiv:1605.03385Progress of Theoretical and Experimental PhysicsSep 26, 20175 citations

Projectification of point group symmetries with a background flux and the Lieb-Schultz-Mattis theorem

Yasuhiro Tada, Masaki Oshikawa

We discuss the Lieb-Schultz-Mattis (LSM) theorem in two-dimensional spin systems with on-site ${\mathrm U}(1)\rtimes {\mathbb Z}_2$ spin rotation symmetry and point group $C_{2v}$ symmetry about a site. We ``twist" the point group symmetry by introducing a small uniform U(1) flux to obtain a projective symmetry, similarly to the familiar magnetic translation symmetry. The LSM theorem is proved in presence of the flux and then it is demonstrated that the theorem holds also for the flux-free system. Besides, the uniform flux enables us to show the LSM theorem for the time-reversal symmetry and the site-centered $C_2$ -rotation symmetry.We discuss the Lieb-Schultz-Mattis (LSM) theorem in two-dimensional spin systems with on-site ${\mathrm U}(1)\rtimes {\mathbb Z}_2$ spin rotation symmetry and point group $C_{2v}$ symmetry about a site. We ``twist" the point group symmetry by introducing a small uniform U(1) flux to obtain a projective symmetry, similarly to the familiar magnetic translation symmetry. The LSM theorem is proved in presence of the flux and then it is demonstrated that the theorem holds also for the flux-free system. Besides, the uniform flux enables us to show the LSM theorem for the time-reversal symmetry and the site-centered $C_2$-rotation symmetry.

arXiv:2505.00927Physical review. B./Physical review. BFeb 23, 20263 citations

We discuss the Lieb-Schultz-Mattis (LSM) theorem in two-dimensional spin systems with on-site

{\mathrm U}(1)\rtimes {\mathbb Z}_2

spin rotation symmetry and point group

C_{2v}

symmetry about a site. We ``twist" the point group symmetry by introducing a small uniform U(1) flux to obtain a projective symmetry, similarly to the familiar magnetic translation symmetry. The LSM theorem is proved in presence of the flux and then it is demonstrated that the theorem holds also for the flux-free system. Besides, the uniform flux enables us to show the LSM theorem for the time-reversal symmetry and the site-centered

C_2

-rotation symmetry.