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arXiv:1503.05662 · Journal of Physics A Mathematical and Theoretical

Extension of the Lieb–Schultz–Mattis theorem

K. Nomura, 順平 森重, Takaichi Isoyama

Published Aug 19, 2015• 11 citations
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Abstract

Lieb, Schultz and Mattis (LSM) (1961 Ann. Phys., NY 16 407 ) studied the S = 1/2 XXZ spin chain. The theorems of LSM's paper can be applied to broader models. In the original LSM theorem the nonfrustrating system was assumed. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers tried to extend the LSM theorem for excited states. In the cases , the lowest energy eigenvalues are continuous for wave number q . But we found that their proofs were insufficient, and improve upon them. In addition, we can prove the LSM theory without the assumption of the discrete symmetry, which means that LSM-type theorems are applicable for Dzyaloshinskii–Moriya type interactions or other nonsymmetric models.

DOI
10.1088/1751-8113/48/37/375001

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Lieb, Schultz and Mattis (LSM) (1961 Ann. Phys., NY 16 407 ) studied the S = 1/2 XXZ spin chain. The theorems of LSM's paper can be applied to broader models. In the original LSM theorem the nonfrustrating system was assumed. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers tried to extend the LSM theorem for excited states. In the cases , the lowest energy eigenvalues are continuous for wave number q . But we found that their proofs were insufficient, and improve upon them. In addition, we can prove the LSM theory without the assumption of the discrete symmetry, which means that LSM-type theorems are applicable for Dzyaloshinskii–Moriya type interactions or other nonsymmetric models.
The theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. Independently, Kolb considered the relation between the wave number qqq and the twisting boundary condition, and he obtained a similar result as LSM. However, in frustrating cases it is known that there exist several exceptions for the assumption of the unique lowest state for the finite size, which is important in the traditional LSM theorem. In our previous paper, without the assumption of the uniqueness, we have extended the LSMK theorem for frustrating and non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work.
The theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. However, in the original LSM theorem the non-frustrating system was assumed. Independently, Kolb considered the relation between the wave number qqq and the twisting boundary condition. In our previous paper, reconsidering the LSM theorem and combining Kolb's discussion, we have extended the LSMK theorem for frustrating or non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work. Keywords: Lieb-Schultz-Mattis, rigorous theorem, frustration, one-dimension, topological aspect, Dzyalosinskii-Moriya
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.
The Lieb-Schultz-Mattis (LSM) theorem asserts that microscopic details of the system can impose non-trivial constraints on the system's low-energy properties. While traditionally applied to short-range interaction systems, where locality ensures a vanishing spectral gap in large system size limit, the impact of long-range interactions on the LSM theorem remains an open question. Long-range interactions are prevalent in experimental platforms such as Rydberg atoms, dipolar quantum gases, polar molecules, optical cavities, and trapped ions, where the interaction decay exponent can be experimentally tuned. We extend the LSM theorem in one dimension to long-range interacting systems and find that the LSM theorem holds for exponentially or power-law two-body interactions with a decay exponent α>2α> 2α>2. However, for power-law interactions with α<2α< 2α<2, the constraints of the LSM theorem on the ground state do not apply. Numerical simulations of long-range versions of the Heisenberg and Majumdar-Ghosh models, both satisfying the LSM symmetry requirements, are also provided. Our results suggest promising directions for experimental validation of the LSM theorem in systems with tunable long-range interactions.
In closed systems, the celebrated Lieb-Schultz-Mattis (LSM) theorem states that a one-dimensional locally interacting half-integer spin chain with translation and spin rotation symmetries cannot have a non-degenerate gapped ground state. However, the applicability of this theorem is diminished when the system interacts with a bath and loses its energy conservation. In this letter, we propose that the LSM theorem can be revived in the entanglement Hamiltonian when the coupling to the bath renders the system short-range correlated. Specifically, we argue that the entanglement spectrum cannot have a non-degenerate minimum, isolated by a gap from other states. We further support the results with numerical examples where a spin-[Formula: see text] system is coupled to another spin-[Formula: see text] chain serving as the bath. Compared with the original LSM theorem that primarily addresses UV-IR correspondence, our findings reveal that the UV data and topological constraints also play a pivotal role in shaping the entanglement in open quantum many-body systems.