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We propose and prove a family of generalized Lieb-Schultz-Mattis (LSM) theorems for symmetry protected topological (SPT) phases on boson/spin models in any dimensions. The "conventional" LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a nontrivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or "higher-order" crystalline SPT phases, characterized by nontrivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.

Objective: This study aimed to compare the outcomes of unilateral biportal endoscopy, unilateral laminectomy bilateral decompression (UBE-ULBD), and open lumbar decompression (OLD) in patients with lumbar epidural lipomatosis (LEL)., Methods: This prospective observational study was conducted from March 2019 to May 2022 and encompassed 33 patients with LEL who underwent lumbar decompression. The study included 15 cases of UBE-ULBD decompression and 18 cases of open decompression, which were followed up for 1 year. The baseline characteristics, initial clinical manifestations, and surgical details [including estimated blood loss (EBL) and preoperative complications] of all patients were recorded. Radiographic evaluation included the cross-sectional area (CSA) of the thecal sac and paraspinal muscles on MRI. Clinical results were analyzed using the Short Form-36 Score (SF-36), the Numeric Pain Rating Scale (NRS) for lumbar and leg pain, creatine kinase, the Roland and Morris Disability Questionnaire (RMDQ), and the Oswestry Disability Index (ODI)., Results: The dural sac CSA increased considerably at the 1-year postoperative follow-up in both groups ( p < 0.001). The operative duration in the OLD group (48.2 ± 7.2 min) was shorter than that in the UBE-ULBD group (67.7 ± 6.3 min, p < 0.001). The OLD group (97.2 ± 19.8 mL) was associated with more EBL than the UBE-ULBD group (40.6 ± 13.6 mL, p < 0.001). The duration of hospitalization in the OLD group (5.4 ± 1.3 days) was significantly longer compared with the UBE-ULBD group (3.5 ± 1.2 days, p < 0.01). The SF-36, NRS, RMDQ, and ODI scores improved in both groups postoperatively ( p < 0.001). Serum creatine kinase values in the UBE-ULBD group (101.7 ± 15.5) were significantly lower than those in the OLD group (330.8 ± 28.1 U/L) 1 day after surgery ( p < 0.001). The degree of paraspinal muscle atrophy in the UBE-ULBD group (4.81 ± 1.94) was significantly lower than that in the OLD group (12.15 ± 6.99) at 1 year ( p < 0.001)., Conclusion: UBE-ULBD and OLD demonstrated comparable clinical outcomes in treating LEL. However, UBE-ULBD surgery was associated with shorter hospital stays, lower rates of incision infection, lighter paravertebral muscle injury, and lower EBL than OLD surgery. Consequently, UBE-ULBD can be recommended in patients with LEL if conservative treatment fails., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2024 Tan, Zheng, Lei, Ouyang, Wen, Shi and Yang.)