Fermionic quantum criticality through the lens of topological holography (2405.09611v2)
Abstract: We utilize the topological holographic framework to characterize and gain insights into the nature of quantum critical points and gapless phases in fermionic quantum systems. Topological holography is a general framework that describes the generalized global symmetry and the symmetry charges of a local quantum system in terms of a slab of a topological order, termed as the symmetry topological field theory (SymTFT), in one higher dimension. In this work, we consider a generalization of the topological holographic picture for $(1+1)d$ fermionic quantum phases of matter. We discuss how spin structures are encoded in the SymTFT and establish the connection between the formal fermionization formula in quantum field theory and the choice of fermionic gapped boundary conditions of the SymTFT. We demonstrate the identification and the characterization of the fermionic gapped phases and phase transitions through detailed analysis of various examples, including the fermionic systems with $\mathbb{Z}{2}{F}$, $\mathbb{Z}{2} \times \mathbb{Z}{2}{F}$, $\mathbb{Z}{4}{F}$, and the fermionic version of the non-invertible $\text{Rep}(S_{3})$ symmetry. Our work uncovers many exotic fermionic quantum critical points and gapless phases, including two kinds of fermionic symmetry enriched quantum critical points, a fermionic gapless symmetry protected topological (SPT) phase, and a fermionic gapless spontaneous symmetry breaking (SSB) phase that breaks the fermionic non-invertible symmetry.
- X. G. WEN, Topological orders in rigid states, International Journal of Modern Physics B 04, 239 (1990), https://doi.org/10.1142/S0217979290000139 .
- L. Kong, X.-G. Wen, and H. Zheng, Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers, arXiv e-prints , arXiv:1502.01690 (2015), arXiv:1502.01690 [cond-mat.str-el] .
- C. L. Douglas and D. J. Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds, arXiv e-prints , arXiv:1812.11933 (2018), arXiv:1812.11933 [math.QA] .
- T. Johnson-Freyd, On the classification of topological orders, Communications in Mathematical Physics 393, 989 (2022).
- L. Bhardwaj, S. Schafer-Nameki, and A. Tiwari, Unifying Constructions of Non-Invertible Symmetries, arXiv e-prints , arXiv:2212.06159 (2022), arXiv:2212.06159 [hep-th] .
- T. Bartsch, M. Bullimore, and A. Grigoletto, Higher representations for extended operators, arXiv e-prints , arXiv:2304.03789 (2023), arXiv:2304.03789 [hep-th] .
- J. McGreevy, Generalized Symmetries in Condensed Matter, Annual Review of Condensed Matter Physics 14, 57 (2023), arXiv:2204.03045 [cond-mat.str-el] .
- J. Fuchs, I. Runkel, and C. Schweigert, Tft construction of rcft correlators i: partition functions, Nuclear Physics B 646, 353 (2002).
- J. Fuchs, I. Runkel, and C. Schweigert, Tft construction of rcft correlators ii: unoriented world sheets, Nuclear Physics B 678, 511 (2004a).
- J. Fuchs, I. Runkel, and C. Schweigert, Tft construction of rcft correlators: Iii: simple currents, Nuclear Physics B 694, 277 (2004b).
- J. Fuchs, I. Runkel, and C. Schweigert, Tft construction of rcft correlators iv:: Structure constants and correlation functions, Nuclear Physics B 715, 539 (2005).
- L. Kong, X.-G. Wen, and H. Zheng, Boundary-bulk relation in topological orders, Nuclear Physics B 922, 62 (2017), arXiv:1702.00673 [cond-mat.str-el] .
- D. S. Freed and C. Teleman, Topological dualities in the Ising model, arXiv e-prints , arXiv:1806.00008 (2018), arXiv:1806.00008 [math.AT] .
- L. Kong and H. Zheng, Gapless edges of 2d topological orders and enriched monoidal categories, Nuclear Physics B 927, 140 (2018).
- R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases, arXiv e-prints , arXiv:1912.02817 (2019), arXiv:1912.02817 [hep-th] .
- L. Kong and H. Zheng, A mathematical theory of gapless edges of 2d topological orders. part i, Journal of High Energy Physics 2020, 150 (2020).
- W. Ji and X.-G. Wen, Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions, Phys. Rev. Res. 2, 033417 (2020).
- D. Gaiotto and J. Kulp, Orbifold groupoids, Journal of High Energy Physics 2021, 132 (2021).
- W. Ji and X.-G. Wen, A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly, arXiv e-prints , arXiv:2106.02069 (2021), arXiv:2106.02069 [cond-mat.str-el] .
- L. Kong and H. Zheng, A mathematical theory of gapless edges of 2d topological orders. part ii, Nuclear Physics B 966, 115384 (2021).
- D. S. Freed, G. W. Moore, and C. Teleman, Topological symmetry in quantum field theory, arXiv e-prints , arXiv:2209.07471 (2022), arXiv:2209.07471 [hep-th] .
- J. Kaidi, K. Ohmori, and Y. Zheng, Symmetry TFTs for Non-Invertible Defects, arXiv e-prints , arXiv:2209.11062 (2022), arXiv:2209.11062 [hep-th] .
- H. Moradi, S. Faroogh Moosavian, and A. Tiwari, Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics, arXiv e-prints , arXiv:2207.10712 (2022), arXiv:2207.10712 [cond-mat.str-el] .
- A. Chatterjee and X.-G. Wen, Algebra of local symmetric operators and braided fusion n𝑛nitalic_n-category – symmetry is a shadow of topological order, arXiv e-prints , arXiv:2203.03596 (2022a), arXiv:2203.03596 [cond-mat.str-el] .
- A. Chatterjee and X.-G. Wen, Holographic theory for the emergence and the symmetry protection of gaplessness and for continuous phase transitions, arXiv e-prints , arXiv:2205.06244 (2022b), arXiv:2205.06244 [cond-mat.str-el] .
- A. Chatterjee, W. Ji, and X.-G. Wen, Emergent maximal categorical symmetry in a gapless state, arXiv e-prints , arXiv:2212.14432 (2022), arXiv:2212.14432 [cond-mat.str-el] .
- L. Kong and H. Zheng, Categories of quantum liquids I, Journal of High Energy Physics 2022, 70 (2022), arXiv:2011.02859 [hep-th] .
- L. Kong, X.-G. Wen, and H. Zheng, One dimensional gapped quantum phases and enriched fusion categories, Journal of High Energy Physics 2022, 22 (2022), arXiv:2108.08835 [cond-mat.str-el] .
- F. Benini, C. Copetti, and L. Di Pietro, Factorization and global symmetries in holography, SciPost Physics 14, 019 (2023), arXiv:2203.09537 [hep-th] .
- L. Bhardwaj and S. Schafer-Nameki, Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT, arXiv e-prints , arXiv:2305.17159 (2023), arXiv:2305.17159 [hep-th] .
- T. D. Brennan and Z. Sun, A SymTFT for Continuous Symmetries, arXiv e-prints , arXiv:2401.06128 (2024), arXiv:2401.06128 [hep-th] .
- F. Apruzzi, F. Bedogna, and N. Dondi, SymTh for non-finite symmetries, arXiv e-prints , arXiv:2402.14813 (2024), arXiv:2402.14813 [hep-th] .
- A. Antinucci and F. Benini, Anomalies and gauging of U(1) symmetries, arXiv e-prints , arXiv:2401.10165 (2024), arXiv:2401.10165 [hep-th] .
- F. Bonetti, M. Del Zotto, and R. Minasian, SymTFTs for Continuous non-Abelian Symmetries, arXiv e-prints , arXiv:2402.12347 (2024), arXiv:2402.12347 [hep-th] .
- R. Wen and A. C. Potter, Classification of 1+1D gapless symmetry protected phases via topological holography, arXiv e-prints , arXiv:2311.00050 (2023a), arXiv:2311.00050 [cond-mat.str-el] .
- A. Chatterjee, Ö. M. Aksoy, and X.-G. Wen, Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries, arXiv e-prints , arXiv:2405.05331 (2024), arXiv:2405.05331 [cond-mat.str-el] .
- Y. Wan and C. Wang, Fermion condensation and gapped domain walls in topological orders, Journal of High Energy Physics 2017, 172 (2017), arXiv:1607.01388 [cond-mat.str-el] .
- D. Aasen, E. Lake, and K. Walker, Fermion condensation and super pivotal categories, Journal of Mathematical Physics 60, 121901 (2019), arXiv:1709.01941 [cond-mat.str-el] .
- D. Gaiotto and A. Kapustin, Spin tqfts and fermionic phases of matter, International Journal of Modern Physics A 31, 1645044 (2016).
- A. Debray, W. Ye, and M. Yu, Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders, arXiv e-prints , arXiv:2312.13341 (2023), arXiv:2312.13341 [math-ph] .
- P. Boyle Smith and Y. Zheng, Backfiring Bosonisation, arXiv e-prints , arXiv:2403.03953 (2024), arXiv:2403.03953 [hep-th] .
- K. Omori, Categorical aspects of symmetry in fermionic systems (2024), pIRSA:24030089 see, https://pirsa.org.
- W. Ji and X.-G. Wen, Noninvertible anomalies and mapping-class-group transformation of anomalous partition functions, Phys. Rev. Res. 1, 033054 (2019).
- A. Chatterjee and X.-G. Wen, Holographic theory for continuous phase transitions: Emergence and symmetry protection of gaplessness, Phys. Rev. B 108, 075105 (2023).
- C. Shen and L.-Y. Hung, Defect verlinde formula for edge excitations in topological order, Phys. Rev. Lett. 123, 051602 (2019).
- T. Scaffidi, D. E. Parker, and R. Vasseur, Gapless symmetry-protected topological order, Phys. Rev. X 7, 041048 (2017).
- L. Li, M. Oshikawa, and Y. Zheng, Decorated Defect Construction of Gapless-SPT States, arXiv e-prints , arXiv:2204.03131 (2022), arXiv:2204.03131 [cond-mat.str-el] .
- R. Wen and A. C. Potter, Classification of 1+1D gapless symmetry protected phases via topological holography, arXiv e-prints , arXiv:2311.00050 (2023b), arXiv:2311.00050 [cond-mat.str-el] .
- R. Thorngren, A. Vishwanath, and R. Verresen, Intrinsically gapless topological phases, Phys. Rev. B 104, 075132 (2021).
- R. Wen and A. C. Potter, Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology, arXiv e-prints , arXiv:2208.09001 (2022), arXiv:2208.09001 [cond-mat.str-el] .
- K. Inamura, Fermionization of fusion category symmetries in 1+1 dimensions, Journal of High Energy Physics 2023, 101 (2023).
- S. X. Cui and Z. Wang, Universal quantum computation with weakly integral anyons, J. Math. Phys. 56, 032202 (2015), arXiv:1401.7096 [quant-ph] .
- C.-T. Hsieh, Y. Nakayama, and Y. Tachikawa, Fermionic minimal models, Phys. Rev. Lett. 126, 195701 (2021).
- D. Aasen, P. Bonderson, and C. Knapp, Characterization and Classification of Fermionic Symmetry Enriched Topological Phases, arXiv e-prints , arXiv:2109.10911 (2021), arXiv:2109.10911 [cond-mat.str-el] .
- R. Wen, W. Ye, and A. C. Potter, Topological holography for fermions, 2404.19004 (2024).