Classifying Phases Protected By Matrix Product Operator Symmetries Using Matrix Product States

Abstract

We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormalization fixed point classifications and facilitate the numerical study of these systems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and we also provide examples of our classification, together with explicit constructions based on groups. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symmetries also play a key role.

► BibTeX data

@article{GarreRubio2023classifyingphases, doi = {10.22331/q-2023-02-21-927}, url = {https://doi.org/10.22331/q-2023-02-21-927}, title = {Classifying phases protected by matrix product operator symmetries using matrix product states}, author = {Garre-Rubio, Jos{‘{e}} and Lootens, Laurens and Moln{‘{a}}r, Andr{‘{a}}s}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {927}, month = feb, year = {2023}<br /> }

► References

[1] M. B. Hastings and Xiao-Gang Wen. Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B, 72: 045141, Jul 2005. 10.1103/PhysRevB.72.045141.
https://doi.org/10.1103/PhysRevB.72.045141

[2] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa. Symmetry protection of topological phases in one-dimensional quantum spin systems. Physical Review B, 85 (7), Feb 2012. ISSN 1550-235X. 10.1103/physrevb.85.075125.
https://doi.org/10.1103/physrevb.85.075125

[3] Frank Pollmann, Ari M. Turner, Erez Berg, and Masaki Oshikawa. Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B, 81: 064439, Feb 2010. 10.1103/PhysRevB.81.064439.
https://doi.org/10.1103/PhysRevB.81.064439

[4] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett., 86: 5188–5191, May 2001. 10.1103/PhysRevLett.86.5188.
https://doi.org/10.1103/PhysRevLett.86.5188

[5] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B, 83: 035107, Jan 2011a. 10.1103/PhysRevB.83.035107.
https://doi.org/10.1103/PhysRevB.83.035107

[6] Norbert Schuch, David Pérez-García, and Ignacio Cirac. Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B, 84: 165139, Oct 2011. 10.1103/PhysRevB.84.165139.
https://doi.org/10.1103/PhysRevB.84.165139

[7] Yoshiko Ogata. Classification of symmetry protected topological phases in quantum spin chains, 2021.

[8] M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated states on quantum spin chains. Communications in Mathematical Physics, 144 (3): 443–490, Mar 1992. ISSN 1432-0916. 10.1007/BF02099178.
https://doi.org/10.1007/BF02099178

[9] M. M.Wolf D. Pérez-García, F. Verstraete and J. I. Cirac. Matrix product state representations. Quant. Inf. Comput., 7 (401), 2007. 10.48550/ARXIV.QUANT-PH/0608197.
https://doi.org/10.48550/ARXIV.QUANT-PH/0608197

[10] M B Hastings. An area law for one-dimensional quantum systems. Journal of Statistical Mechanics: Theory and Experiment, 2007 (08): P08024, 2007a. 10.1088/1742-5468/2007/08/P08024.
https://doi.org/10.1088/1742-5468/2007/08/P08024

[11] M. B. Hastings. Entropy and entanglement in quantum ground states. Phys. Rev. B, 76: 035114, Jul 2007b. 10.1103/PhysRevB.76.035114.
https://doi.org/10.1103/PhysRevB.76.035114

[12] D. Gross, J. Eisert, N. Schuch, and D. Perez-Garcia. Measurement-based quantum computation beyond the one-way model. Phys. Rev. A, 76: 052315, Nov 2007. 10.1103/PhysRevA.76.052315.
https://doi.org/10.1103/PhysRevA.76.052315

[13] Jutho Haegeman, David Pérez-García, Ignacio Cirac, and Norbert Schuch. Order parameter for symmetry-protected phases in one dimension. Phys. Rev. Lett., 109: 050402, Jul 2012. 10.1103/PhysRevLett.109.050402.
https://doi.org/10.1103/PhysRevLett.109.050402

[14] Frank Pollmann and Ari M. Turner. Detection of symmetry-protected topological phases in one dimension. Phys. Rev. B, 86: 125441, Sep 2012. 10.1103/PhysRevB.86.125441.
https://doi.org/10.1103/PhysRevB.86.125441

[15] Andreas Elben, Jinlong Yu, Guanyu Zhu, Mohammad Hafezi, Frank Pollmann, Peter Zoller, and Benoît Vermersch. Many-body topological invariants from randomized measurements in synthetic quantum matter. Science Advances, 6 (15), Apr 2020. ISSN 2375-2548. 10.1126/sciadv.aaz3666.
https://doi.org/10.1126/sciadv.aaz3666

[16] Gertian Roose, Laurens Vanderstraeten, Jutho Haegeman, and Nick Bultinck. Anomalous domain wall condensation in a modified ising chain. Phys. Rev. B, 99: 195132, May 2019. 10.1103/PhysRevB.99.195132.
https://doi.org/10.1103/PhysRevB.99.195132

[17] Juven C. Wang, Luiz H. Santos, and Xiao-Gang Wen. Bosonic anomalies, induced fractional quantum numbers, and degenerate zero modes: The anomalous edge physics of symmetry-protected topological states. Phys. Rev. B, 91: 195134, May 2015. 10.1103/PhysRevB.91.195134.
https://doi.org/10.1103/PhysRevB.91.195134

[18] Adrian Feiguin, Simon Trebst, Andreas W. W. Ludwig, Matthias Troyer, Alexei Kitaev, Zhenghan Wang, and Michael H. Freedman. Interacting anyons in topological quantum liquids: The golden chain. Phys. Rev. Lett., 98: 160409, Apr 2007. 10.1103/PhysRevLett.98.160409.
https://doi.org/10.1103/PhysRevLett.98.160409

[19] Simon Trebst, Eddy Ardonne, Adrian Feiguin, David A. Huse, Andreas W. W. Ludwig, and Matthias Troyer. Collective states of interacting fibonacci anyons. Phys. Rev. Lett., 101: 050401, Jul 2008. 10.1103/PhysRevLett.101.050401.
https://doi.org/10.1103/PhysRevLett.101.050401

[20] C. Gils, E. Ardonne, S. Trebst, D. A. Huse, A. W. W. Ludwig, M. Troyer, and Z. Wang. Anyonic quantum spin chains: Spin-1 generalizations and topological stability. Phys. Rev. B, 87: 235120, Jun 2013. 10.1103/PhysRevB.87.235120.
https://doi.org/10.1103/PhysRevB.87.235120

[21] Thomas Quella. Symmetry-protected topological phases beyond groups: The $q$-deformed affleck-kennedy-lieb-tasaki model. Physical Review B, 102 (8), aug 2020. 10.1103/physrevb.102.081120.
https://doi.org/10.1103/physrevb.102.081120

[22] Thomas Quella. Symmetry-protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain. Physical Review B, 103 (5), feb 2021. 10.1103/physrevb.103.054404.
https://doi.org/10.1103/physrevb.103.054404

[23] Mehmet Burak Şahinoğlu, Dominic Williamson, Nick Bultinck, Michaël Mariën, Jutho Haegeman, Norbert Schuch, and Frank Verstraete. Characterizing topological order with matrix product operators. Annales Henri Poincaré, 22 (2): 563–592, jan 2021. 10.1007/s00023-020-00992-4.
https://doi.org/10.1007/s00023-020-00992-4

[24] N. Bultinck, M. Mariën, D.J. Williamson, M.B. Şahinoğlu, J. Haegeman, and F. Verstraete. Anyons and matrix product operator algebras. Annals of Physics, 378: 183–233, 2017. ISSN 0003-4916. https://doi.org/10.1016/j.aop.2017.01.004.
https://doi.org/10.1016/j.aop.2017.01.004

[25] Laurens Lootens, Jürgen Fuchs, Jutho Haegeman, Christoph Schweigert, and Frank Verstraete. Matrix product operator symmetries and intertwiners in string-nets with domain walls. SciPost Phys., 10: 53, 2021a. 10.21468/SciPostPhys.10.3.053.
https://doi.org/10.21468/SciPostPhys.10.3.053

[26] A. Kitaev and L. Kong. Models for gapped boundaries and domain walls. Commun. Math. Phys, 2012. doi.org/10.1007/s00220-012-1500-5.
https://doi.org/10.1007/s00220-012-1500-5

[27] Markus Hauru, Glen Evenbly, Wen Wei Ho, Davide Gaiotto, and Guifre Vidal. Topological conformal defects with tensor networks. Physical Review B, 94 (11): 115125, 2016. 10.1103/physrevb.94.115125.
https://doi.org/10.1103/physrevb.94.115125

[28] David Aasen, Roger SK Mong, and Paul Fendley. Topological defects on the lattice: I. the ising model. Journal of Physics A: Mathematical and Theoretical, 49 (35): 354001, 2016. 10.1088/1751-8113/49/35/354001.
https://doi.org/10.1088/1751-8113/49/35/354001

[29] Robijn Vanhove, Matthias Bal, Dominic J Williamson, Nick Bultinck, Jutho Haegeman, and Frank Verstraete. Mapping topological to conformal field theories through strange correlators. Physical review letters, 121 (17): 177203, 2018. 10.1103/physrevlett.121.177203.
https://doi.org/10.1103/physrevlett.121.177203

[30] R. Thorngren and Y. Wang. Fusion category symmetry i: Anomaly in-flow and gapped phases. arXiv:1912.02817, 2019. 10.48550/ARXIV.1912.02817.
https://doi.org/10.48550/ARXIV.1912.02817
arXiv:1912.02817

[31] David Aasen, Paul Fendley, and Roger S. K. Mong. Topological defects on the lattice: Dualities and degeneracies. 2020. 10.48550/ARXIV.2008.08598.
https://doi.org/10.48550/ARXIV.2008.08598

[32] W. Ji and X.-G. Wen. A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly. arXiv:2106.02069, 2021. 10.48550/ARXIV.2106.02069.
https://doi.org/10.48550/ARXIV.2106.02069
arXiv:2106.02069

[33] Robijn Vanhove, Laurens Lootens, Hong-Hao Tu, and Frank Verstraete. Topological aspects of the critical three-state potts model. Journal of Physics A: Mathematical and Theoretical, 55 (23): 235002, may 2022. 10.1088/1751-8121/ac68b1.
https://doi.org/10.1088/1751-8121/ac68b1

[34] Laurens Lootens, Clement Delcamp, Gerardo Ortiz, and Frank Verstraete. Category-theoretic recipe for dualities in one-dimensional quantum lattice models. arXiv preprint arXiv:2112.09091, 2021b. 10.48550/ARXIV.2112.09091.
https://doi.org/10.48550/ARXIV.2112.09091
arXiv:2112.09091

[35] Kansei Inamura. On lattice models of gapped phases with fusion category symmetries. Journal of High Energy Physics, 2022 (3), mar 2022. 10.1007/jhep03(2022)036.
https://doi.org/10.1007/jhep03(2022)036

[36] Andras Molnar, Alberto Ruiz de Alarcón, José Garre-Rubio, Norbert Schuch, J. Ignacio Cirac, and David Pérez-García. Matrix product operator algebras i: representations of weak hopf algebras and projected entangled pair states. 2022. 10.48550/ARXIV.2204.05940.
https://doi.org/10.48550/ARXIV.2204.05940

[37] Mikel Sanz, David Pérez-García, Michael M. Wolf, and J. Ignacio Cirac. A quantum version of wielandt’s inequality. IEEE Transactions on Information Theory, 56: 4668–4673, 2010. 10.1109/tit.2010.2054552.
https://doi.org/10.1109/tit.2010.2054552

[38] J.I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete. Matrix product density operators: Renormalization fixed points and boundary theories. Annals of Physics, 378: 100 – 149, 2017. ISSN 0003-4916. https://doi.org/10.1016/j.aop.2016.12.030.
https://doi.org/10.1016/j.aop.2016.12.030

[39] Xie Chen, Zheng-Xin Liu, and Xiao-Gang Wen. Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations. Phys. Rev. B, 84: 235141, Dec 2011b. 10.1103/PhysRevB.84.235141.
https://doi.org/10.1103/PhysRevB.84.235141

[40] J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete. Matrix product states and projected entangled pair states: Concepts, symmetries, theorems. Reviews of Modern Physics, 93 (4), dec 2021. 10.1103/revmodphys.93.045003.
https://doi.org/10.1103/revmodphys.93.045003

[41] Sergey Bravyi and Matthew B. Hastings. A short proof of stability of topological order under local perturbations. Communications in Mathematical Physics, 307 (3): 609–627, sep 2011. 10.1007/s00220-011-1346-2.
https://doi.org/10.1007/s00220-011-1346-2

[42] Victor Ostrik. Module categories over the drinfeld double of a finite group, 2006.

[43] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Complete classification of one-dimensional gapped quantum phases in interacting spin systems. Phys. Rev. B, 84: 235128, Dec 2011c. 10.1103/PhysRevB.84.235128.
https://doi.org/10.1103/PhysRevB.84.235128

[44] Andras Molnar, Yimin Ge, Norbert Schuch, and J. Ignacio Cirac. A generalization of the injectivity condition for projected entangled pair states. Journal of Mathematical Physics, 59 (2): 021902, 2018a. 10.1063/1.5007017.
https://doi.org/10.1063/1.5007017

[45] Andras Molnar, José Garre-Rubio, David Pérez-García, Norbert Schuch, and J Ignacio Cirac. Normal projected entangled pair states generating the same state. New Journal of Physics, 20 (11): 113017, nov 2018b. 10.1088/1367-2630/aae9fa.
https://doi.org/10.1088/1367-2630/aae9fa

[46] Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. On fusion categories, 2017.

[47] Viktor Ostrik. Module categories, weak hopf algebras and modular invariants, 2001.

[48] D. Nikshych P. Etingof, S. Gelaki and V. Ostrik. Tensor categories, volume 205. Mathematical Soc., 2015. 10.1090/surv/205.
https://doi.org/10.1090/surv/205

[49] Gabriella Bòhm and Korníl Szlachónyi. A coassociativec *-quantum group with nonintegral dimensions. Letters in Mathematical Physics, 38 (4): 437–456, Dec 1996. ISSN 1573-0530. 10.1007/bf01815526.
https://doi.org/10.1007/bf01815526

[50] Pavel Etingof and Viktor Ostrik. Finite tensor categories, 2003.

[51] M. Sanz, M. M. Wolf, D. Pérez-García, and J. I. Cirac. Matrix product states: Symmetries and two-body hamiltonians. Phys. Rev. A, 79: 042308, Apr 2009. 10.1103/PhysRevA.79.042308.
https://doi.org/10.1103/PhysRevA.79.042308

[52] S. B. Bravyi and A. Yu. Kitaev. Quantum codes on a lattice with boundary. 1998. 10.48550/ARXIV.QUANT-PH/9811052.
https://doi.org/10.48550/ARXIV.QUANT-PH/9811052

[53] Salman Beigi, Peter W. Shor, and Daniel Whalen. The quantum double model with boundary: Condensations and symmetries. Communications in Mathematical Physics, 306 (3): 663–694, jun 2011. 10.1007/s00220-011-1294-x.
https://doi.org/10.1007/s00220-011-1294-x

[54] F. Verstraete and J. I. Cirac. Renormalization algorithms for quantum-many body systems in two and higher dimensions. ArXiv: cond-mat/0407066. 10.48550/ARXIV.COND-MAT/0407066.
https://doi.org/10.48550/ARXIV.COND-MAT/0407066

[55] Norbert Schuch, Ignacio Cirac, and David Perez-Garcia. Peps as ground states: Degeneracy and topology. Annals of Physics, 325 (10): 2153 – 2192, 2010. ISSN 0003-4916. https://doi.org/10.1016/j.aop.2010.05.008.
https://doi.org/10.1016/j.aop.2010.05.008

[56] M. de Wild Propitius. Topological interactions in broken gauge theories. PhD thesis, arXiv:hep-th/9511195., 1995. 10.48550/ARXIV.HEP-TH/9511195.
https://doi.org/10.48550/ARXIV.HEP-TH/9511195
arXiv:hep-th/9511195

[57] Eddy Ardonne and Joost Slingerland. Clebsch–gordan and 6j-coefficients for rank 2 quantum groups. Journal of Physics A: Mathematical and Theoretical, 43 (39): 395205, Aug 2010. ISSN 1751-8121. 10.1088/1751-8113/43/39/395205.
https://doi.org/10.1088/1751-8113/43/39/395205

[58] Ingo Runkel. Structure constants for the d-series virasoro minimal models. Nuclear Physics B, 579 (3): 561–589, Jul 2000. ISSN 0550-3213. 10.1016/s0550-3213(99)00707-5.
https://doi.org/10.1016/s0550-3213(99)00707-5

[59] Liang Kong, Xiao-Gang Wen, and Hao Zheng. Boundary-bulk relation in topological orders. Nuclear Physics B, 922: 62–76, Sep 2017. ISSN 0550-3213. 10.1016/j.nuclphysb.2017.06.023.
https://doi.org/10.1016/j.nuclphysb.2017.06.023

[60] Dominic V. Else and Chetan Nayak. Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge. Phys. Rev. B, 90: 235137, Dec 2014. 10.1103/PhysRevB.90.235137.
https://doi.org/10.1103/PhysRevB.90.235137

[61] Michael P. Zaletel. Detecting two-dimensional symmetry-protected topological order in a ground-state wave function. Physical Review B, 90 (23), Dec 2014. ISSN 1550-235X. 10.1103/physrevb.90.235113.
https://doi.org/10.1103/physrevb.90.235113

[62] Kyle Kawagoe and Michael Levin. Anomalies in bosonic symmetry-protected topological edge theories: Connection to f symbols and a method of calculation. Physical Review B, 104 (11), Sep 2021. ISSN 2469-9969. 10.1103/physrevb.104.115156.
https://doi.org/10.1103/physrevb.104.115156

[63] Dominic V. Else, Ilai Schwarz, Stephen D. Bartlett, and Andrew C. Doherty. Symmetry-protected phases for measurement-based quantum computation. Physical Review Letters, 108 (24), Jun 2012. ISSN 1079-7114. 10.1103/physrevlett.108.240505.
https://doi.org/10.1103/physrevlett.108.240505

Cited by

[1] John McGreevy, “Generalized Symmetries in Condensed Matter”, arXiv:2204.03045, (2022).

[2] José Garre Rubio and Ilya Kull, “Gauging quantum states with non-anomalous matrix product operator symmetries”, arXiv:2209.07355, (2022).

[3] Kansei Inamura, “Fermionization of fusion category symmetries in 1+1 dimensions”, arXiv:2206.13159, (2022).

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This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Classifying phases protected by matrix product operator symmetries using matrix product states

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Classifying phases protected by matrix product operator symmetries using matrix product states

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