2000 character limit reached
Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension (2405.05323v1)
Published 8 May 2024 in cond-mat.str-el, hep-th, and quant-ph
Abstract: We propose a systematic wave function based approach to construct topological invariants for families of lattice systems that are short-range entangled using local parameter spaces. This construction is particularly suitable when given a family of tensor networks that can be viewed as the ground states of $d$ dimensional lattice systems, for which we construct the closed $(d+2)$-form higher Berry curvature, which is a generalization of the well known 2-form Berry curvature. Such $(d+2)$-form higher Berry curvature characterizes a flow of $(d+1)$-form higher Berry curvature in the system. Our construction is equally suitable for constructing other higher pumps, such as the (higher) Thouless pump in the presence of a global on-site $U(1)$ symmetry, which corresponds to a closed $d$-form. The cohomology classes of such higher differential forms are topological invariants and are expected to be quantized for short-range entangled states. We illustrate our construction with exactly solvable lattice models that are in nontrivial higher Berry classes in $d=2$.
- M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 392, 45–57 (1984).
- D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
- A. Kitaev, “Differential forms on the space of statistical mechanics models,” (2019), talk at the conference in celebration of Dan Freed’s 60th birthdayhttps://web.ma.utexas.edu/topqft/talkslides/kitaev.pdf.
- Anton Kapustin and Lev Spodyneiko, “Higher-dimensional generalizations of berry curvature,” Physical Review B 101 (2020a), 10.1103/physrevb.101.235130.
- Anton Kapustin and Lev Spodyneiko, “Higher-dimensional generalizations of the thouless charge pump,” (2020b), arXiv:2003.09519 [cond-mat.str-el] .
- Po-Shen Hsin, Anton Kapustin, and Ryan Thorngren, “Berry phase in quantum field theory: Diabolical points and boundary phenomena,” Physical Review B 102 (2020), 10.1103/physrevb.102.245113.
- Clay Cordova, Daniel Freed, Ho Tat Lam, and Nathan Seiberg, “Anomalies in the space of coupling constants and their dynamical applications i,” SciPost Physics 8 (2020a), 10.21468/scipostphys.8.1.001.
- Clay Cordova, Daniel Freed, Ho Tat Lam, and Nathan Seiberg, “Anomalies in the space of coupling constants and their dynamical applications II,” SciPost Physics 8 (2020b), 10.21468/scipostphys.8.1.002.
- Dominic V. Else, “Topological goldstone phases of matter,” Physical Review B 104 (2021), 10.1103/physrevb.104.115129.
- Yichul Choi and Kantaro Ohmori, “Higher berry phase of fermions and index theorem,” Journal of High Energy Physics 2022 (2022), 10.1007/jhep09(2022)022.
- David Aasen, Zhenghan Wang, and Matthew B. Hastings, “Adiabatic paths of hamiltonians, symmetries of topological order, and automorphism codes,” Physical Review B 106 (2022), 10.1103/physrevb.106.085122.
- Xueda Wen, Marvin Qi, Agnès Beaudry, Juan Moreno, Markus J. Pflaum, Daniel Spiegel, Ashvin Vishwanath, and Michael Hermele, “Flow of (higher) berry curvature and bulk-boundary correspondence in parametrized quantum systems,” (2022), arXiv:2112.07748 [cond-mat.str-el] .
- Po-Shen Hsin and Zhenghan Wang, “On topology of the moduli space of gapped hamiltonians for topological phases,” Journal of Mathematical Physics 64, 041901 (2023).
- Anton Kapustin and Nikita Sopenko, “Local Noether theorem for quantum lattice systems and topological invariants of gapped states,” Journal of Mathematical Physics 63, 091903 (2022), arXiv:2201.01327 [math-ph] .
- Ken Shiozaki, “Adiabatic cycles of quantum spin systems,” Physical Review B 106 (2022), 10.1103/physrevb.106.125108.
- Sven Bachmann, Wojciech De Roeck, Martin Fraas, and Tijl Jappens, “A classification of G𝐺Gitalic_G-charge Thouless pumps in 1D invertible states,” arXiv e-prints , arXiv:2204.03763 (2022), arXiv:2204.03763 [math-ph] .
- Shuhei Ohyama, Ken Shiozaki, and Masatoshi Sato, “Generalized thouless pumps in (1+1)11(1+1)( 1 + 1 )-dimensional interacting fermionic systems,” Phys. Rev. B 106, 165115 (2022).
- Shuhei Ohyama, Yuji Terashima, and Ken Shiozaki, “Discrete higher berry phases and matrix product states,” (2023), arXiv:2303.04252 [cond-mat.str-el] .
- Adam Artymowicz, Anton Kapustin, and Nikita Sopenko, “Quantization of the higher Berry curvature and the higher Thouless pump,” arXiv e-prints , arXiv:2305.06399 (2023), arXiv:2305.06399 [math-ph] .
- Agnes Beaudry, Michael Hermele, Juan Moreno, Markus Pflaum, Marvin Qi, and Daniel Spiegel, “Homotopical foundations of parametrized quantum spin systems,” (2023), arXiv:2303.07431 [math-ph] .
- Shuhei Ohyama and Shinsei Ryu, “Higher structures in matrix product states,” arXiv e-prints , arXiv:2304.05356 (2023), arXiv:2304.05356 [cond-mat.str-el] .
- Marvin Qi, David T. Stephen, Xueda Wen, Daniel Spiegel, Markus J. Pflaum, Agnès Beaudry, and Michael Hermele, “Charting the space of ground states with tensor networks,” arXiv e-prints , arXiv:2305.07700 (2023), arXiv:2305.07700 [cond-mat.str-el] .
- Ken Shiozaki, Niclas Heinsdorf, and Shuhei Ohyama, “Higher Berry curvature from matrix product states,” arXiv e-prints , arXiv:2305.08109 (2023), arXiv:2305.08109 [quant-ph] .
- Lev Spodyneiko, “Hall conductivity pump,” arXiv e-prints , arXiv:2309.14332 (2023), arXiv:2309.14332 [cond-mat.mes-hall] .
- Arun Debray, Sanath K. Devalapurkar, Cameron Krulewski, Yu Leon Liu, Natalia Pacheco-Tallaj, and Ryan Thorngren, “A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases,” arXiv e-prints , arXiv:2309.16749 (2023), arXiv:2309.16749 [hep-th] .
- A. Kitaev, “Toward a topological classification of many-body quantum states with short-range entanglement,” (2011), talk at Simons Center for Geometry and Physics http://scgp.stonybrook.edu/archives/1087.
- A. Kitaev, “On the classification of short-range entangled states,” (2013), talk at Simons Center for Geometry and Physics http://scgp.stonybrook.edu/archives/16180.
- A. Kitaev, “Homotopy-theoretic approach to spt phases in action: Z16subscript𝑍16Z_{1}6italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 6 classification of three-dimensional superconductors,” (2015), talk at Institute for Pure and Applied Mathematics http://www.ipam.ucla.edu/programs/workshops/symmetry-and-topology-in-quantum-matter/.
- D. J. Thouless, “Quantization of particle transport,” Phys. Rev. B 27, 6083–6087 (1983).
- Ophelia Evelyn Sommer, Ashvin Vishwanath, and Xueda Wen, “Higher berry curvature from the wave function i: Schmidt decomposition and matrix product states,” (2024), to appear.
- J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete, “Matrix product states and projected entangled pair states: Concepts, symmetries, theorems,” Rev. Mod. Phys. 93, 045003 (2021).
- Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, “Classification of gapped symmetric phases in one-dimensional spin systems,” Phys. Rev. B 83, 035107 (2011).
- Frank Pollmann, Erez Berg, Ari M. Turner, and Masaki Oshikawa, “Symmetry protection of topological phases in one-dimensional quantum spin systems,” Phys. Rev. B 85, 075125 (2012).
- Norbert Schuch, David Pérez-Garcia, and Ignacio Cirac, “Classifying quantum phases using matrix product states and projected entangled pair states,” Phys. Rev. B 84, 165139 (2011).
- Román Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117–158 (2014).
- John Roe, Lectures on coarse geometry, 31 (American Mathematical Soc., 2003).
- Xie Chen, Arpit Dua, Michael Hermele, David T. Stephen, Nathanan Tantivasadakarn, Robijn Vanhove, and Jing-Yu Zhao, “Sequential quantum circuits as maps between gapped phases,” Physical Review B 109 (2024), 10.1103/physrevb.109.075116.
- Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett, “Generalized global symmetries,” Journal of High Energy Physics 2015, 172 (2015).
- Anton Kapustin and Ryan Thorngren, “Higher Symmetry and Gapped Phases of Gauge Theories,” in Algebra, Geometry, and Physics in the 21st Century: Kontsevich Festschrift, Progress in Mathematics, edited by Denis Auroux, Ludmil Katzarkov, Tony Pantev, Yan Soibelman, and Yuri Tschinkel (Springer International Publishing, Cham, 2017) pp. 177–202.
- Daniel Aloni, Eduardo García-Valdecasas, Matthew Reece, and Motoo Suzuki, “Spontaneously broken (−1)1(-1)( - 1 )-form u(1) symmetries,” (2024), arXiv:2402.00117 [hep-th] .
- Jacob McNamara and Cumrun Vafa, “Baby universes, holography, and the swampland,” (2020), arXiv:2004.06738 [hep-th] .
- Yuya Tanizaki and Mithat Ünsal, “Modified instanton sum in QCD and higher-groups,” Journal of High Energy Physics 2020, 123 (2020).
- Thomas Vandermeulen, “Lower-form symmetries,” (2022), arXiv:2211.04461 [hep-th] .
- Michael P. Zaletel and Frank Pollmann, “Isometric tensor network states in two dimensions,” Phys. Rev. Lett. 124, 037201 (2020).
- 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, 021902 (2018).
- Shuhei Ohyama and Shinsei Ryu, (2024), to appear.
Collections
Sign up for free to add this paper to one or more collections.
Paper Prompts
Sign up for free to create and run paper prompts using GPT-5.