1Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
2Department of Mathematics, University of Tübingen, 72076 Tübingen, Germany
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We consider random translation-invariant frustration-free quantum spin Hamiltonians on $mathbb Z^D$ in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.
Featured image: Assignment of the local interactions in 2D.
Popular summary
Here we study “typical” frustration-free translation-invariant Hamiltonians on any graph and their spectral gaps. We randomly generate translation-invariant Hamiltonians on any graph and prove that they are automatically frustration-free under a small rank constraint. Then we prove that these Hamiltonians are gapped with a positive probability.
The argument automatically controls the size of all gaps on intermediate system sizes as well (these are called “local gaps”). This allows us to verify a recent criterion for an area law for the entanglement of the ground state in 2D by Anshu-Arad-Gosset, again with positive probability.
► BibTeX data
► References
[1] Itai Arad, Zeph Landau, Umesh Vazirani, and Thomas Vidick. “Rigorous RG algorithms and area laws for low energy eigenstates in 1D”. Communications in Mathematical Physics 356, 65–105 (2017).
https://doi.org/10.1007/s00220-017-2973-z
[2] Matthew B Hastings. “An area law for one-dimensional quantum systems”. Journal of statistical mechanics: theory and experiment 2007, P08024 (2007).
https://doi.org/10.1088/1742-5468/2007/08/p08024
[3] Matthew B Hastings and Tohru Koma. “Spectral gap and exponential decay of correlations”. Communications in mathematical physics 265, 781–804 (2006).
https://doi.org/10.1007/s00220-006-0030-4
[4] Bruno Nachtergaele and Robert Sims. “Lieb-Robinson bounds and the exponential clustering theorem”. Communications in mathematical physics 265, 119–130 (2006).
https://doi.org/10.1007/s00220-006-1556-1
[5] Sven Bachmann, Spyridon Michalakis, Bruno Nachtergaele, and Robert Sims. “Automorphic equivalence within gapped phases of quantum lattice systems”. Communications in Mathematical Physics 309, 835–871 (2012).
https://doi.org/10.1007/s10955-015-1260-7
[6] Matthew B Hastings. “Lieb-Schultz-Mattis in higher dimensions”. Physical review b 69, 104431 (2004).
https://doi.org/10.1103/physrevb.69.104431
[7] Bruno Nachtergaele, Robert Sims, and Amanda Young. “Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms”. Journal of Mathematical Physics 60, 061101 (2019).
https://doi.org/10.1063/1.5095769
[8] Sergey Bravyi, Matthew B Hastings, and Spyridon Michalakis. “Topological quantum order: stability under local perturbations”. Journal of mathematical physics 51, 093512 (2010).
https://doi.org/10.1063/1.3490195
[9] Simone Del Vecchio, Jürg Fröhlich, Alessandro Pizzo, and Stefano Rossi. “Lie–schwinger block-diagonalization and gapped quantum chains with unbounded interactions”. Communications in Mathematical Physics 381, 1115–1152 (2021).
https://doi.org/10.1007/s00220-020-03878-y
[10] Wojciech De Roeck and Manfred Salmhofer. “Persistence of exponential decay and spectral gaps for interacting fermions”. Communications in Mathematical Physics 365, 773–796 (2019).
https://doi.org/10.1007/s00220-018-3211-z
[11] Spyridon Michalakis and Justyna P Zwolak. “Stability of frustration-free Hamiltonians”. Communications in Mathematical Physics 322, 277–302 (2013).
https://doi.org/10.1007/s00220-013-1762-6
[12] Bruno Nachtergaele, Robert Sims, and Amanda Young. “Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems” (2021).
[13] F Duncan M Haldane. “Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model”. Physics letters a 93, 464–468 (1983).
https://doi.org/10.1016/0375-9601(83)90631-x
[14] F Duncan M Haldane. “Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state”. Physical review letters 50, 1153 (1983).
https://doi.org/10.1103/physrevlett.50.1153
[15] Marius Lemm and Evgeny Mozgunov. “Spectral gaps of frustration-free spin systems with boundary”. Journal of Mathematical Physics 60, 051901 (2019).
https://doi.org/10.1063/1.5089773
[16] Alexei Kitaev. “Anyons in an exactly solved model and beyond”. Annals of Physics 321, 2–111 (2006).
https://doi.org/10.1016/j.aop.2005.10.005
[17] Michael A Levin and Xiao-Gang Wen. “String-net condensation: A physical mechanism for topological phases”. Physical Review B 71, 045110 (2005).
https://doi.org/10.1103/physrevb.71.045110
[18] Ian Affleck, Tom Kennedy, Elliott H Lieb, and Hal Tasaki. “Valence bond ground states in isotropic quantum antiferromagnets”. Communications in Mathematical Physics 115, 477–528 (1988).
https://doi.org/10.1007/bf01218021
[19] Akimasa Miyake. “Quantum computational capability of a 2D valence bond solid phase”. Annals of Physics 326, 1656–1671 (2011).
https://doi.org/10.1016/j.aop.2011.03.006
[20] Frank Verstraete and J Ignacio Cirac. “Valence-bond states for quantum computation”. Physical Review A 70, 060302 (2004).
https://doi.org/10.1103/physreva.70.060302
[21] Tzu-Chieh Wei, Ian Affleck, and Robert Raussendorf. “Affleck-Kennedy-Lieb-Tasaki state on a honeycomb lattice is a universal quantum computational resource”. Physical review letters 106, 070501 (2011).
https://doi.org/10.1103/physrevlett.106.070501
[22] Tzu-Chieh Wei, Poya Haghnegahdar, and Robert Raussendorf. “Hybrid valence-bond states for universal quantum computation”. Physical Review A 90, 042333 (2014).
https://doi.org/10.1103/physreva.90.042333
[23] Ramis Movassagh and Peter W Shor. “Supercritical entanglement in local systems: Counterexample to the area law for quantum matter”. Proceedings of the National Academy of Sciences 113, 13278–13282 (2016).
https://doi.org/10.1073/pnas.1605716113
[24] Zhao Zhang, Amr Ahmadain, and Israel Klich. “Novel quantum phase transition from bounded to extensive entanglement”. Proceedings of the National Academy of Sciences 114, 5142–5146 (2017).
https://doi.org/10.1073/pnas.1702029114
[25] Sven Bachmann and Bruno Nachtergaele. “Product vacua with boundary states and the classification of gapped phases”. Communications in Mathematical Physics 329, 509–544 (2014).
https://doi.org/10.1007/s00220-014-2025-x
[26] Sven Bachmann, Eman Hamza, Bruno Nachtergaele, and Amanda Young. “Product Vacua and Boundary State Models in $ d $ Dimensions”. Journal of Statistical Physics 160, 636–658 (2015).
https://doi.org/10.1007/s10955-015-1260-7
[27] Michael Bishop, Bruno Nachtergaele, and Amanda Young. “Spectral gap and edge excitations of d-dimensional PVBS models on half-spaces”. Journal of Statistical Physics 162, 1485–1521 (2016).
https://doi.org/10.1007/s10955-016-1457-4
[28] Bruno Nachtergaele. “The spectral gap for some spin chains with discrete symmetry breaking”. Communications in mathematical physics 175, 565–606 (1996).
https://doi.org/10.1007/bf02099509
[29] Houssam Abdul-Rahman, Marius Lemm, Angelo Lucia, Bruno Nachtergaele, and Amanda Young. “A class of two-dimensional AKLT models with a gap”. Analytic Trends in Mathematical Physics 741, 1–21 (2020).
https://doi.org/10.1090/conm/741/14917
[30] Marius Lemm and Bruno Nachtergaele. “Gapped PVBS models for all species numbers and dimensions”. Reviews in Mathematical Physics 31, 1950028 (2019).
https://doi.org/10.1142/s0129055x19500284
[31] Marius Lemm, Anders W Sandvik, and Ling Wang. “Existence of a spectral gap in the Affleck-Kennedy-Lieb-Tasaki model on the hexagonal lattice”. Physical review letters 124, 177204 (2020).
https://doi.org/10.1103/physrevlett.124.177204
[32] Marius Lemm, Anders W Sandvik, and Sibin Yang. “The AKLT model on a hexagonal chain is gapped”. Journal of Statistical Physics 177, 1077–1088 (2019).
https://doi.org/10.1007/s10955-019-02410-4
[33] Nicholas Pomata and Tzu-Chieh Wei. “AKLT models on decorated square lattices are gapped”. Physical Review B 100, 094429 (2019).
https://doi.org/10.1103/physrevb.100.094429
[34] Nicholas Pomata and Tzu-Chieh Wei. “Demonstrating the Affleck-Kennedy-Lieb-Tasaki spectral gap on 2d degree-3 lattices”. Physical review letters 124, 177203 (2020).
https://doi.org/10.1103/physrevlett.124.177203
[35] Bruno Nachtergaele, Simone Warzel, and Amanda Young. “Spectral Gaps and Incompressibility in a $nu=1/3$ Fractional Quantum Hall System”. Communications in Mathematical Physics 383, 1093–1149 (2021).
https://doi.org/10.1007/s00220-021-03997-0
[36] Simone Warzel and Amanda Young. “A bulk spectral gap in the presence of edge states for a truncated pseudopotential”. In Annales Henri Poincaré. Pages 1–46. Springer (2022).
https://doi.org/10.1007/s00023-022-01210-z
[37] Sergey Bravyi and David Gosset. “Gapped and gapless phases of frustration-free spin-$1/2$ chains”. Journal of Mathematical Physics 56, 061902 (2015).
https://doi.org/10.1063/1.4922508
[38] Marius Lemm. “Gaplessness is not generic for translation-invariant spin chains”. Physical Review B 100, 035113 (2019).
https://doi.org/10.1103/physrevb.100.035113
[39] Ramis Movassagh. “Generic local Hamiltonians are gapless”. Physical review letters 119, 220504 (2017).
https://doi.org/10.1103/physrevlett.119.220504
[40] Ramis Movassagh, Edward Farhi, Jeffrey Goldstone, Daniel Nagaj, Tobias J Osborne, and Peter W Shor. “Unfrustrated qudit chains and their ground states”. Physical Review A 82, 012318 (2010).
https://doi.org/10.1103/physreva.82.012318
[41] Or Sattath, Siddhardh C Morampudi, Chris R Laumann, and Roderich Moessner. “When a local Hamiltonian must be frustration-free”. Proceedings of the National Academy of Sciences 113, 6433–6437 (2016).
https://doi.org/10.1073/pnas.1519833113
[42] Roman Koteckỳ and David Preiss. “Cluster expansion for abstract polymer models”. Communications in Mathematical Physics 103, 491–498 (1986).
https://doi.org/10.1007/bf01211762
[43] Maciej Koch-Janusz, DI Khomskii, and Eran Sela. “Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice from $t_{2g}$ Orbitals”. Physical Review Letters 114, 247204 (2015).
https://doi.org/10.1103/physrevlett.114.247204
[44] Anurag Anshu, Itai Arad, and David Gosset. “An area law for 2D frustration-free spin systems”. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Pages 12–18. (2022).
https://doi.org/10.1145/3519935.3519962
[45] DA Yarotsky. “Uniqueness of the ground state in weak perturbations of non-interacting gapped quantum lattice systems”. Journal of statistical physics 118, 119–144 (2005).
https://doi.org/10.1007/s10955-004-8780-x
[46] Johannes Bausch, Toby S Cubitt, Angelo Lucia, and David Perez-Garcia. “Undecidability of the spectral gap in one dimension”. Physical Review X 10, 031038 (2020).
https://doi.org/10.1103/physrevx.10.031038
[47] Toby S Cubitt, David Perez-Garcia, and Wolf Michael M. “Undecidability of the spectral gap”. Nature 528, 207–211 (2015).
https://doi.org/10.1038/nature16059
[48] Anurag Anshu and Zeph Landau. personal communication.
[49] Nilin Abrahamsen. “Sharp implications of AGSPs for degenerate ground spaces” (2020).
[50] Stefan Knabe. “Energy gaps and elementary excitations for certain VBS-quantum antiferromagnets”. Journal of statistical physics 52, 627–638 (1988).
https://doi.org/10.1007/bf01019721
[51] Anurag Anshu. “Improved local spectral gap thresholds for lattices of finite size”. Physical Review B 101, 165104 (2020).
https://doi.org/10.1103/physrevb.101.165104
[52] David Gosset and Evgeny Mozgunov. “Local gap threshold for frustration-free spin systems”. Journal of Mathematical Physics 57, 091901 (2016).
https://doi.org/10.1063/1.4962337
[53] Marius Lemm. “Finite-size criteria for spectral gaps in D-dimensional quantum spin systems”. Analytic trends in mathematical physics 741, 121 (2020).
https://doi.org/10.1090/conm/741/14923
[54] Marius Lemm and David Xiang. “Quantitatively improved finite-size criteria for spectral gaps”. Journal of Physics A: Mathematical and Theoretical 55, 295203 (2022).
https://doi.org/10.1088/1751-8121/ac7989
[55] Joseph E Mayer. “The statistical mechanics of condensing systems. I”. The Journal of Chemical Physics 5, 67–73 (1937).
https://doi.org/10.1063/1.1749933
[56] HD Ursell. “The evaluation of Gibbs’ phase-integral for imperfect gases”. In Mathematical Proceedings of the Cambridge Philosophical Society. Volume 23, pages 685–697. Cambridge University Press (1927).
https://doi.org/10.1017/s0305004100011191
[57] J Groeneveld. “Two theorems on classical many-particle systems”. Phys. Letters 3 (1962).
https://doi.org/10.1016/0031-9163(62)90198-1
[58] David Ruelle. “Statistical mechanics: Rigorous results”. World Scientific. (1999).
https://doi.org/10.1142/4090
[59] Mark Fannes, Bruno Nachtergaele, and Reinhard F Werner. “Finitely correlated states on quantum spin chains”. Communications in mathematical physics 144, 443–490 (1992).
https://doi.org/10.1007/bf02099178
[60] J Von Neumann. “Functional Operators, Vol. II. The Geometry of Orthogonal Spaces (this is a reprint of mimeographed lecture notes first distributed in 1933) Annals of Math”. Studies Nr. 22 Princeton Univ. Press (1950).
https://doi.org/10.1515/9781400882250
Cited by
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.
