Exact holographic tensor networks for the Motzkin spin chain

Rafael N. Alexander1,2, Glen Evenbly3, and Israel Klich4

1Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia
2Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA
3School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
4Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network's utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice. We discuss connections to renormalization and holography.

► BibTeX data

► References

[1] Kenneth G. Wilson. The renormalization group and critical phenomena. Rev. Mod. Phys., 55: 583–600, Jul 1983. 10.1103/​RevModPhys.55.583.

[2] Michael E. Fisher. Renormalization group theory: Its basis and formulation in statistical physics. Rev. Mod. Phys., 70: 653–681, Apr 1998. 10.1103/​RevModPhys.70.653.

[3] Subir Sachdev. Quantum Phase Transitions. Cambridge University Press, 2 edition, 2011. 10.1017/​CBO9780511973765.

[4] H. v. Löhneysen, T. Pietrus, G. Portisch, H. G. Schlager, A. Schröder, M. Sieck, and T. Trappmann. Non-fermi-liquid behavior in a heavy-fermion alloy at a magnetic instability. Phys. Rev. Lett., 72: 3262–3265, May 1994. 10.1103/​PhysRevLett.72.3262.

[5] S R Julian, C Pfleiderer, F M Grosche, N D Mathur, G J McMullan, A J Diver, I R Walker, and G G Lonzarich. The normal states of magnetic d and f transition metals. Journal of Physics: Condensed Matter, 8 (48): 9675–9688, nov 1996. 10.1088/​0953-8984/​8/​48/​002.

[6] S. A. Grigera, R. S. Perry, A. J. Schofield, M. Chiao, S. R. Julian, G. G. Lonzarich, S. I. Ikeda, Y. Maeno, A. J. Millis, and A. P. Mackenzie. Magnetic field-tuned quantum criticality in the metallic ruthenate $\text{Sr}_3\text{Ru}_2\text{O}_7$. Science, 294 (5541): 329–332, 2001. 10.1126/​science.1063539.

[7] Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349: 117–158, 2014. 10.1016/​j.aop.2014.06.013.

[8] Jacob C Bridgeman and Christopher T Chubb. Hand-waving and interpretive dance: an introductory course on tensor networks. Journal of Physics A: Mathematical and Theoretical, 50 (22): 223001, may 2017. 10.1088/​1751-8121/​aa6dc3.

[9] Mark Fannes, Bruno Nachtergaele, and Reinhard F Werner. Finitely correlated states on quantum spin chains. Communications in mathematical physics, 144 (3): 443–490, 1992. 10.1007/​BF02099178.

[10] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69: 2863–2866, Nov 1992. 10.1103/​PhysRevLett.69.2863.

[11] G. Vidal. Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett., 101: 110501, Sep 2008. 10.1103/​PhysRevLett.101.110501.

[12] G. Vidal. Entanglement renormalization. Phys. Rev. Lett., 99: 220405, Nov 2007. 10.1103/​PhysRevLett.99.220405.

[13] Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett., 59: 799–802, Aug 1987. 10.1103/​PhysRevLett.59.799.

[14] Glen Evenbly and Steven R. White. Entanglement renormalization and wavelets. Phys. Rev. Lett., 116: 140403, Apr 2016. 10.1103/​PhysRevLett.116.140403.

[15] Jutho Haegeman, Brian Swingle, Michael Walter, Jordan Cotler, Glen Evenbly, and Volkher B. Scholz. Rigorous free-fermion entanglement renormalization from wavelet theory. Phys. Rev. X, 8: 011003, Jan 2018. 10.1103/​PhysRevX.8.011003.

[16] 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 (47): 13278–13282, 2016. 10.1073/​pnas.1605716113.

[17] Sergey Bravyi, Libor Caha, Ramis Movassagh, Daniel Nagaj, and Peter W. Shor. Criticality without frustration for quantum spin-1 chains. Phys. Rev. Lett., 109: 207202, Nov 2012. 10.1103/​PhysRevLett.109.207202.

[18] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, 2003. ISSN 0003-4916. https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0.

[19] Daniel S. Rokhsar and Steven A. Kivelson. Superconductivity and the quantum hard-core dimer gas. Phys. Rev. Lett., 61: 2376–2379, Nov 1988. 10.1103/​PhysRevLett.61.2376.

[20] David Perez-García, Frank Verstraete, Michael M Wolf, and J Ignacio Cirac. $\text{PEPS}$ as unique ground states of local $\text{Hamiltonians}$. Quantum Information & Computation, 8 (6-7): 650–663, 2008.

[21] Norbert Schuch, Ignacio Cirac, and David Pérez-García. Peps as ground states: Degeneracy and topology. Annals of Physics, 325 (10): 2153–2192, 2010. https:/​/​doi.org/​10.1016/​j.aop.2010.05.008.

[22] 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.

[23] Zhao Zhang, Amr Ahmadain, and Israel Klich. Novel quantum phase transition from bounded to extensive entanglement. Proceedings of the National Academy of Sciences, 114 (20): 5142–5146, 2017. 10.1073/​pnas.1702029114.

[24] Olof Salberger and Vladimir Korepin. Entangled spin chain. Reviews in Mathematical Physics, 29 (10): 1750031, 2017. 10.1142/​S0129055X17500313.

[25] L. Dell'Anna, O. Salberger, L. Barbiero, A. Trombettoni, and V. E. Korepin. Violation of cluster decomposition and absence of light cones in local integer and half-integer spin chains. Phys. Rev. B, 94: 155140, Oct 2016. 10.1103/​PhysRevB.94.155140.

[26] Olof Salberger, Takuma Udagawa, Zhao Zhang, Hosho Katsura, Israel Klich, and Vladimir Korepin. Deformed fredkin spin chain with extensive entanglement. Journal of Statistical Mechanics: Theory and Experiment, 2017 (6): 063103, jun 2017. 10.1088/​1742-5468/​aa6b1f.

[27] Zhao Zhang and Israel Klich. Entropy, gap and a multi-parameter deformation of the fredkin spin chain. Journal of Physics A: Mathematical and Theoretical, 50 (42): 425201, sep 2017. 10.1088/​1751-8121/​aa866e.

[28] Takuma Udagawa and Hosho Katsura. Finite-size gap, magnetization, and entanglement of deformed fredkin spin chain. Journal of Physics A: Mathematical and Theoretical, 50 (40): 405002, sep 2017. 10.1088/​1751-8121/​aa85b5.

[29] Fumihiko Sugino and Pramod Padmanabhan. Area law violations and quantum phase transitions in modified motzkin walk spin chains. Journal of Statistical Mechanics: Theory and Experiment, 2018 (1): 013101, jan 2018. 10.1088/​1742-5468/​aa9dcb.

[30] M. Bal, M. M. Rams, V. Zauner, J. Haegeman, and F. Verstraete. Matrix product state renormalization. Phys. Rev. B, 94: 205122, Nov 2016. 10.1103/​PhysRevB.94.205122.

[31] Fernando G. S. L. Brandão, Elizabeth Crosson, M. Burak Şahinoğlu, and John Bowen. Quantum error correcting codes in eigenstates of translation-invariant spin chains. Phys. Rev. Lett., 123: 110502, Sep 2019. 10.1103/​PhysRevLett.123.110502.

[32] Xiao Chen, Eduardo Fradkin, and William Witczak-Krempa. Gapless quantum spin chains: multiple dynamics and conformal wavefunctions. Journal of Physics A: Mathematical and Theoretical, 50 (46): 464002, oct 2017. 10.1088/​1751-8121/​aa8dbc.

[33] Sukhwinder Singh, Robert N. C. Pfeifer, and Guifre Vidal. Tensor network states and algorithms in the presence of a global u(1) symmetry. Phys. Rev. B, 83: 115125, Mar 2011. 10.1103/​PhysRevB.83.115125.

[34] I Klich, S. H. Lee, and K Iida. Glassiness and exotic entropy scaling induced by quantum fluctuations in a disorder-free frustrated magnet. Nature communications, 5 (3497), 2014. 10.1038/​ncomms4497.

[35] G. Evenbly and G. Vidal. Algorithms for entanglement renormalization. Phys. Rev. B, 79: 144108, Apr 2009. 10.1103/​PhysRevB.79.144108.

[36] Andrew J. Ferris. Fourier transform for fermionic systems and the spectral tensor network. Phys. Rev. Lett., 113: 010401, Jul 2014. 10.1103/​PhysRevLett.113.010401.

[37] Alexander Jahn, Zoltán Zimborás, and Jens Eisert. Central charges of aperiodic holographic tensor-network models. Phys. Rev. A, 102: 042407, Oct 2020. 10.1103/​PhysRevA.102.042407.

[38] Robert N. C. Pfeifer, Glen Evenbly, and Guifré Vidal. Entanglement renormalization, scale invariance, and quantum criticality. Phys. Rev. A, 79: 040301, Apr 2009. 10.1103/​PhysRevA.79.040301.

[39] J Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete. Matrix product unitaries: structure, symmetries, and topological invariants. Journal of Statistical Mechanics: Theory and Experiment, 2017 (8): 083105, aug 2017. 10.1088/​1742-5468/​aa7e55.

[40] M. Burak Şahinoğlu, Sujeet K. Shukla, Feng Bi, and Xie Chen. Matrix product representation of locality preserving unitaries. Phys. Rev. B, 98: 245122, Dec 2018. 10.1103/​PhysRevB.98.245122.

[41] Brian Swingle. Entanglement renormalization and holography. Phys. Rev. D, 86: 065007, Sep 2012. 10.1103/​PhysRevD.86.065007.

[42] Marika Taylor. Non-relativistic holography. arXiv preprint arXiv:0812.0530, 2008. https:/​/​arxiv.org/​abs/​0812.0530.

[43] Koushik Balasubramanian and John McGreevy. Gravity duals for nonrelativistic conformal field theories. Phys. Rev. Lett., 101: 061601, Aug 2008. 10.1103/​PhysRevLett.101.061601.

[44] D. T. Son. Toward an ads/​cold atoms correspondence: A geometric realization of the schrödinger symmetry. Phys. Rev. D, 78: 046003, Aug 2008. 10.1103/​PhysRevD.78.046003.

[45] Marika Taylor. Lifshitz holography. Classical and Quantum Gravity, 33 (3): 033001, jan 2016. 10.1088/​0264-9381/​33/​3/​033001.

[46] Rafael N. Alexander, Amr Ahmadain, Zhao Zhang, and Israel Klich. Exact rainbow tensor networks for the colorful motzkin and fredkin spin chains. Phys. Rev. B, 100: 214430, Dec 2019. 10.1103/​PhysRevB.100.214430.

Cited by

[1] Glen Evenbly, "Number-State Preserving Tensor Networks as Classifiers for Supervised Learning", arXiv:1905.06352.

[2] Rafael N. Alexander, Amr Ahmadain, Zhao Zhang, and Israel Klich, "Exact rainbow tensor networks for the colorful Motzkin and Fredkin spin chains", Physical Review B 100 21, 214430 (2019).

[3] Thomas Schuster, Bryce Kobrin, Ping Gao, Iris Cong, Emil T. Khabiboulline, Norbert M. Linke, Mikhail D. Lukin, Christopher Monroe, Beni Yoshida, and Norman Y. Yao, "Many-body quantum teleportation via operator spreading in the traversable wormhole protocol", arXiv:2102.00010.

[4] Jacob Miller, Guillaume Rabusseau, and John Terilla, "Tensor Networks for Probabilistic Sequence Modeling", arXiv:2003.01039.

[5] A. Ahmadain and I. Klich, "Emergent geometry and path integral optimization for a Lifshitz action", Physical Review D 103 10, 105013 (2021).

[6] Fumihiko Sugino, "Highly Entangled Spin Chains and 2D Quantum Gravity", Symmetry 12 6, 916 (2020).

The above citations are from SAO/NASA ADS (last updated successfully 2021-10-22 18:01:04). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2021-10-22 18:01:02).