Locality estimates for complex time evolution in 1D

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It is a generalized belief that there are no thermal phase transitions in short range 1D quantum systems. However, the only known case for which this is rigorously proven is for the particular case of finite range translational invariant interactions. The proof was obtained by Araki in his seminal paper of 1969 as a consequence of pioneering locality estimates for the time-evolution operator that allowed him to prove its analiticity on the whole complex plane, when applied to a local observable. However, as for now there is no mathematical proof of the abscence of 1D thermal phase transitions if one allows exponential tails in the interactions. In this work we extend Araki’s result to include exponential (or faster) tails. Our main result is the analyticity of the time-evolution operator applied on a local observable on a suitable strip around the real line. As a consequence we obtain that thermal states in 1D exhibit exponential decay of correlations above a threshold temperature that decays to zero with the exponent of the interaction decay, recovering Araki’s result as a particular case. Our result however still leaves open the possibility of 1D thermal short range phase transitions. We conclude with an application of our result to the spectral gap problem for Projected Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to Cirac et al.
[1] H. Araki Gibbs states of the one-dimensional quantum spin chain. Comm. Math. Phys. 14, 120–157 (1969). [2] H. Araki Expansional in Banach algebras. Ann. scient. Ec. Norm. Sup. , 4 Serie, t. 6, 67–84 (1973). [3] H. Araki On uniqueness of KMS states of one-dimensional quantum lattice systems. Comm. Math. Phys. 44, no. 1, 1–7 (1975). [4] V. Baladi Positive transfer operators and decay of correlations. World Scientific, Singapore, 2000. [5] S. Bachmann, S. Michalakis, B. Nachtergaele, R. Sims Automorphic equivalence within gapped phases of quantum lattice systems. Comm. Math. Phys. 309, 835–871 (2012) [6] G. Bouch Complex-time singularity and locality estimates for quantum lattice systems. J.Math. Phys. 56, 123303 (2015). [7] O. Bratteli and D. Robinson Operator Algebras and Quantum Statistical Mechanics I. 2nd edition, Springer, 1987. [8] O. Bratteli and D. Robinson Operator Algebras and Quantum Statistical Mechanics II. 2nd edition, Springer, 1997. [9] J. I. Cirac, D. Poilblanc, N. Schuch, and F. Verstraete Entanglement spectrum and boundary theories with projected entangled-pair states. Phys. Rev. B 83, 245134 (2011). [10] R. L. Dobrushin, Analyticity of correlation functions in one-dimensional classical systems with slowly decreasing potentials. Comm. Math. Phys. 32, no 4, 269–289 (1973). [11] C. Domb and M.S. Green (Eds.) Phase Transitions and Critical Phenomena. Vol. I, Aca�demic Press, London (1972). [12] F. Dyson Existence of a phase transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys. 12, 91–107 (1969). [13] V. Golodets and S. V Neshveyev Gibbs states for AF-algebras. J. Math. Phys. 234, 6329–6344 (1998). [14] J. Haah, M. B. Hastings, R. Kothari, and G. Hao Low Quantum algorithm for simulat�ing real time evolution of lattice Hamiltonians. In 59th Annual IEEE Symposium on Founda�tions of Computer Science – FOCS 2018, pages 350–360. IEEE Computer Soc., Los Alamitos, CA, 2018. [15] M. B. Hastings Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004). [16] M. B. Hastings Quantum belief propagation: an algorithm for thermal quantum systems. Phys. Rev. B 76, 201102 (2007). [17] M. B. Hastings An area law for one dimensional quantum systems. J. Stat. Mech. 2007, P08024. [18] M. B. Hastings Locality in quantum systems. In Lecture Notes of the Les Houches Summer School 95. Quantum Theory from Small to Large Scales, pages 171–211. Oxford University Press (2012). [19] M. B. Hastings and T. Koma Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006). [20] M. J. Kastoryano, A. Lucia Divide and conquer method for proving gaps of frustration free Hamiltonians. J. Stat. Mech. Theory Exp., vol. 2018, 033105, 1–23 (2018). [21] M. J. Kastoryano, A. Lucia and D. Perez-Garcia Locality at the boundary implies gap in the bulk for 2D PEPS. Comm. Math. Phys. 366, 895–926 (2019). [22] A. Kishimoto On uniqueness of KMS states of one-dimensional quantum lattice systems. Comm. Math. Phys. 47, no. 2, 167–170 (1976). [23] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert Locality of temperature. Phys. Rev. X 4, 031019 (2014) . [24] T. Kuwahara, K. Kato and F. Brandao˜ Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature. arXiv preprint arXiv:1910.09425 (2019). [25] M. Lenci and L. Rey-Bellet Large deviations in quantum lattice systems: one-phase region. J. Stat. Phys., 119 (3-4), 715–746 (2005). [26] E. Lieb and D. Robinson The finite group velocity of quantum spin systems. Comm. Math. Phys. 28, 251–257 (1972). [27] T. Matsui On non-commutative Ruelle transfer operator. Rev. Math. Phys. 13, no. 10, 1183–1201 (2001). [28] T. Matsui Bosonic central limit theorem for the one-dimensional XY model. Rev. Math. Phys., 14 (7-8), 675–700, (2002). Dedicated to Professor Huzihiro Araki on the occasion of his 70th birthday. [29] T. Matsui On the algebra of fluctuation in quantum spin chains. Ann. Henri Poincar, 4 (1), 63–83 (2003). [30] B. Nachtergaele, Y. Ogata, and R. Sims Propagation of correlations in quantum lattice systems. J. Stat. Phys., 124 (1), 1–13 (2006). [31] B. Nachtergaele and R. Sims Lieb-Robinson bounds and the exponential clustering theo�rem. Comm. Math. Phys., 265 (1), 119–130 (2006) [32] B. Nachtergaele and R. Sims Much ado about something - Why Lieb-Robinson bounds are useful. IAMP News Bull. 22–29 (2010). [33] B. Nachtergaele and R. Sims Lieb-Robinson bounds in quantum many-body physics. In Entropy and the quantum, volume 529 of Contemp. Math., pages 141–176. Amer. Math. Soc., Providence, RI (2010). [34] B. Nachtergaele, R. Sims and A. Young Quasi-locality bounds for quantum lattice sys�tems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms. J. Math. Phys. 60, 061101 (2019). [35] Y. Ogata Large deviations in quantum spin chains. Comm. Math. Phys., 296 (1), 35–68 (2010). [36] Y. Ogata and L. Rey-Bellet Ruelle-Lanford functions and large deviations for asymptot�ically decoupled quantum systems. Rev. Math. Phys., 23 (2), 211–232 (2011). [37] D. Pérez-García, F. Verstraete, I. Cirac, and M. Wolf PEPS as unique ground states of local Hamiltonians. Quant. Inf. Comp. 8, 0650 (2008). [38] D. W. Robinson Statistical mechanics of quantum spin systems. II. Comm. Math. Phys. 7, no. 4, 337–348 (1968). [39] D. Ruelle Statistical Mechanics of a One-dimensional Lattice Gas. Comm. Math. Phys. 9, no. 4, 267–278 (1968). [40] D. Ruelle Statistical Mechanics. Benjamin, New York-Amsterdam (1969) [41] D. Ruelle Thermodynamic formalism: the mathematical structure of equilibrium statistical mechanics. Cambridge University Press (2004)