Classical hydrodynamics is a remarkably versatile description of the coarse-grained behaviour of many-particle systems once local equilibrium has been established1. The form of the hydrodynamical equations is determined primarily by the conserved quantities present in a system. Some quantum spin chains are known to possess, even in the simplest cases, a greatly expanded set of conservation laws, and recent work suggests that these laws strongly modify collective spin dynamics, even at high temperature2,3. Here, by probing the dynamical exponent of the one-dimensional Heisenberg antiferromagnet KCuF3 with neutron scattering, we find evidence that the spin dynamics are well described by the dynamical exponent z = 3/2, which is consistent with the recent theoretical conjecture that the dynamics of this quantum system are described by the Kardar–Parisi–Zhang universality class4,5. This observation shows that low-energy inelastic neutron scattering at moderate temperatures can reveal the details of emergent quantum fluid properties like those arising in non-Fermi liquids in higher dimensions.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Nature Communications Open Access 02 October 2022
Subscribe to Nature+
Get immediate online access to the entire Nature family of 50+ journals
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
All plotted experimental data are publicly available at https://doi.org/10.13139/ORNLNCCS/1668822.
Landau, L. D. & Lifshitz, E. M. Fluid Mechanics 2nd edn (Butterworth-Heinemann, 1987).
Castro-Alvaredo, O. A., Doyon, B. & Yoshimura, T. Emergent hydrodynamics in integrable quantum systems out of equilibrium. Phys. Rev. X 6, 041065 (2016).
Bertini, B., Collura, M., De Nardis, J. & Fagotti, M. Transport in out-of-equilibrium XXZ chains: exact profiles of charges and currents. Phys. Rev. Lett. 117, 207201 (2016).
Kardar, M., Parisi, G. & Zhang, Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986).
Ljubotina, M., Žnidarič, M. & Prosen, Tcv Kardar–Parisi–Zhang physics in the quantum Heisenberg magnet. Phys. Rev. Lett. 122, 210602 (2019).
Giamarchi, T., Rüegg, C. & Tchernyshyov, O. Bose–Einstein condensation in magnetic insulators. Nat. Phys. 4, 198–204 (2008).
Haldane, F. D. M. Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983).
Sachdev, S. Quantum magnetism and criticality. Nat. Phys. 4, 173–185 (2008).
Faure, Q. et al. Topological quantum phase transition in the Ising-like antiferromagnetic spin chain BaCo2V2O8. Nat. Phys. 14, 716–722 (2018).
Giamarchi, T. Quantum Physics in One Dimension (Clarendon Press, 2004).
Caux, J.-S. & Hagemans, R. The four-spinon dynamical structure factor of the Heisenberg chain. J. Stat. Mech. Theory Exp. 2006, P12013–P12013 (2006).
Bulchandani, V. B., Vasseur, R., Karrasch, C. & Moore, J. E. Bethe-Boltzmann hydrodynamics and spin transport in the XXZ chain. Phys. Rev. B 97, 045407 (2018).
Schemmer, M., Bouchoule, I., Doyon, B. & Dubail, J. Generalized hydrodynamics on an atom chip. Phys. Rev. Lett. 122, 090601 (2019).
Dupont, M. & Moore, J. E. Universal spin dynamics in infinite-temperature one-dimensional quantum magnets. Phys. Rev. B 101, 121106 (2020).
De Nardis, J., Medenjak, M., Karrasch, C. & Ilievski, E. Universality classes of spin transport in one-dimensional isotropic magnets: the onset of logarithmic anomalies. Phys. Rev. Lett. 124, 210605 (2020).
Karrasch, C., Bardarson, J. H. & Moore, J. E. Finite-temperature dynamical density matrix renormalization group and the drude weight of spin-1/2 chains. Phys. Rev. Lett. 108, 227206 (2012).
Zotos, X., Naef, F. & Prelovsek, P. Transport and conservation laws. Phys. Rev. B 55, 11029–11032 (1997).
Prosen, T. Open XXZ spin chain: nonequilibrium steady state and a strict bound on ballistic transport. Phys. Rev. Lett. 106, 217206 (2011).
Ilievski, E., De Nardis, J., Medenjak, M. & Prosen, T. Superdiffusion in one-dimensional quantum lattice models. Phys. Rev. Lett. 121, 230602 (2018).
Gopalakrishnan, S. & Vasseur, R. Kinetic theory of spin diffusion and superdiffusion in XXZ spin chains. Phys. Rev. Lett. 122, 127202 (2019).
De Nardis, J., Medenjak, M., Karrasch, C. & Ilievski, E. Anomalous spin diffusion in one-dimensional antiferromagnets. Phys. Rev. Lett. 123, 186601 (2019).
Bulchandani, V. B. Kardar–Parisi–Zhang universality from soft gauge modes. Phys. Rev. B 101, 041411 (2020).
Takeuchi, K. A. & Sano, M. Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Phys. Rev. Lett. 104, 230601 (2010).
Somoza, A. M., Ortuño, M. & Prior, J. Universal distribution functions in two-dimensional localized systems. Phys. Rev. Lett. 99, 116602 (2007).
Kulkarni, M., Huse, D. A. & Spohn, H. Fluctuating hydrodynamics for a discrete Gross–Pitaevskii equation: mapping onto the Kardar–Parisi–Zhang universality class. Phys. Rev. A 92, 043612 (2015).
Nahum, A., Ruhman, J., Vijay, S. & Haah, J. Quantum entanglement growth under random unitary dynamics. Phys. Rev. X 7, 031016 (2017).
de Gier, J., Schadschneider, A., Schmidt, J. & Schütz, G. M. Kardar–Parisi–Zhang universality of the Nagel–Schreckenberg model. Phys. Rev. E 100, 052111 (2019).
Prähofer, M. & Spohn, H. Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255–279 (2004).
Hirakawa, K. & Kurogi, Y. One-dimensional antiferromagnetic properties of KCuF3. Prog. Theor. Phys. Suppl. 46, 147–161 (1970).
Tennant, D. A., Perring, T. G., Cowley, R. A. & Nagler, S. E. Unbound spinons in the S = 1/2 antiferromagnetic chain KCuF3. Phys. Rev. Lett. 70, 4003–4006 (1993).
Lake, B. et al. Multispinon continua at zero and finite temperature in a near-ideal Heisenberg chain. Phys. Rev. Lett. 111, 137205 (2013).
Dupont, M., Capponi, S., Laflorencie, N. & Orignac, E. Dynamical response and dimensional crossover for spatially anisotropic antiferromagnets. Phys. Rev. B 98, 094403 (2018).
Chakravarty, S., Halperin, B. I. & Nelson, D. R. Two-dimensional quantum Heisenberg antiferromagnet at low temperatures. Phys. Rev. B 39, 2344–2371 (1989).
Müller, G., Thomas, H., Beck, H. & Bonner, J. C. Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field. Phys. Rev. B 24, 1429–1467 (1981).
Granroth, G. E., Vandergriff, D. H. & Nagler, S. E. SEQUOIA: a fine resolution chopper spectrometer at the SNS. Physica B 385–386, 1104–1106 (2006).
Granroth, G. E. et al. SEQUOIA: a newly operating chopper spectrometer at the SNS. J. Phys. Conf. Ser. 251, 012058 (2010).
Azuah, R. T. et al. Dave: a comprehensive software suite for the reduction, visualization and analysis of low energy neutron spectroscopic data. J. Res. Natl Inst. Stand. Technol. 114, 341–358 (2009).
Schollwock, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011).
Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations Preprint at https://arxiv.org/pdf/2007.14822v1.pdf (2020).
Verstraete, F., García-Ripoll, J. J. & Cirac, J. I. Matrix product density operators: simulation of finite-temperature and dissipative systems. Phys. Rev. Lett. 93, 207204 (2004).
Holzner, A., Weichselbaum, A., McCulloch, I. P., Schollwöck, U. & von Delft, J. Chebyshev matrix product state approach for spectral functions. Phys. Rev. B 83, 195115 (2011).
This manuscript has been authored by UT-Batelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide licence to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). N.E.S., M.D. and J.E.M. were supported by the DOE, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05-CH11231 through the Scientific Discovery through Advanced Computing (SciDAC) programme (KC23DAC Topological and Correlated Matter via Tensor Networks and Quantum Monte Carlo). During the last period of the work, N.E.S. was supported by the DOE, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05-CH11231 through the Theory Institute for Molecular Spectroscopy (TIMES). J.E.M. was also supported by a Simons Investigatorship. This research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. D.A.T. and J.E.M. were supported by the DOE, Office of Science, National Quantum Information Science Research Centers. This research also used the Lawrencium computational cluster resource provided by the IT Division at the Lawrence Berkeley National Laboratory (supported by the Director, Office of Science, Office of Basic Energy Sciences, of the DOE under contract no. DE-AC02-05-CH11231). This research used resources of the Compute and Data Environment for Science (CADES) at the Oak Ridge National Laboratory, which is supported by the Office of Science of the DOE under contract no. DE-AC05-00OR22725. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility operated under contract no. DE-AC02-05CH11231.
The authors declare no competing interests.
Peer review information Nature Physics thanks Andrew Boothroyd, Igor Zaliznyak and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Scheie, A., Sherman, N.E., Dupont, M. et al. Detection of Kardar–Parisi–Zhang hydrodynamics in a quantum Heisenberg spin-1/2 chain. Nat. Phys. 17, 726–730 (2021). https://doi.org/10.1038/s41567-021-01191-6
Nature Communications (2022)