Abstract
The generalized hydrodynamic (GHD) approach has been extremely successful in describing the out-of-equilibrium properties of a great variety of integrable many-body quantum systems. It naturally extracts the large-scale dynamical degrees of freedom of the system, and is thus a particularly good probe for emergent phenomena. One such phenomenon is the presence of unstable particles, traditionally seen via special analytic structures of the scattering matrix. Because of their finite lifetime and energy threshold, these are especially hard to study. In this paper we apply the GHD approach to a model possessing both unstable excitations and quantum integrability. The largest family of relativistic integrable quantum field theories known to have these features are the homogeneous sine-Gordon models. We consider the simplest non-trivial example of such theories and investigate the effect of an unstable excitation on various physical quantities, both at equilibrium and in the non-equilibrium state arising from the partitioning protocol. The hydrodynamic approach sheds new light onto the physics of the unstable particle, going much beyond its definition via the analytic structure of the scattering matrix, and clarifies its effects both on the equilibrium and out-of-equilibrium properties of the theory. Crucially, within this dynamical perspective, we identify unstable particles as finitely-lived bound states of co-propagating stable particles of different types, and observe how stable populations of unstable particles emerge in large-temperature thermal baths.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Eisert, M. Friesdorf and C. Gogolin, Quantum many-body systems out of equilibrium, Nature Phys. 11 (2015) 124.
T. Kinoshita, T. Wenger and D. Weiss, A Quantum Newton’s Cradle, Nature 440 (2006) 900.
M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons, Phys. Rev. Lett. 98 (2007) 050405 [cond-mat/0604476].
B. Pozsgay, M. Mestyán, M.A. Werner, M. Kormos, G. Zaránd and G. Takács, Correlations after Quantum Quenches in the X X Z Spin Chain: Failure of the Generalized Gibbs Ensemble, Phys. Rev. Lett. 113 (2014) 117203 [arXiv:1405.2843].
M. Mierzejewski, P. Prelovšek and T. Prosen, Breakdown of the Generalized Gibbs Ensemble for Current-Generating Quenches, Phys. Rev. Lett. 113 (2014) 020602 [arXiv:1405.2557].
T. Prosen, Quasilocal conservation laws in XXZ spin-1/2 chains: Open, periodic and twisted boundary conditions, Nucl. Phys. B 886 (2014) 1177 [arXiv:1406.2258] [INSPIRE].
M. Mierzejewski, P. Prelovšek and T. Prosen, Identifying Local and Quasilocal Conserved Quantities in Integrable Systems, Phys. Rev. Lett. 114 (2015) 140601 [arXiv:1412.6974].
E. Ilievski, M. Medenjak and T. Prosen, Quasilocal Conserved Operators in the Isotropic Heisenberg Spin-1/2 Chain, Phys. Rev. Lett. 115 (2015) 120601 [arXiv:1506.05049] [INSPIRE].
B. Doyon, Thermalization and Pseudolocality in Extended Quantum Systems, Commun. Math. Phys. 351 (2017) 155 [arXiv:1512.03713].
E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F.H.L. Essler and T. Prosen, Complete Generalized Gibbs Ensembles in an Interacting Theory, Phys. Rev. Lett. 115 (2015) 157201 [arXiv:1507.02993].
P. Calabrese, H. Essler and G. Mussardo, Introduction to ‘Quantum Integrability in Out-of-Equilibrium Systems’, J. Stat. Mech. 06 (2016) 4001.
O.A. Castro-Alvaredo, B. Doyon and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6 (2016) 041065 [arXiv:1605.07331] [INSPIRE].
B. Bertini, M. Collura, J. De Nardis and M. Fagotti, Transport in Out-of-Equilibrium X X Z Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117 (2016) 207201 [arXiv:1605.09790] [INSPIRE].
M. Fagotti, Charges and currents in quantum spin chains: late-time dynamics and spontaneous currents, J. Phys. A 50 (2017) 034005 [arXiv:1608.02869].
A. Urichuk, Y. Oez, A. Klümper and J. Sirker, The spin Drude weight of the XXZ chain and generalized hydrodynamics, SciPost Phys. 6 (2019) 005 [arXiv:1808.09033] [INSPIRE].
D.-L. Vu and T. Yoshimura, Equations of state in generalized hydrodynamics, SciPost Phys. 6 (2019) 023 [arXiv:1809.03197] [INSPIRE].
J. De Nardis, D. Bernard and B. Doyon, Diffusion in generalized hydrodynamics and quasiparticle scattering, SciPost Phys. 6 (2019) 049 [arXiv:1812.00767] [INSPIRE].
Z. Bajnok and I. Vona, Exact finite volume expectation values of conserved currents, Phys. Lett. B 805 (2020) 135446 [arXiv:1911.08525] [INSPIRE].
H. Spohn, The collision rate ansatz for the classical Toda lattice, Phys. Rev. E 101 (2020) 060103 [arXiv:2004.03802] [INSPIRE].
T. Yoshimura and H. Spohn, Collision rate ansatz for quantum integrable systems, arXiv:2004.07113 [INSPIRE].
M. Borsi, B. Pozsgay and L. Pristyák, Current Operators in Bethe Ansatz and Generalized Hydrodynamics: An Exact Quantum-Classical Correspondence, Phys. Rev. X 10 (2020) 011054 [arXiv:1908.07320] [INSPIRE].
B. Pozsgay, Current operators in integrable spin chains: lessons from long range deformations, SciPost Phys. 8 (2020) 016 [arXiv:1910.12833] [INSPIRE].
B. Pozsgay, Algebraic construction of current operators in integrable spin chains, Phys. Rev. Lett. 125 (2020) 070602 [arXiv:2005.06242] [INSPIRE].
D. Bernard and B. Doyon, Conformal field theory out of equilibrium: a review, J. Stat. Mech. 1606 (2016) 064005 [arXiv:1603.07765] [INSPIRE].
R. Vasseur and J.E. Moore, Nonequilibrium quantum dynamics and transport: from integrability to many-body localization, J. Stat. Mech. 1606 (2016) 064010 [arXiv:1603.06618] [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
T.R. Klassen and E. Melzer, The Thermodynamics of purely elastic scattering theories and conformal perturbation theory, Nucl. Phys. B 350 (1991) 635 [INSPIRE].
J. Mossel and J.-S. Caux, Generalized TBA and generalized Gibbs, J. Phys. A 45 (2012) 255001 [arXiv:1203.1305] [INSPIRE].
B. Doyon and T. Yoshimura, A note on generalized hydrodynamics: inhomogeneous fields and other concepts, SciPost Phys. 2 (2017) 014 [arXiv:1611.08225] [INSPIRE].
A. Bastianello and A. De Luca, Integrability-Protected Adiabatic Reversibility in Quantum Spin Chains, Phys. Rev. Lett. 122 (2019) 240606 [arXiv:1811.07922] [INSPIRE].
A. Bastianello, V. Alba and J.-S. Caux, Generalized Hydrodynamics with Space-Time Inhomogeneous Interactions, Phys. Rev. Lett. 123 (2019) 130602 [arXiv:1906.01654] [INSPIRE].
J. De Nardis, D. Bernard and B. Doyon, Hydrodynamic Diffusion in Integrable Systems, Phys. Rev. Lett. 121 (2018) 160603 [arXiv:1807.02414] [INSPIRE].
S. Gopalakrishnan, D.A. Huse, V. Khemani and R. Vasseur, Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems, Phys. Rev. B 98 (2018) 220303 [arXiv:1809.02126] [INSPIRE].
M. Fagotti, Locally quasi-stationary states in noninteracting spin chains, SciPost Phys. 8 (2020) 048 [arXiv:1910.01046] [INSPIRE].
A. Bastianello, J. De Nardis and A. De Luca, Generalised hydrodynamics with dephasing noise, arXiv:2003.01702 [INSPIRE].
X. Cao, V.B. Bulchandani and J.E. Moore, Incomplete Thermalization from Trap-Induced Integrability Breaking: Lessons from Classical Hard Rods, Phys. Rev. Lett. 120 (2018) 164101 [arXiv:1710.09330] [INSPIRE].
A.J. Friedman, S. Gopalakrishnan and R. Vasseur, Diffusive hydrodynamics from integrability breaking, Phys. Rev. B 101 (2020) 180302 [arXiv:1912.08826] [INSPIRE].
J. Durnin, M.J. Bhaseen and B. Doyon, Non-Equilibrium Dynamics and Weakly Broken Integrability, arXiv:2004.11030 [INSPIRE].
M. Schemmer, I. Bouchoule, B. Doyon and J. Dubail, Generalized Hydrodynamics on an Atom Chip, Phys. Rev. Lett. 122 (2019) 090601 [arXiv:1810.07170].
B. Doyon, Lecture notes on Generalised Hydrodynamics, arXiv:1912.08496 [INSPIRE].
C.R. Fernandez-Pousa, M.V. Gallas, T.J. Hollowood and J. Miramontes, Solitonic integrable perturbations of parafermionic theories, Nucl. Phys. B 499 (1997) 673 [hep-th/9701109] [INSPIRE].
C.R. Fernandez-Pousa, M.V. Gallas, T.J. Hollowood and J. Miramontes, The Symmetric space and homogeneous sine-Gordon theories, Nucl. Phys. B 484 (1997) 609 [hep-th/9606032] [INSPIRE].
C.R. Fernandez-Pousa and J. Miramontes, Semiclassical spectrum of the homogeneous sine-Gordon theories, Nucl. Phys. B 518 (1998) 745 [hep-th/9706203] [INSPIRE].
J. Miramontes and C.R. Fernandez-Pousa, Integrable quantum field theories with unstable particles, Phys. Lett. B 472 (2000) 392 [hep-th/9910218] [INSPIRE].
B. Doyon, T. Yoshimura and J.-S. Caux, Soliton Gases and Generalized Hydrodynamics, Phys. Rev. Lett. 120 (2018) 045301 [arXiv:1704.05482] [INSPIRE].
M. Karowski and P. Weisz, Exact Form-Factors in (1+1)-Dimensional Field Theoretic Models with Soliton Behavior, Nucl. Phys. B 139 (1978) 455 [INSPIRE].
F. Smirnov, Form factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14, World Scientific, Singapore (1992).
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
E. Witten, Non-abelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 455.
S.P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Sov. Math., Dokl. 24 (1981) 222.
S.P. Novikov, The Hamiltonian formalism and a many-valued analogue of Morse theory, Russ. Math. Sur. 37 (1982) 1.
O.A. Castro-Alvaredo, A. Fring, C. Korff and J.L. Miramontes, Thermodynamic Bethe ansatz of the homogeneous sine-Gordon models, Nucl. Phys. B 575 (2000) 535 [hep-th/9912196] [INSPIRE].
O.A. Castro-Alvaredo, J. Dreissig and A. Fring, Integrable scattering theories with unstable particles, Eur. Phys. J. C 35 (2004) 393 [hep-th/0211168] [INSPIRE].
P. Dorey and J. Miramontes, Mass scales and crossover phenomena in the homogeneous sine-Gordon models, Nucl. Phys. B 697 (2004) 405 [hep-th/0405275] [INSPIRE].
O.A. Castro-Alvaredo, A. Fring and C. Korff, Form-factors of the homogeneous sine-Gordon models, Phys. Lett. B 484 (2000) 167 [hep-th/0004089] [INSPIRE].
O.A. Castro-Alvaredo and A. Fring, Identifying the operator content, the homogeneous sine-Gordon models, Nucl. Phys. B 604 (2001) 367 [hep-th/0008044] [INSPIRE].
O.A. Castro-Alvaredo and A. Fring, Renormalization group flow with unstable particles, Phys. Rev. D 63 (2001) 021701 [hep-th/0008208] [INSPIRE].
O.A. Castro-Alvaredo and A. Fring, Decoupling the SU(N )(2) homogeneous sine-Gordon model, Phys. Rev. D 64 (2001) 085007 [hep-th/0010262] [INSPIRE].
Z. Bajnok, J. Balog, K. Ito, Y. Satoh and G.Z. Toth, On the mass-coupling relation of multi-scale quantum integrable models, JHEP 06 (2016) 071 [arXiv:1604.02811] [INSPIRE].
Z. Bajnok, J. Balog, K. Ito, Y. Satoh and G.Z. Tóth, Exact mass-coupling relation for the homogeneous sine-Gordon model, Phys. Rev. Lett. 116 (2016) 181601 [arXiv:1512.04673] [INSPIRE].
A.B. Zamolodchikov, Resonance factorized scattering and roaming trajectories, J. Phys. A 39 (2006) 12847 [INSPIRE].
P. Dorey, G. Siviour and G. Takács, Form factor relocalisation and interpolating renormalisation group flows from the staircase model, JHEP 03 (2015) 054 [arXiv:1412.8442] [INSPIRE].
D.X. Horvath, P.E. Dorey and G. Takács, Roaming form factors for the tricritical to critical Ising flow, JHEP 07 (2016) 051 [arXiv:1604.05635] [INSPIRE].
P. Dorey and F. Ravanini, Generalizing the staircase models, Nucl. Phys. B 406 (1993) 708 [hep-th/9211115] [INSPIRE].
D.X. Horváth, Hydrodynamics of massless integrable RG flows and a non-equilibrium c-theorem, JHEP 10 (2019) 020 [arXiv:1905.08590] [INSPIRE].
L. Bonnes, F.H.L. Essler and A.M. Läuchli, “Light-Cone” Dynamics After Quantum Quenches in Spin Chains, Phys. Rev. Lett. 113 (2014) 187203 [arXiv:1404.4062] [INSPIRE].
D. Bernard and B. Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A 45 (2012) 362001 [arXiv:1202.0239] [INSPIRE].
D. Bernard and B. Doyon, Non-Equilibrium Steady States in Conformal Field Theory, Ann. Henri Poincaré 16 (2014) 113.
D. Bernard and B. Doyon, Time-reversal symmetry and fluctuation relations in non-equilibrium quantum steady states, J. Phys. A 46 (2013) 372001 [arXiv:1306.3900] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
O. Castro-Alvaredo, Y. Chen, B. Doyon and M. Hoogeveen, Thermodynamic Bethe ansatz for non-equilibrium steady states: exact energy current and fluctuations in integrable QFT, J. Stat. Mech. 1403 (2014) P03011 [arXiv:1310.4779] [INSPIRE].
D. Bernard, B. Doyon and J. Viti, Non-Equilibrium Conformal Field Theories with Impurities, J. Phys. A 48 (2015) 05FT01 [arXiv:1411.0470] [INSPIRE].
S. Fischer, C. Karrasch, D. Schuricht and L. Fritz, Energy transport between critical one-dimensional systems with different central charges, Phys. Rev. B 101 (2020) 205146 [arXiv:2002.10844] [INSPIRE].
L. Mazza, J. Viti, M. Carrega, D. Rossini and A. De Luca, Energy transport in an integrable parafermionic chain via generalized hydrodynamics, Phys. Rev. B 98 (2018) 075421 [arXiv:1804.04476] [INSPIRE].
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and F. Ravanini, Evolution of the equilibrium effective velocities of the SU3 (2)-HSG Model, 8 May 2020, https://youtu.be/lvWd4qxMShQ.
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and F. Ravanini, Evolution of the equilibrium spectral densities of the SU3 (2)-HSG Model, 8 May 2020, https://youtu.be/7jX0HFa1cgs.
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and F. Ravanini, Evolution of the out-of-equilibrium effective velocities of the SU3 (2)-HSG Model, 22 May 2020, https://youtu.be/m2ApWaQkcHE.
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and F. Ravanini, Evolution of the out-of-Equilibrium Spectral Densities of the SU3 (2)-HSG Model, 22 May 2020, https://youtu.be/suIftU1oNcw.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2005.11266
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Castro-Alvaredo, O.A., De Fazio, C., Doyon, B. et al. On the hydrodynamics of unstable excitations. J. High Energ. Phys. 2020, 45 (2020). https://doi.org/10.1007/JHEP09(2020)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2020)045