Abstract
Quarks cannot be observed as free particles in nature because they are confined into baryons and mesons, as a result of the fact that the strong interaction between them increases with their separation. However, it is less known that this phenomenon also occurs in condensed matter and statistical physics as experimentally proved in several quasi1D compounds^{1,2}. Most of the theoretical and experimental studies so far concentrated on understanding the consequences of confinement for the equilibrium physics of both highenergy and condensed matter systems. Here, instead we show that confinement has dramatic consequences for the nonequilibrium dynamics following a quantum quench and that these effects could be exploited as a quantitative probe of confinement.
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Main
A global quantum quench is the nonequilibrium dynamics initiated by a sudden change of a parameter in the Hamiltonian of an isolated quantum system, a protocol that is routinely engineered in coldatom experiments^{3,4,5,6,7,8,9,10,11}. According to a by now standard picture^{12}, the initial state acts as a source of quasiparticle excitations. A quasiparticle of momentum p moves with velocity v_{p} and carries quantum correlations through the systems. If there is a maximum speed of propagation v_{max} ≥ v_{p} (for example, as a consequence of the Lieb–Robinson bound^{13}), all connected correlations at distance ℓ vanish for times t < ℓ/2v_{max} (ref. 12) and the entanglement entropy of an interval of length ℓ grows linearly in the same time window^{14}.
How can confinement change qualitatively the spreading of correlations? Exactly as in the standard scenario, the initial state emits pairs of quasiparticles starting off in opposite directions. However, these are eventually turned back by the confining potential, leading to an oscillatory behaviour as shown in Fig. 1. In analogy to strong interaction, the bound states formed by this process are called mesons. Mesons produced after a quench typically move much slower than the elementary quasiparticles due to their larger mass.
Confinement takes place in one of the paradigmatic models of statistical mechanics, the Ising chain in both transverse (h_{z}) and longitudinal (h_{x}) magnetic fields with Hamiltonian
where the coupling J sets the energy scale and σ_{j}^{α} are the Pauli matrices. This model has been engineered with cold atoms^{15} and quench protocols have also been implemented^{9}.
For h_{x} = 0, the Hamiltonian (1) is diagonalized by a mapping to free fermions (see Supplementary Information). For h_{z} < 1 the system is in the ferromagnetic phase where the massive fermions are freely propagating domain walls separating domains of opposite magnetization. Switching on a nonzero h_{x} induces a linear potential between pairs of domain walls confining them into mesons^{16}. In this Letter, we report numerical results for several observables showing that the nonequilibrium quench dynamics is drastically altered by confinement.
Expectation value of the order parameter
We first consider the time evolution of the magnetization 〈σ^{x}(t)〉 that in the integrable case (h_{x} = 0) decays exponentially to zero^{17} (see Supplementary Information). For nonzero h_{x}, we report the numerical data for 〈σ^{x}(t)〉 in Fig. 2 for two representative quenches. It is evident that even a small longitudinal field alters substantially the dynamics, turning the exponential relaxation into an oscillatory behaviour with numerous frequencies^{18}. The power spectrum presented in Fig. 2 shows that the dominant frequencies are compatible, to a surprising high degree of accuracy, with the masses of the mesons and their differences.
The twopoint function shows the clearest signature of confinement manifested in a dramatically altered lightcone structure. In Fig. 3a we report the equaltime connected correlation function 〈σ_{1}^{x}σ_{m+1}^{x}〉_{c} for the quench from the fully ferromagnetic state to h_{z} = 0.25 and varying h_{x} from 0 to 0.4. If h_{x} = 0, we recover the integrable dynamics with a clear lightcone spreading. For small h_{x} = 0.025 the correlation follows qualitatively the integrable behaviour for short times, but then it drastically slows down and bounces back. Further increasing h_{x}, the region with lightcone propagation shrinks to an almost invisible portion of the spacetime. This effect can be explained by considering that for a quench with h_{x} = 0 the initial state can be written as^{17}
in terms of the postquench domain wall excitation a_{k}^{†}. This formula quantifies the fact that the initial state can be written in terms of pairs of particles with opposite momenta and K(k) is the amplitude of production of these pairs (the exact dependence of K(k) on k is given in Supplementary Information, but this is inessential for the following). Adding a field h_{x} confines the domain walls into mesons. When K(k) is small, the state is dominated by the linear terms that contain only (k, −k) domain walls pairs that get confined into mesons at rest. Quadratic terms (and higher order ones) lead to propagating mesons, but they are substantial only when K(k) is large enough. These have velocities that are significantly different from those of the domain walls (see Supplementary Information).
For the quench in Fig. 3a, K(k) ≪ 1 for all momenta and practically only zeromomentum mesons are formed. Zooming in the ‘white’ region apparently without signal in Fig. 3a, traces of mesons with nonzero velocities should be visible. This is done in Fig. 3b where the same connected correlation is displayed on a different intensity scale. The signal in Fig. 3b is three orders of magnitude smaller than that in Fig. 3a and shows a feeble light cone characterized by a velocity different from that of the domain walls. The maximum meson velocity describes incredibly well the slope of the light cone for all values of h_{x}.
As further supporting evidence for the above scenario, in Fig. 3c we report connected correlations for a large quench (with K(k) ≫ 1) from the paramagnetic phase to the ferromagnetic confining one. In this case, given that mesons with nonzero velocities are formed with high probability, the light cones are visible without zooming, and their slopes correspond to the maximal velocity of the mesons. A crucial consistency check of the validity of the overall scenario is that for quenches to the paramagnetic (nonconfining) phase with h_{x} ≠ 0 there is no radical change in the light cone as shown in Fig. 3d.
Similarly, the connected correlation function of the transverse component of the spin also reflects the change of the light cone due to the modified velocity of the mesons (see Supplementary Information). Furthermore, for quenches from and to several other values of the magnetic fields the overall picture is unchanged, so our conclusions are very general and not limited to the reported cases.
Entanglement entropy provides another smoking gun for the quasiparticle propagation and lightcone effects^{14}. For zero h_{x}, we observe in Fig. 4 a linear growth in time of the halfchain entanglement entropy in agreement with the known exact results^{19}. By turning on the interaction h_{x}, the growth of the entanglement entropy for quenches within the ferromagnetic phase is strongly slowed down and, after a transient, it appears to oscillate around a saturation value. The frequencies of these oscillations are in rough agreement with the meson masses and their differences as reported in Supplementary Information. This is a consequence of the fact that mesons are predominantly produced at rest and it is consistent with the strong suppression of the light cone of the twopoint function. Actually, the small fraction of mesons with nonnegligible velocities should produce a very slow increase of the entanglement that, however, is too small to be observed numerically. Conversely, for a quench across the critical point, the growth of the entanglement entropy is not negligible, but it is considerably reduced due to the production of many mesons with nonvanishing velocities. Overall, the data for the entanglement are compatible with the confinement scenario drawn for the correlations.
Discussions
We have given compelling arguments and numerical evidence proving that confinement strongly affects the nonequilibrium dynamics following a quantum quench. The main effect is a dramatic change of the lightcone structure of correlation functions and entanglement entropy. At the same time, the onepoint function oscillates in time with frequencies equal to the meson masses. These effects should be easily measurable in coldatom experiments; corrections due to the trapping harmonic potential are expected to be as small as the finitesize effects we observed. We also showed that the quench dynamics provides direct access to the meson masses via the power spectrum of the onepoint functions. This ‘quench spectroscopy’ could turn out to be more powerful than standard equilibrium methods to measure the spectrum.
To conclude, we speculate on a few other consequences and applications of our work. It was noticed some time ago^{20} that in some quenches within the Hamiltonian (1) the system does not approach asymptotically a thermal stationary state as expected from the nonintegrability of the model. A possible explanation is that due to confinement there are rare states in the spectrum that prevent the eigenstate thermalization hypothesis^{21} from being applied. Along the same line of thought, even if these confined systems eventually thermalize, the standard prethermalization scenario^{22,23} for weak integrability breaking cannot be applied, since a small perturbation changes not only the longtime asymptotic expectation values, but completely alters the dynamics even at short timescales.
Finally, confinement is expected to have similar effects also in higher dimensions and so it is natural to wonder what the consequences are for realistic nonequilibrium situations in quantum chromodynamics such as the quark–gluon plasma in hadron colliders. While even approximate fieldtheoretical calculations for strong interactions are beyond our reach, holographic methods that have been successfully applied to the study of standard lightcone phenomena^{24} and to quenches in confining theories^{25} could be insightful to understand this fascinating problem.
Methods
The Hamiltonian (1) for h_{x} ≠ 0 and h_{z} ≠ 0 is not integrable and the spectrum can be described only by resorting to various approximations, such as, for example, fieldtheoretical ones^{26,27,28} that are valid in the vicinity of the critical point h_{z} = 1 and h_{x} = 0. Here we use a complementary approach: the lowdensity approximation of ref. 29, which describes the energy levels very accurately when the system is far away from the critical point. In the Supplementary Information we explain how this approach works and we report several pieces of evidence for its applicability for the values of magnetic fields of interest. This approximation allows us to calculate all the required properties of the mesons, namely their number, masses and velocities.
To simulate the time evolution after the quantum quench we use an iTEBD^{30} algorithm, details of which are reported in the Supplementary Information. Finally, the entanglement entropy is defined as the von Neumann entropy S_{A} = −Trρ_{A} lnρ_{A} of the reduced density matrix ρ_{A} of a subsystem A.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the authors on request.
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Acknowledgements
We are grateful to R. M. Konik and J. Cardy for helpful discussions. This work was supported by the ERC under Starting Grant 279391 EDEQS (P.C. and M.C.). M.K. and G.T. thank SISSA for hospitality. M.K. was partially supported by a Janos Bolyai Research Scholarship of the HAS. This work was also partially supported by the CNRHAS bilateral grant SNK84/2013.
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Kormos, M., Collura, M., Takács, G. et al. Realtime confinement following a quantum quench to a nonintegrable model. Nature Phys 13, 246–249 (2017). https://doi.org/10.1038/nphys3934
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DOI: https://doi.org/10.1038/nphys3934
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