Ballistic transport in the one-dimensional Hubbard model:
the hydrodynamic approach
We outline a general formalism of hydrodynamics for quantum systems with multiple particle species which undergo completely elastic scattering. In the thermodynamic limit, the complete kinematic data of the problem consists of the particle content, the dispersion relations, and a universal dressing transformation which accounts for interparticle interactions. We consider quantum integrable models and we focus on the one-dimensional fermionic Hubbard model. By linearizing hydrodynamic equations, we provide exact closed-form expressions for Drude weights, generalized static charge susceptibilities and charge-current correlators valid on hydrodynamic scale, represented as integral kernels operating diagonally in the space of mode numbers of thermodynamic excitations. We find that, on hydrodynamic scales, Drude weights manifestly display Onsager reciprocal relations even for generic (i.e. non-canonical) equilibrium states, and establish a generalized detailed balance condition for a general quantum integrable model. We present the first exact analytic expressions for the general Drude weights in the Hubbard model, and explain how to reconcile different approaches for computing Drude weights from the previous literature.
In past few years, a lot of interest has been devoted to studying various paradigms of non-ergodic many-body physics, such as quantum quenches, equilibration to generalized Gibbs ensembles and phenomenon of pre-thermalization Polkovnikov et al. (2011); Calabrese et al. (2016); Gogolin and Eisert (2016). One of the prominent recent results is the formalism of generalized hydrodynamics developed in Bertini et al. (2016); Castro-Alvaredo et al. (2016), with a large number of subsequent studies investigating its various aspects and applications Doyon and Yoshimura (2017); B. Doyon, and H. Spohn, and T. Yoshimura (2017); B. Doyon and H. Spohn (2017); B. Doyon, T. Yoshimura, and J.-S. Caux (2017); B. Doyon, J. Dubail, R. Konik, and T. Yoshimura (2017); V. B. Bulchandani, R. Vasseur, C. Karrasch, and J. E. Moore (2017); Piroli, Lorenzo and Nardis, Jacopo De and Collura, Mario and Bertini, Bruno and Fagotti, Maurizio (2017); Alba, Vincenzo (2017), including the exact computation of Drude weights in the Heisenberg model XXZ spin-1/2 chain Ilievski and De Nardis (2017). In analogy to the conventional theory of hydrodynamics Spohn (1991), the authors of Doyon and Spohn (2017) just recently obtained a closed formula for Drude weights expressed in terms of local equilibrium state functions for the case of integrable Bose gas (Lieb–Liniger model) and conjectured that similar formulae may hold in quantum integrable models more generally. In this work, we go a step further and extend the formalism to integrable models which possess physical particles with internal degrees of freedom and are solvable by nested Bethe Ansatz. Nesting is referred to the situation when physical degrees of freedom are associated with a higher rank symmetry group, leading to eigenfunctions with a hierarchical structure of internal quantum numbers and elementary excitations of different flavours. While studies of such models has been traditionally focused on Gibbs equilibrium Essler et al. (1991, 1992); Essler and Korepin (1994); Fabian H. L. Essler and Holger Frahm and Frank Göhmann and Andreas Klümper and Vladimir E. Korepin (2005); Beisert et al. (2011); Frolov and Quinn (2012), they have also been recently studied in the nonequilibrium context Mestyan, Marton and Bertini, Bruno and Piroli, Lorenzo and Calabrese, Pasquale (2017); Robinson, Neil J. and Caux, Jean-Sébastien and Konik, M. Robert (2016).
The chief aspect in which interacting quantum integrable theories differ from widely studied noninteracting systems is the dressing of (quasi)particle excitations, i.e. a process in which bare properties of the particle-hole type of excitations renormalize in the presence of interactions with a non-trivial reference (vacuum) state. The task of classifying excitations has been traditionally restricted to ground states for some of the simplest Bethe Ansatz solvable models Korepin et al. (1993), and subsequently extended to some important examples of exactly solvable models of correlated electrons Bares et al. (1991); Essler and Korepin (1994); Fabian H. L. Essler and Holger Frahm and Frank Göhmann and Andreas Klümper and Vladimir E. Korepin (2005); Quinn and Frolov (2013). A comprehensive exposition of the dressing formalism for grand canonical ensembles in nested Bethe Ansatz models can be found in Quinn and Frolov (2013).
Integrable theories exhibit a completely elastic (factorizable) scattering of particle-like excitations Zamolodchikov and Zamolodchikov (1979). Properties
of such excitations represent the kinematic data of the theory. In particular, in Bethe Ansatz solvable models (see
e.g. Korepin et al. (1993); Fabian H. L. Essler and Holger Frahm and Frank
Göhmann and Andreas Klümper and Vladimir E.
Korepin (2005)) thermodynamic excitations relative to a bare vacuum 111In fermionic interacting
integrable models there exist distinct inequivalent possibilities of choosing a bare vacuum.
Despite this results in different sets of excitations, various choices have no effect on the values of physical
observables. can be inferred from the solutions to (nested) Bethe equations. The latter in a finite volume take the form
imposing single-valuedness of many-body eigenstates.
Here the sets of quantum numbers are called the Bethe roots and represent rapidity variables for
distinct species (or flavours) of elementary excitations. The number and types of excitations depends on the model and can be
inferred with aid of representation theory of the underlying quantized Lie (super)algebra.
Elementary excitations typically form complexes which are interpreted as bound states.
The emergent thermodynamic particle content, which can be inferred by e.g. analysing the limit of Bethe equations,
is generally different from elementary excitations and is labeled by a pair of mode numbers,
a particle type index and a real rapidity variable . The complete kinematic data
are obtained from the bare momenta and energies , and interparticle scattering phase
shifts . Once given these functions, no explicit operator representation of the Hamiltonian and its conservation
laws is ever required. In this work we present the details of the entire formalism for the non-trivial case of the (fermionic)
A distinguished feature of integrable systems is a macroscopic number of local conservation laws which can be formally expressed in terms of a discrete basis of local charges , with labelling lattice sites. The associated currents are defined with aid of the continuity equation, . The key concept of the hydrodynamic approach is the dressing of bare energies and momenta of particle excitations, which can be presented in a compact form
with convolution . In interacting quantum integrable models solvable by (nested) Bethe Ansatz, the matrix convolution kernel takes a universal form
with kernels defined as derivatives of the scattering phase shifts , , and . The (Fermi) filling functions specify the fraction of occupied modes with rapidities inside a small interval around .
Dispersion relations of excitations depend on a many-body vacuum which is uniquely specified by the rapidity distributions . In terms of (thermodynamic) particle excitations, the equilibrium averages of charge and current densities decompose as , , where are the current densities per mode Bertini et al. (2016); Castro-Alvaredo et al. (2016). The group velocities of propagating particles are thus state-dependent, .
We furthermore introduce the effective charges as the bare charges renormalized under transformation , namely the effective value of a local charge density is obtained as
Here parameters are the chemical potentials of a local (generalized) equilibrium ensemble parametrized as Vidmar and Rigol (2016); Essler and Fagotti (2016); Ilievski et al. (2015). It is important to emphasize that despite the derivatives of dressed energies satisfying , the effective charges are not the proper dressed charges associated with an excitation, and specifically . We moreover note that with aid of fusion identities among the scattering kernels, the transformation (2) can be decoupled to a quasi-local form in the mode space, cf. Supplemental Material SM () (SM) for explicit form for Hubbard model.
In this work, we shall mainly be concerned with general off-diagonal Drude weights
which represent magnitudes of the singular parts of the zero-frequency generalized conductivities Kubo (1957); Mahan (2000), . We use to denote the connected part of the equilibrium expectation values. Although we shall restrict ourselves to grand canonical equilibria, our formalism applies (without modifications) to general local equilibrium states.
An exact representation for can be given in terms of the static covariance matrix , , with diagonal components representing (generalized) static susceptibilities, and charge-current correlators (overlaps) , . Explicit expressions in terms of thermodynamic state functions can be found in SM (). The time-averaged current–current correlator Eq. (4) can be projected onto the subspace formed by local conserved quantities which yields the well-known Mazur–Suzuki equality Mazur (1969); Suzuki (1971) and proves useful for bounding dynamical susceptibilities Zotos et al. (1997). In matrix notation the latter reads Ilievski and Prosen (2012).
A central result of our work is that on hydrodynamic scale, static charge-charge, charge-current correlations, and generic Drude weights, all assume a universal mode decomposition (writing formally )
which has exactly the same form as in the case of a single-component interacting integrable Bose gas derived in a recent paper Doyon and Spohn (2017). Importantly, in the above formula the kernels and effective charges are expressible in terms of properties of equilibrium states which can be efficiently computed within Thermodynamic Bethe Ansatz (TBA) method Yang and Yang (1969); Takahashi (1971); Gaudin (1971). It is noteworthy that Eq. (5) is written solely in the mode space, i.e. it acts (diagonally) on particle labels and rapidities, and that no explicit knowledge of a complete set of local charges is ever required in a computation. Indeed, thermodynamic expectation values of local charges are expressible as linear functionals of particles’ rapidity distributions (see e.g. Ilievski et al. (2015, 2016)) which are a natural extension of momentum distribution functions of free theories Ilievski et al. (2017).
The hydrodynamic approach Bertini et al. (2016); Castro-Alvaredo et al. (2016) is based on the notion of local quasi-stationary states, characterized by the local continuity equation in the mode space . In the simplest scenario, one can think of a quantum quench in which an inhomogeneous initial state is initialized as two homogeneous equilibrated macroscopic regions brought in contact at , see Spohn and Lebowitz (1977); Bernard and Doyon (2014, 2016). In such a scenario, an emergent nonequilibrium state remains confined to the light cone region determined by particles’ dressed velocities, leading eventually to a quasi-stationary state which depends on the ray coordinate and is determined by the condition of vanishing convective derivative .
The setting proves particularly useful for studying nonequilibrium transport properties and, in particular, computation of Drude weights. The latter can be conveniently defined as asymptotic current rates in the limit of vanishing bias (while keeping other chemical potentials fixed),
The above prescription has been initially used in Vasseur et al. (2015) and employed in a recent numerical study Karrasch (2017a), while an analogous formula already appeared in an earlier work Ilievski and Prosen (2012). Equation (6) has been recently evaluated in Ilievski and De Nardis (2017); Bulchandani et al. (2017) using the hydrodynamic approach, transforming it first in the light cone coordinates, , and then computing quasi-stationary currents which are generated by joining together two nearly identical equilibrium states, i.e. imposing a small chemical potential drop at the origin and . Here has the role of a thermodynamic force, e.g. to study energy transport we identify .
Just very recently in Doyon and Spohn (2017) the authors applied Eq. (6) to the Lieb–Liniger model and obtained closed-form expressions analogous to Eq. (5). Below we generalize this result to interacting quantum models which involve multiple species of excitations and internal degrees of freedom. It is quite remarkable however that the final outcome remains a bilinear functional operating diagonally in the mode-number space, while the effect of interparticle interactions gets absorbed into a universal renormalization of bare charges, see Eq. (3).
Eq. (6) indicates that Drude weights are expressible as the variation of the equilibrium expectation values of total current Ilievski and De Nardis (2017) with respect to thermodynamic forces , , being the susceptibility of a system to develop ballistic currents. On each ray , the averages of particle current densities are given by Bertini et al. (2016) , where rapidity dependence has been suppressed for brevity. Given the filling functions inside the light cone , with the left/right boundary conditions , and neglecting corrections of order , one can integrate out the dependence on the light cone coordinates (see SM SM () for details). This leads to the form of Eq. (5), with
On detailed balance.
The symmetry under exchanging indices an in representation (5), , indicates that the Onsager reciprocal relations Onsager (1931) remain valid for any stationary state, not only in thermal Gibbs equilibrium. This is indeed a general property of the hydrodynamic equation of motion Spohn (1991). Moreover we here show that in a general local equilibrium state of an integrable quantum model, there exist a generalized detailed balance condition on the hydrodynamic scale (i.e for small and ), similarly as in the Lieb-Liniger model found recently in Nardis and Panfil (2016); De Nardis et al. (2017). More specifically, given a conserved quantity of the model , the corresponding dynamical structure factor defined as decomposes in terms of individual particle contributions, . In the low-momentum limit , each term is determined by a single matrix element of a particle-hole excitation on a reference equilibrium state SM (). Therefore, following the logic presented in Nardis and Panfil (2016), we derive the following generalized reversibility property
with , with fixed by the energy constraint . In the case of thermal (canonical) equilibrium, given by , we have , which is the usual detailed balance relation.
where is electron hopping and is the Coulomb interaction. This model has received a lot of attention in the past decadesShastry (1988); Woynarovich (1989); Frahm and Korepin (1990); Ogata and Shiba (1990); Essler et al. (1991) as well as in the last years Prosen and Žnidarič (2012); Karrasch et al. (2014); Prosen (2014); Seabra et al. (2014); Neumayer et al. (2015); Reiner et al. (2016); Veness and Essler (2016); Veness et al. (2016); Tiegel et al. (2016); Karrasch et al. (2017); Lin et al. (2017); Karrasch (2017a, b); Bolens et al. (2017). We consider the repulsive case , featuring a -dependent charge gap and gapless spin degrees of freedom.
The Hubbard model is diagonalized by means of nested Bethe Ansatz Essler et al. (1991); Fabian H. L. Essler and Holger Frahm and Frank Göhmann and Andreas Klümper and Vladimir E. Korepin (2005). Eigenstates in a finite system of length are characterized by quantum numbers which are solutions to Lieb–Wu equations Lieb and Wu (1968) (cf. SM SM ()) The model involves two elementary degrees of freedom; the physical particles are momentum-carrying electrons, while spin degrees of freedom represent internal (non-dynamical) excitations described by auxiliary quantum numbers. In a thermodynamic system one finds various types of charge and/or spin-carrying bound states. Specifically, the thermodynamic particle content of the Hubbard model has been derived in Takahashi (1972) (see also Fabian H. L. Essler and Holger Frahm and Frank Göhmann and Andreas Klümper and Vladimir E. Korepin (2005); Quinn and Frolov (2013)) and comprises of (i) spin-up momentum-carrying electronic excitations which carry unit bare (electronic) charge (ii) spin-singlet electronic bound states and (iii) charge-neutral non-dynamical spin-carrying magnonic excitations. A detailed description of the particle content and other information, including explicit expressions for their bare momenta, energies, scattering phases and the dressing transformation, are reported in SM SM ().
We present temperature dependence of charge and spin, see Fig. 1, and thermal Drude weights, see Fig. 2, in grand canonical equilibrium , where is total electron charge, and , total magnetization. We compared our data with the recent DMRG computation presented in Karrasch (2017b, a). Most notably, at low temperatures appreciably below the charge gap we confirm the ‘Hubbard to Heisenberg crossover’ in the thermal Drude weight observed previously in Karrasch et al. (2016); Karrasch (2017a), see Fig. 2. In SM () we also present an exact computation of the asymptotic charge and current profiles inside a light cone and make comparisons with the numerical results of Karrasch (2017b).
We presented a general theoretical and computational framework to access the singular components (Drude weights) of generalized transport coefficients in quantum integrable models. We exemplified our approach by computing exact numerical values of (diagonal) charge, spin and thermal Drude weights in the one-dimensional fermionic Hubbard model in grand canonical equilibrium at finite temperatures and chemical potentials. Using the two-partition protocol, we additionally computed the quasi-stationary energy and charge density profile and the corresponding current SM ().
Our results finally permit to establish the equivalence of various approaches for computing the spin Drude weight employed in the previous literature: (i) using projections onto local conserved subspaces by virtue of Mazur–Suzuki equality Zotos et al. (1997); Prosen (2011); Prosen and Ilievski (2013), (ii) taking the linear-response limit of the asymptotic current rates Ilievski and De Nardis (2017); Bulchandani et al. (2017) and (iii) computing the energy-level curvatures Shastry and Sutherland (1990); Fujimoto and Kawakami (1998); Zotos (1999); Benz et al. (2005) under the twisted boundary conditions in accordance with Kohn formula Kohn (1964). The latter has been evaluated within the TBA framework in Fujimoto and Kawakami (1998); Zotos (1999), yielding a closed formula expressed in terms of filling functions, magnonic dispersion relations and corrections to the Bethe spectrum induced by the twist. Remarkably however, it is easy to see that the twist-dependence of the energy levels can be directly linked to the effective spin as given by Eq. (2). This in turn reconciles the results of Zotos (1999) with Eq. (7), representing the equilibrium analogue of definition (6) used previously in refs. Ilievski and De Nardis (2017); Bulchandani et al. (2017) (further details are given in SM SM (), which also includes refs. Narozhny et al. (1998); Peres et al. (1999); Alvarez and Gros (2002); Heidrich-Meisner et al. (2003); Fujimoto and Kawakami (2003); Herbrych et al. (2011); Sirker et al. (2011); Žnidarič (2011); Gromov and Kazakov (2011); Karrasch et al. (2012); Cavaglià et al. (2015); De Luca et al. (2016)).
Finally, our results show that a generalized version of the detailed balance Nardis and Panfil (2016); De Nardis et al. (2017); Foini et al. (2017) is valid on hydrodynamic scales in any stationary state.
Both authors contributed equally to the theory. J. De Nardis performed the numerical computations.
We are grateful to C. Karrasch for providing the tDMRG data for the Drude weights in Hubbard model and thank M. Van Caspel, M. Fagotti, E. Quinn and H. Spohn for valuable discussions and/or reading the manuscript. E.I. is supported by VENI grant number 680-47-454 by the Netherlands Organisation for Scientific Research (NWO). J.D.N. acknowledge support by LabEx ENS-ICFP:ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*.
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Ballistic transport in the one-dimensional Hubbard model:
the hydrodynamic approach
In this Supplementary Material we collect the most important technical results, present the detailed derivations and provide additional numerical results. The structure is as follows:
Appendix A covers the technical background of the nested Bethe Ansatz technique for solving the one-dimensional Hubbard model. We follow closely the presentation of Frolov and Quinn (2012); Quinn and Frolov (2013) which employs rapidity parametrization. A quasi-local formulation of TBA equations and the dressing transformation presented here appear to be new.
In Appendix E we briefly revisit the exceptional case of spin Drude weight in the anisotropic Heisenberg spin- chain. We explain how to unify the three apriori different definitions for computing the spin Drude weigh employ in the previous literature.
In Appendix F we present a general solution to the hydrodynamic equations for Hubbard model for the evolution from a bipartite initial equilibrium state. As an example, we compute the energy density and energy current quasi-stationary profiles and compare them with the results of tDMRG simulation.
Appendix A Thermodynamic Bethe Ansatz for Hubbard model
The Hamiltonian of 1D Hubbard model is of the form
The model possesses two globally conserved charges associated with symmetries, the total electron charge and the total spin ,
which are sometimes included in the definition Hamiltonian, .
Bethe equations for a finite system of length with periodic boundary conditions have been derived by Lieb and Wu Lieb and Wu (1968) and take the nested form,
with . Bethe roots (rapidities) are related to electron (quasi)momenta, while are associated with their spin. The number of Bethe roots in Eqs. (13) in terms of the total charge and spin is and , respectively. Bethe roots are associated the bare charge , , and the bare spin , . We note that parametrization of Lieb–Wu equations (13) in terms of -roots is different from the conventional one given in terms of electron (quasi)momenta as in Takahashi (1972). While the two are simply related by , rapidity parametrization proves more convenient since it renders all scattering amplitudes manifestly rational functions depending only on the difference of particles’ rapidities. A downside is that the momentum-dependent phase as a function of momentum-carrying roots then becomes a double-valued function, meaning that each root gives two distinct values of momenta. It is thus convenient to introduce a new type of roots, referred to as the -roots, by virtue of Zhukovsky transform
The corresponding functional equation has two solutions (branches), and presently we adopt
with a square-root branch cut on the interval . For any the two branches correspond to the values given by
When rapidity is taken from the branch cut, , we adopt the following prescription
i.e. we take the two values just above and just below the cut . Since we have
the two branches of momenta are given by
Thermodynamic solutions to Eqs. (13) – taking the limit while keeping ratios and finite – can be inferred from the stability condition of the asymptotic solutions, and comprise of self-conjugate string-like patterns of regularly displaced complex-valued rapidities with equal real parts centred on the real axis. These are identified with the thermodynamic particle content of the model which in the Hubbard model and comprise of:
-particles, which are spin-up momentum-carrying electronic excitations which carry unit electron charge . The -particle excitations are split into two branches denoted by with the corresponding rapidities . The -particles do not form bound states on their own. Their bare momenta are denoted by , and satisfy and , with . The lower and upper momentum branches of the -particle are
with the corresponding derivatives
The bare energies are and read
-strings, which are bound states of -roots and -roots, carrying charge and no spin . An -string is parametrized by and comprises of rapidities
To find the corresponding -roots we assign and . The corresponding momenta and energies are obtained by summing over all constituent -roots (recall that -roots carry no momentum)
The derivative of their momenta satisfy , reading explicitly
-strings, which are chargless () compounds made of -roots, with spin . They are parametrized by , and are of the form
The total charge and total spin in a Bethe eigenstate in terms of numbers of string excitations () is
In the thermodynamic limit, the solutions to Eqs. (13) become densely distributed on the rapidity axis and can be expressed in terms of particle densities which are defined as smooth densities of Bethe strings ,
Given a set of string solutions , the unoccupied solutions to Bethe equations are understood as the holes. The hole densities in the thermodynamic limit are denoted by , while the total densities of a state are denoted by .
In the Hubbard model, the densities of - and -strings, denoted by and , respectively, are supported on the whole real axis, . On the other hand, the rapidity distributions of the special -particles are split into two separate densities which are compactly supported on the branch cut . To this end, we define two types of integral transformations. First, we introduce the standard convolution as
taken with the convention that one drop and/or when and/or depend only on a single variable, and adopting the implicit summation convention for convolution expressions of the form and , namely summing and integrating over the domain of a -string. Since the -particles’ rapidity variable have a bounded integration domain, i.e. , we introduced a restricted convolution operation . The densities of -particles satisfy the sum rule,
Denoting , , and , the electron charge and spin densities are expressed as
Energy density of a macroscopic state is obtained by adding contributions of all energy-carrying particles,
Takahashi’s equations for the densities in rapidity parametrization take the form
where is the Kronecker delta, the Dirac delta, and the is the adjacency (incidence) matrix for the model,
Local statistical ensembles.
Thermodynamic Bethe Ansatz method is based on expressing the free energy density of a local statistical ensemble (a generalized Gibbs ensemble) as a set of coupled non-linear integral equations for the thermodynamic variables (e.g. Fermi filling functions of the thermodynamic excitations). Generalized Gibbs ensembles are conventionally expressed in the form
for a suitable (discrete) basis of local conserved quantities and the corresponding chemical potentials . By accounting for the fact that particles’ mode distributions essentially contain the complete information about local correlations functions, it is convenient to consider as a starting point an analytic parametrization Ilievski et al. (2017)
where correspond formally to a continuous family of local conserved operators whose eigenvalues coincide with the particles’ rapidity distributions, and are the chemical potentials pertaining to individual modes. The partition sum in the limit is then evaluated with a saddle-point integration using Yang–Yang approach Yang and Yang (1969), where the entropy density per particle is the logarithm of the number states occupying an infinitesimal rapidity interval which (in models obeying the Fermi statistics) takes a universal form
A solution to the variational problem , with , yields canonical TBA equations
where the TBA -functions are as usual defined as ratios of hole and particle densities for each thermodynamic excitation in the spectrum,
The set of -functions is equivalent to the set of Fermi filling functions , defined as . For later purposes we moreover introduce the filling functions of the holes, .
For instance, in canonical Gibbs equilibrium, , the canonical source terms in terms of particles’ bare energies and chemical potentials for the electronic charge and spin read
Using the fusion identities (cf. A.2), the above set of equations can be brought to an equivalent quasi-local form, reading explicitly
supplemented with the asymptotic conditions
a.1 Dressing of excitations and effective charges
Excited states with respect to a reference macrosopic state (representing a many-body vacuum) are characterized in terms of the particle-hole type of excitations and a background of non-excited modes (quantum numbers) which experience a shift as a back-reaction to creating excitations. The difference between the rapidities of excited and reference states induced by particle-type of excitations of type can be expressed as
while the hole-type excitations experience the same the shift of the opposite sign. The shift functions , describing the back-flow of non-excited rapidities, satisfy a closed set of integral equations
In the thermodynamic limit, Eq. (46) can be expressed as an integral equation which governs the dressing of bare quantities (suppressing rapidity parameters)
Differentiating this expression with respect to rapidity variable we find
where the inverse of the dressing convolution kernel explicitly reads
Two special (but central) examples of the above transformation are the dressed energies and dressed momenta , providing dispersion relations of the particle-hole excitations with respect to a reference macrostate. The dressed velocities yield the group velocity of propagation and are given by