Classification of Zamolodchikov–Faddeev algebra transformations for quantum quenches in interacting integrable quantum field theories and the first-principles construction of the quench initial state

In a (1+1)-dimensional integrable quantum field theory, a quantum quench changes the parameters of a Hamiltonian whose excitations scatter factorially; the pre-quench and post-quench creation/annihilation operators are then related by a transformation that must preserve the Zamolodchikov–Faddeev (ZF) exchange algebra. Sotiriadis, Fioretto and Mussardo (2012) showed that linear Bogoliubov transformations preserve the ZF algebra only at free points, and constructed two simple classes of admissible transformations — one that shifts the two-body S-matrix and one that 'dresses' each particle with pairs of opposite momenta, reducing to the usual Bogoliubov map in the free limit and turning the ground state into a generalized (interacting) squeezed state. They left open the classification of the full solution set, whose generic elements are infinite operator series that do not truncate even for infinitesimal quenches, together with controlled truncation or resummation schemes to compute the initial-state amplitudes from them, the role of infrared singularities at coincident momenta near free points (where the zero-momentum S-matrix $S(0)$ is discontinuous in the coupling), and application to physically important quenches such as the Lieb–Liniger model from a filled pre-quench ground state.

Background

A quantum quench is the sudden change of a parameter of an isolated quantum system's Hamiltonian — for instance the mass or coupling of a relativistic field theory — after which the pre-quench ground state evolves unitarily under the new Hamiltonian. Predicting the subsequent dynamics and the steady state requires expanding the pre-quench state in the eigenbasis of the post-quench Hamiltonian; the overlaps with many-particle eigenstates are the essential input of the exact methods available for integrable models, namely the quench action variational principle and form-factor (matrix-element) expansions.

In $1+1$ dimensions an integrable quantum field theory has infinitely many conserved charges, and scattering is purely elastic and factorized into two-body collisions described by a scattering matrix (S-matrix) $S(p_1,p_2)$, which for diagonal scattering is a scalar function of the two momenta. The natural creation/annihilation operators of the asymptotic particles are the Zamolodchikov–Faddeev (ZF) operators $Z_p$, $Z_p^\dagger$, whose defining exchange algebra is $Z_{p_1}Z_{p_2}=S(p_1,p_2)Z_{p_2}Z_{p_1}$; local fields are series in these operators. Because particles of different momenta do not commute in this algebra, relating two integrable theories' operators is highly constrained.

In a free theory a quench is handled by a linear Bogoliubov transformation mixing creation and annihilation operators, and the initial state becomes a squeezed coherent state (an exponential of back-to-back particle pairs acting on the vacuum). Sotiriadis, Fioretto and Mussardo (J. Stat. Mech. 2012, building on Fioretto–Mussardo 2010) asked what replaces this when the post-quench theory is interacting. Writing the transformed operators and S-matrix as $Z'_p=Z_p+\epsilon W_p$ and $S'=S+\epsilon T$, preservation of the ZF algebra imposes the linear conditions $W_{p_1}Z_{p_2}+Z_{p_1}W_{p_2}=T(p_1,p_2)Z_{p_2}Z_{p_1}+S(p_1,p_2)(Z_{p_2}W_{p_1}+W_{p_2}Z_{p_1})$ and its $Z^\dagger$ analogue, with $T$ constrained by unitarity, and $W_p$ admits a general expansion (eq. (19) of that paper) in normal-ordered products of $Z$'s and $Z^\dagger$'s. They proved that a linear Bogoliubov ansatz fails except for free bosons or fermions, and constructed two simple classes of solutions: eq. (20), which shifts the S-matrix but preserves the vacuum and conserved charges, and eq. (40), which leaves the S-matrix unchanged and 'dresses' each particle with pairs of opposite momenta — the interacting generalization of the Bogoliubov transformation, under which the ground state becomes a generalized squeezed state (at least for real coefficients).

The authors themselves stressed what remains unresolved: the generic solution of the ZF-preservation conditions is an infinite series that does not truncate even for infinitesimal quenches; no truncation or resummation scheme (for example a small-density reorganization) was established to make such operator maps computable and to extract initial-state amplitudes; the perturbative coefficients of the constructed maps carry infrared singularities at coincident momenta, reflecting the discontinuity of the zero-momentum S-matrix $S(0)$ in the coupling at free points (in interacting theories $S(0)=-1$, so the transition from a free-boson point is non-analytic); and the physically important Lieb–Liniger quench — the Lieb–Liniger model being the one-dimensional Bose gas with contact interactions of strength $c$ — where the pre-quench ground state is not the empty Fock vacuum but contains a macroscopic number of particles, requires construction beyond the empty-vacuum perturbation.

After 2012 the field largely worked around the operator-transformation problem rather than solving it. The initial state is typically assumed to be a squeezed pair state of Ghoshal–Zamolodchikov boundary-integrable form (i.e. of the structure of the boundary state describing an integrable boundary in a bulk-integrable theory), with the pair amplitude proportional to a Dirichlet boundary reflection amplitude; this ansatz has strong numerical support in sinh-Gordon quenches (Sotiriadis–Takács–Mussardo 2014; Horváth–Sotiriadis–Takács 2016, who reformulated the determination as an infinite integral-equation hierarchy with a controlled truncation property but left the pairing argument heuristic). The quench action method (Caux–Essler; reviewed by Caux 2016) then determines the steady state from these overlaps, and truncated conformal space approach (TCSA) computations supply numerical benchmarks. For the Lieb–Liniger model, exact results exist only for quenches whose initial state is the free-endpoint Bose-Einstein condensate (De Nardis et al. 2014; Kormos–Collura–Calabrese 2014; Rylands–Calabrese–Bertini 2023), and perturbative overlap expansions in the quench size were developed by Hódsági–Kormos–Takács (2019). Notably, numerical work on the $E_8$ Ising field theory (the critical Ising model perturbed by a magnetic field, whose excitation spectrum contains eight massive particles; Hódsági–Kormos–Takács 2018) shows that interacting quenches need not produce pure pair states — multiparticle components are physically real, which is precisely the structure the unconstructed higher transformation classes would describe.

Problem Statement

Classify and make computable the transformations of Zamolodchikov–Faddeev (ZF) operators that preserve the ZF algebra in interacting integrable quantum field theories, and use them to determine the quench initial state from first principles.

Concretely, for a diagonal integrable theory with two-body S-matrix $S(p_1,p_2)$ and ZF exchange algebra $Z_{p_1}Z_{p_2}=S(p_1,p_2)Z_{p_2}Z_{p_1}$, consider infinitesimal transformations $Z'_p=Z_p+\epsilon W_p$ with $S'=S+\epsilon T$ such that the transformed operators satisfy the ZF algebra, i.e. $W_p$ obeys the conditions of eqs. (16)-(18) of Sotiriadis–Fioretto–Mussardo (2012), with $W_p$ expanded in the general series (eq. (19)) in products of $Z$'s and $Z^\dagger$'s. The problem comprises: (i) the classification and explicit construction of the full set of solutions beyond the two exhibited classes — the S-matrix-shifting class (eq. (20)) and the S-preserving dressing class (eq. (40)) — in particular the solutions in which the series (19) does not truncate and which arise already for infinitesimal quenches, including a determination of whether the two simple classes exhaust all terminating solutions; (ii) a controlled truncation or resummation scheme (for example a small-density reorganization) that renders these infinite-series operator maps practically computable and delivers the initial-state amplitudes (overlaps with post-quench multiparticle eigenstates) for generic interacting quenches; (iii) an analysis of the infrared non-analyticities at coincident momenta — the discontinuity of $S(0)$ in the coupling at free points and the singular coefficients of the known infinitesimal maps — and of their impact on the convergence and physical interpretation of the series; and (iv) application of the resulting machinery to quench protocols of physical interest, in particular the Lieb–Liniger quench from a pre-quench ground state with macroscopic particle density (filled vacuum, with pre-quench coupling $c_0>0$), beyond the exactly-free $c_0=0$ endpoint already solved by other methods. Parts (i)–(ii) constitute the required core of the problem, i.e. the classification together with a computable truncation or resummation scheme; parts (iii)–(iv) are independent sub-problems, each acceptable as a partial contribution on its own, and neither need be resolved for a solution of the core.

References

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