Universality conditions for the charge full counting statistics of quenched one-dimensional quantum gases

When the Lieb-Liniger or Gaudin-Yang model of a one-dimensional interacting gas is quenched far from equilibrium from a (fermionic) Bose-Einstein condensate initial state, the scaled cumulants of the particle number in a large subsystem relax to values fixed by the initial state alone through the doubling relation $\kappa_m^s(\infty)=\kappa_m^s(0)/2^{m-1}$, equivalently $C_0^s(\beta)=\tfrac{1}{2}C_{\mathrm{GGE}}^s(2\beta)$ for the scaled cumulant generating functions, with no dependence on the interaction strength. The open problem is to establish the precise necessary and sufficient conditions under which this universal relation holds and to put it on rigorous footing: to replace the formal diagonal-ensemble and quench-action arguments, and the recent hydrodynamic heuristics whose key assumptions (commutator estimates, spatial and temporal clustering, local relaxation, a large-deviation principle) are uncontrolled, by a proof; to determine whether integrability of the post-quench Hamiltonian, pair structure of the initial state, or hydrodynamic-mode structure is required, and in particular whether the interaction-independence of the steady-state extensive statistics survives in non-integrable models or for initial states beyond pure quasi-particle pairs; and to characterize the failure of the relation for initial states with vanishing extensive cumulants together with the long-range-correlation corrections expected at higher cumulant orders. A follow-up hydrodynamic theory by the same first author with Doyon and Ruggiero explains the doubling mechanism and gives sufficient, partly conjectural conditions, but explicitly leaves the rigor, the necessity direction, the non-integrable extension, and the higher-cumulant corrections open.

Background

Full counting statistics (FCS) refers to the complete probability distribution of the outcomes of repeated measurements of an observable on identically prepared systems, rather than only its mean. Here the observable is the particle number $\hat N_A$ in a spatial subsystem $A$ of length $\ell$ of a one-dimensional quantum gas. In the limit of large $\ell$ the distribution is encoded in the scaled cumulant generating function (SCGF) $C(\beta)=\lim_{\ell\to\infty}\ell^{-1}\log\langle e^{\beta\hat N_A}\rangle$, whose derivatives at $\beta=0$ give the scaled cumulants $\kappa_m^s$, i.e. the fluctuations per unit length; for an interacting gas these generally depend on the interaction strength.

The paradigmatic systems are the Lieb-Liniger (LL) model of bosons with contact interaction of strength $c$ and the Gaudin-Yang (GY) model of spin-1/2 fermions with a contact interaction between the two spin species. Both are integrable, meaning they possess infinitely many (quasi)local conserved charges and are exactly solvable by the Bethe ansatz, a construction of exact many-body energy eigenstates as superpositions of plane waves parametrized by quasi-momenta called rapidities; in the thermodynamic limit their steady-state thermodynamics is expressed by the thermodynamic Bethe ansatz (TBA), an integral equation for the root density, the density of occupied rapidities. The nonequilibrium protocol is a quench: the gas is prepared in a simple reference state (for LL the BEC, the ground state of the non-interacting model) and then evolved by the full interacting Hamiltonian. A Bose-Einstein condensate (BEC) initial state for LL, and its fermionic analog for GY (a condensate of correlated opposite-momentum fermion pairs), allows an exact treatment of the subsequent quench dynamics: at long times local observables relax to a generalized Gibbs ensemble (GGE), a density matrix built from all conserved charges with Lagrange multipliers fixed by the initial state, which the quench action (QA) method constructs as the thermodynamic saddle point of the overlap-weighted sum over Bethe eigenstates. Only parity-invariant eigenstates (invariant under rapidity reflection $\lambda\to-\lambda$) overlap the BEC state, which halves the entropy contribution at the saddle. A related object is the diagonal ensemble, the dephased density matrix that keeps only the squared overlaps of the initial state with post-quench energy eigenstates.

Horvath and Rylands (Phys. Rev. A 109, 043302 (2024)) computed the charge FCS of both models over the entire interaction range and found two facts. First, the steady-state scaled cumulants are independent of $c$ and are fixed by the initial state through

$$\kappa_m^s(\infty)=\frac{\kappa_m^s(0)}{2^{m-1}},$$

equivalently the initial-state SCGF is half the steady-state GGE SCGF at doubled counting field,

$$C_0^s(\beta)=\tfrac{1}{2}\,C_{\mathrm{GGE}}^s(2\beta),$$

where $C_0^s$ is computed in the initial state and $C_{\mathrm{GGE}}^s$ in the steady-state GGE. For BEC initial states, whose charge is Poisson distributed with $\kappa_m^s(0)=d$ (the density), this gives $\kappa_m^s(\infty)=d/2^{m-1}$ for all $m$. The relation was established by explicit TBA evaluation and a formal diagonal-ensemble argument, and has an intuitive quasi-particle interpretation: the initial state is a gas of correlated opposite-momentum pairs each carrying charge 2, while at long times both members of a pair cannot remain in the subsystem, so fluctuations are carried by single quasi-particles of charge 1 with the same momentum distribution. Second, the full time evolution of the cumulants is not universal and depends strongly on $c$.

This sits in a broader program. The pair structure of the excited quasi-particles is the defining property of an integrable quench (Piroli, Pozsgay and Vernier). A parallel line based on space-time duality conjectured a closed formula for nonequilibrium FCS; space-time duality is an exact mapping, available in special integrable lattice models, that exchanges the roles of space and time, converting the time evolution of a subsystem into the equilibrium statistics of a dual model. The formula is exact in the rule-54 cellular automaton, a deterministic and explicitly solvable one-dimensional lattice dynamics, and was checked numerically in the spin-1/2 XXZ Heisenberg chain, an integrable anisotropic spin model (Bertini, Calabrese, Collura, Klobas and Rylands). Subsequently Horvath, Doyon and Ruggiero developed a hydrodynamic theory of nonequilibrium FCS based on ballistic fluctuation theory (BFT), the theory of large-scale fluctuations of ballistically (velocity-bounded, diffusion-free) transported conserved quantities, which controls long-range correlations at the Euler, i.e. hydrodynamic, scale: the FCS on a large interval decomposes into a thermodynamic piece reproducing and explaining the doubling relation, plus a dynamical piece given by current fluctuations in a biased non-equilibrium steady state (NESS) of a partitioning protocol (joining two reservoirs prepared in different states). They identified conditions under which the formula is expected to be exact (at most two hydrodynamic velocities, linear hydrodynamics, or an initial state that is an eigenstate of the conserved charge) and argued that in interacting integrable models with fermionic quasi-particle statistics and a continuum of modes, such as LL, a hydrodynamic waft effect (curving of ballistic trajectories in the biased fluid) violates temporal clustering and modifies cumulants of order six and above in a way that cannot yet be evaluated. What is still missing is mathematical rigor for these steps, any necessity result for the conditions, and any controlled example beyond integrable quenches with pair-structured initial states.

Problem Statement

Determine, with mathematical rigor, the conditions under which the charge full counting statistics of a quenched one-dimensional many-body system is universal, in the precise sense of the relations

$$\kappa_m^s(\infty)=\frac{\kappa_m^s(0)}{2^{m-1}}\quad(m=1,2,\dots),\qquad C_0^s(\beta)=\tfrac{1}{2}\,C_{\mathrm{GGE}}^s(2\beta),$$

for the scaled cumulants and scaled cumulant generating functions of the subsystem particle number in the initial state and in the long-time steady state. Concretely, the following linked questions, posed by Horváth and Rylands and sharpened by the subsequent hydrodynamic theory, remain open:

(a) Rigorous derivation. Replace the formal diagonal-ensemble and quench-action argument (with its selection of parity-invariant eigenstates and the half-entropy factor in the saddle-point evaluation) and the hydrodynamic derivation (with its uncontrolled neglect of Baker-Campbell-Hausdorff commutators, the correction terms generated when exponentials of non-commuting operators are multiplied, its spatial and temporal clustering assumptions, its local-relaxation assumption, and its assumed large-deviation principle) by a proof, or by sharp bounds, that establishes the relations at leading extensive order for the Lieb-Liniger and Gaudin-Yang Bose-Einstein-condensate quenches and for whatever wider class of models and initial states the argument covers. The cumulant-order range to be covered is the full range over which the relations hold for these models; the hydrodynamic derivation presently reproduces them exactly only up to and including order five, and the order-six-and-higher long-range-correlation (hydrodynamic-waft) corrections that may terminate that range in fermionic-statistics models with a continuum of modes are treated separately in (c).

(b) Necessary and sufficient conditions. Ascertain which ingredients are actually required: integrability of the post-quench Hamiltonian; the pair structure of the initial state (only correlated opposite-momentum quasi-particle pairs excited, the defining property of an integrable quench); the structure of the hydrodynamic spectrum (for example, at most two ballistic velocities, or fermionic statistics of the quasi-particles); or weaker combinations of these. In particular, determine whether the interaction-independence of the steady-state extensive FCS persists in non-integrable models or under other quench protocols, and whether the relations extend to initial states beyond a pure pair structure, such as quasi-particle multiplets.

(c) Failure modes and corrections. Characterize how the relations degenerate or fail for initial states with vanishing extensive charge cumulants, where the scaling prediction is vacuous or wrong and the physically relevant fluctuations are sub-extensive, and quantify the long-range-correlation (hydrodynamic waft) corrections expected to modify the FCS at scaled-cumulant order six and higher in fermionic-statistics integrable models with a continuum of hydrodynamic modes.

Parts (a)-(c) are facets of this single objective, not independent alternatives: a necessity-and-sufficiency result that covers the Lieb-Liniger and Gaudin-Yang Bose-Einstein-condensate quenches also discharges (a) for those models, while (c) is a distinct question about the degenerate regimes and the higher-order corrections. A complete resolution must address all three.

References

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