Typical finite-energy-density eigenstates of nonintegrable systems are expected to obey the eigenstate thermalization hypothesis and to reproduce thermal local observables. In integrable systems, typical states are instead finite-entropy Bethe macrostates, or generalized Gibbs ensembles, with smooth quasiparticle occupations. Here we show that the spin- 1 2 XXX Heisenberg chain contains an exactly solvable exception. By biasing mutually commuting conserved charges, we use the ground state of a selector Hamiltonian to construct rare eigenstates of the undeformed XXX Hamiltonian. These states are atypical, ordered, chiral, current-carrying, critical, zero-entropy Bethe macrostates at tunable XXX energy densities. They lie outside the finite-entropy manifold dominating generalized Gibbs ensembles, yet remain exact XXX eigenstates with sharp Bethe occupations, finite scalar chirality, nonthermal local observables, and gapless Luttinger-liquid correlations. A finite-interval thermodynamic Bethe ansatz reveals an asymmetric chiral Bethe sea whose zero-field limit selects an extensive spin sector through S U ( 2 ) symmetry breaking. A commuting exchange bias moves the same ordered macrostate through the full thermodynamic XXX energy band, including its Hilbert-space trace center. Finite-size DMRG and exact-diagonalization benchmarks indicate that the magnetochiral signatures persist under weak integrability breaking over prethermal time scales. The construction provides a controlled integrable realization of ETH-violating, scar-like large-deviation eigenstates and a route to ordered critical matter deep inside a many-body spectrum.
Typical finite-energy-density eigenstates of nonintegrable systems are expected to obey the eigenstate thermalization hypothesis and to reproduce thermal local observables. In integrable systems, typical states are instead finite-entropy Bethe macrostates, or generalized Gibbs ensembles, with smooth quasiparticle occupations. Here we show that the spin- 1 2 XXX Heisenberg chain contains an exactly solvable exception. By biasing mutually commuting conserved charges, we use the ground state of a selector Hamiltonian to construct rare eigenstates of the undeformed XXX Hamiltonian. These states are atypical, ordered, chiral, current-carrying, critical, zero-entropy Bethe macrostates at tunable XXX energy densities. They lie outside the finite-entropy manifold dominating generalized Gibbs ensembles, yet remain exact XXX eigenstates with sharp Bethe occupations, finite scalar chirality, nonthermal local observables, and gapless Luttinger-liquid correlations. A finite-interval thermodynamic Bethe ansatz reveals an asymmetric chiral Bethe sea whose zero-field limit selects an extensive spin sector through S U ( 2 ) symmetry breaking. A commuting exchange bias moves the same ordered macrostate through the full thermodynamic XXX energy band, including its Hilbert-space trace center. Finite-size DMRG and exact-diagonalization benchmarks indicate that the magnetochiral signatures persist under weak integrability breaking over prethermal time scales. The construction provides a controlled integrable realization of ETH-violating, scar-like large-deviation eigenstates and a route to ordered critical matter deep inside a many-body spectrum.
We study the spin-1 Babujian-Takhtajan chain deformed by its third conserved charge Q3. We derive Q3 and show that it is a dimensionless energy current and that its local density is a dressed scalar-chirality operator rather than bare chirality alone, as is the case for the spin-1/2 Heisenberg chain. The deformation Hα=H+αQ3 therefore provides a local, exactly solvable current bias: it leaves the eigenstates of the original Hamiltonian unchanged, but reorders them so that selected high-energy current-carrying states become ground states of the tilted problem. Using the thermodynamic Bethe ansatz and confirming the analytical calculations with DMRG, we find a quantum phase transition at αc=J/(8π). For α<αc, the ground-state remains the undeformed Babujian-Takhtajan phase whose low-energy effective field theory is described by the SU(2) Wess-Zumino-Witten (WZW) model at level k=2 representing a critical phase characterized by a central charge c=3/2 and ⟨Q3⟩=0. For α>αc, a finite rapidity interval forms, and the system enters a gapless chiral current-carrying sector described by a c=3/2 CFT. Near the threshold, the free energy starts quadratically as a function of α−αc, while the energy current turn on linearly. The scalar chirality turns on at the same threshold, showing that the postcritical sector is simultaneously current-carrying and chiral. The most immediate experimental routes are composite spin-1 bosons in optical lattices, and programmable qutrit simulators based on trapped ions or superconducting circuits.
There is a dichotomy in the nonequilibrium dynamics of quantum many-body systems. In the presence of integrability, expectation values of local operators equilibrate to values described by a generalized Gibbs ensemble, which retains extensive memory about the initial state of the system. On the other hand, in generic systems such expectation values relax to stationary values described by the thermal ensemble, fixed solely by the energy of the state. At the heart of understanding, this dichotomy is the eigenstate thermalization hypothesis (ETH): individual eigenstates in nonintegrable systems are thermal, in the sense that expectation values agree with the thermal prediction at a temperature set by the energy of the eigenstate. In systems where ETH is violated, thermalization can be avoided. Thus, establishing the range of validity of ETH is crucial in understanding whether a given quantum system thermalizes. Here, we study a simple model with confinement, the quantum Ising chain with a longitudinal field, in which ETH is violated. Despite an absence of integrability, there exist rare (nonthermal) states that persist far into the spectrum. These arise as a direct consequence of confinement: pairs of particles are confined, forming new ``meson'' excitations whose energy can be extensive in the system size. We show that such states are nonthermal in both the continuum and in the low-energy spectrum of the corresponding lattice model. We highlight that the presence of such states within the spectrum has important consequences, with certain quenches leading to an absence of thermalization and local observables evolving anomalously.
Pumping a finite energy density into a quantum system typically leads to ‘melted’ states characterized by exponentially-decaying correlations, as is the case for finite-temperature equilibrium situations. An important exception to this rule are states which, while being at high energy, maintain a low entropy. Such states can interestingly still display features of quantum criticality, especially in one dimension. Here, we consider high-energy states in anisotropic Heisenberg quantum spin chains obtained by splitting the ground state’s magnon Fermi sea into separate pieces. Using methods based on integrability, we provide a detailed study of static and dynamical spin-spin correlations. These carry distinctive signatures of the Fermi sea splittings, which would be observable in eventual experimental realizations. Going further, we employ a multi-component Tomonaga-Luttinger model in order to predict the asymptotics of static correlations. For this effective field theory, we fix all universal exponents from energetics, and all non-universal correlation prefactors using finite-size scaling of matrix elements. The correlations obtained directly from integrability and those emerging from the Luttinger field theory description are shown to be in extremely good correspondence, as expected, for the large distance asymptotics, but surprisingly also for the short distance behavior. Finally, we discuss the description of dynamical correlations from a mobile impurity model, and clarify the relation of the effective field theory parameters to the Bethe Ansatz solution.
We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}\ensuremath{-}\frac{1}{2}$ isotropic Heisenberg (XXX) chain. We provide numerical evidence that the ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as ${L}^{\ensuremath{-}1/2}$, with L the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the XXX chain. We numerically verify that typical midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).
A bstract We study the low-energy properties of the one-dimensional spin-1/2 XXZ chain with time-reversal symmetry-breaking pseudo-scalar chiral interaction and propose a phase diagram for the model. In the integrable case of the isotropic Heisenberg model with the chiral interaction, we employ the thermodynamic Bethe ansatz to find “chiralization”, the response of the ground state versus the strength of the pseudo-scalar chiral interaction of a chiral Heisenberg chain. Unlike the magnetization case, the chirality of the ground state remains zero until the transition point corresponding to critical coupling α c = 2 J/π with J being the antiferromagnetic spin-exchange interaction. The central-charge c = 1 conformal field theories (CFTs) describe the two phases with zero and finite chirality. We show for this particular case and conjecture more generally for similar phase transitions that the difference between two emergent CFTs with identical central charges lies in the symmetry of their ground state (lightest weight) primary fields, i.e., the two phases are symmetry-enriched CFTs. At finite but small temperatures, the non-chiral Heisenberg phase acquires a finite chirality that scales with the temperature quadratically. We show that the finite-size effect around the transition point probes the transition.
We study the spin-1 XY model on a hypercubic lattice in d dimensions and show that this well-known nonintegrable model hosts an extensive set of anomalous finite-energy-density eigenstates with remarkable properties. Namely, they exhibit subextensive entanglement entropy and spatiotemporal long-range order, both believed to be impossible in typical highly excited eigenstates of nonintegrable quantum many-body systems. While generic initial states are expected to thermalize, we show analytically that the eigenstates we construct lead to weak ergodicity breaking in the form of persistent oscillations of local observables following certain quantum quenches-in other words, these eigenstates provide an archetypal example of so-called quantum many-body scars. This Letter opens the door to the analytical study of the microscopic origin, dynamical signatures, and stability of such phenomena.