The material copper pyrazine dinitrate realizes an instructive paradigm of quantum criticality.

We demonstrate the existence of a special chiral “phantom” mode with some analogy to a Goldstone mode in the anisotropic quantum $XXZ$ Heisenberg spin chain. The phantom excitations contribute zero energy to the eigenstate, but a finite fixed quantum of momentum $k_0$. The mode exists not due to symmetry principles, but results from nontrivial scattering properties of magnons with momentum $k_0$ given by the anisotropy via $cos{k}_0=\mathrm{\Delta }$. Different occupations of the phantom mode lead to energetical degeneracies between different magnetization sectors in the periodic case. This mode originates from special string-type solutions of the Bethe ansatz equations with unbounded rapidities, the phantom Bethe roots (PBRs). We derive criteria under which the spectrum contains eigenstates with PBRs, both in open and periodically closed integrable systems, for spin-$1/2$ and higher spins, and discuss the respective chiral eigenstates. The simplest of such eigenstates, the spin helix state, which is a periodically modulated state of chiral nature, is built up from the phantom excitations exclusively. Implications of our results for experiments are discussed.

We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable XXZ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We derive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe ansatz equations for regular roots, one for each invariant subspace. The respective “semiphantom” Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case, “semiphantom” Bethe states carry nonzero magnetic current and are characterized by quasiperiodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating “semiphantom” Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with a simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors and calculate the simplest correlation functions, including the spin-current for all the states in the single-particle multiplet.

We construct the coordinate Bethe ansatz for all eigenstates of the open $\mathrm{spin}\text{−}\frac{1}{2}$ XXZ chain that fulfill the phantom roots criterion (PRC). Under the PRC, the Hilbert space splits into two invariant subspaces and there are two sets of homogeneous Bethe ansatz equations (BAE) to characterize the subspaces in each case. We propose two sets of vectors with chiral shocks to span the invariant subspaces and expand the corresponding eigenstates. All the vectors are factorized and have symmetrical and simple structures. Using several simple cases as examples, we present the core elements of our generalized coordinate Bethe ansatz method. The eigenstates are expanded in our generating set and show clear chirality and certain symmetry properties. The bulk scattering matrices, the reflection matrices on the two boundaries and the BAE are obtained, which demonstrates the agreement with other approaches. Some hypotheses are formulated for the generalization of our approach.