The ability to manipulate single atoms has opened up the door to constructing interesting and useful quantum structures from the ground up 1 . On the one hand, nanoscale arrangements of magnetic atoms are at the heart of future quantum computing and spintronic devices 2,3 ; on the other hand, they can be used as fundamental building blocks for the realization of textbook many-body quantum models 4 , illustrating key concepts such as quantum phase transitions, topological order or frustration as a function of system size. Here, we use low-temperature scanning tunnelling microscopy to construct arrays of magnetic atoms on a surface, designed to behave like spin-1/2 XXZ Heisenberg chains in a transverse field, for which a quantum phase transition from an antiferromagnetic to a paramagnetic phase is predicted in the thermodynamic limit 5. Site-resolved measurements on these finite-size realizations reveal a number of sudden ground state changes when the field approaches the critical value, each corresponding to a new domain wall entering the chains. We observe that these state crossings become closer for longer chains, suggesting the onset of critical behaviour. Our results present opportunities for further studies on quantum behaviour of many-body systems, as a function of their size and structural complexity.Since the birth of quantum mechanics, lattice spin systems 6 have represented a natural starting point for understanding collective quantum dynamics. Today, scanning tunnelling microscopy (STM) techniques allow one to experimentally build and probe realizations of exchange-coupled lattice spins in different geometries [7][8][9] . In linear arrangements, quantum effects are strongest 10 and notions such as quantum phase transitions 11 are most easily understood, the simplest illustration being the Ising model in a transverse field 12,13 . In this work, using STM, we construct finite-size versions of a model in the same universality class, namely the spin-1/2 XXZ chain in a transverse field 5 , which has previously been realized in the bulk material Cs 2 CoCl 4 (refs 14,15). Our set-up allows us to probe the chains with single-spin resolution while tuning an externally applied transverse field through the critical regime.The chains are created by manipulating Co atoms evaporated onto a Cu 2 N/Cu(100) surface (see Methods), which provides efficient decoupling for the magnetic d-shell electrons from the underlying bulk electrons 7 . Employing inelastic electron tunnelling spectroscopy (IETS) 16,17 at sufficiently low temperature (330 mK) allows us to determine the magnetic anisotropy vector of each atom 18 as well as the strength of the exchange coupling between neighbouring atoms 19 . It was previously demonstrated that Co atoms on this surface behave as spin S = 3/2 objects experiencing a strong uniaxial hard-axis anisotropy pointing inplane, perpendicular to the bond with the neighbouring N atoms 20 . As a result, the m z = ±3/2 states split off approximately 5.5 meV above the m z = ±1/2 doublet (see Fig. 1a)...

We reformulate the topological symmetry breaking scheme for phase transitions in systems with anyons in a grahical manner. A new set of quantities called vertex lifiting coefficients (VLCs) is introduced and used to specify the the full operator content of the broken phase. First, it is shown how the assumption that a set of charges behaves like the vacuum of a new theory naturally leads to diagrammatic consistency conditions for a condensate. This recovers the notion of a condensate used in earlier aproaches and uncovers the connection to pure mathematics. The VLCs are needed to solve the consistency conditions and establish the mapping of the fusion and splitting spaces of the broken theory into the parent phase. This enables one to calculate the full set of topological data (S-, T -, R-and F -matrices) for the condensed phase and closed form expressions in terms of the VLCs are provided. We furthermore furnish a cocrete recipe to lift arbitrary diagrams directly from the condensed phase to the original phase using only a limited number of VLCs and we describe a method for the explicit calculation of VLCs for a large class of bosonic condensates. This allows for the explicit calculation of condensed-phase diagrams in many physically relevant cases and representative examples are worked out in detail.

Split Fermi seas in one-dimensional Bose fluidsFokkema, T.B.; Eliëns, I.S.; Caux, J.S. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. For the one-dimensional repulsive Bose gas (Lieb-Liniger model), we study a special class of highly excited states obtained by giving a finite momentum to subgroups of particles. These states, which correspond to "splitting" the ground-state Fermi-sea-like quantum number configuration, are zero-entropy states which display interesting properties more normally associated with ground states. Using a numerically exact method based on integrability, we study these states' excitation spectrum, density correlations, and momentum distribution functions. These correlations display power-law asymptotics and are shown to be accurately described by an effective multicomponent Tomonaga-Luttinger liquid theory whose parameters are obtained from the Bethe ansatz. The nonuniversal correlation prefactors are moreover obtained from integrability, yielding a completely parameter-free fit of the correlator asymptotics.

Abstract. We present a general derivation of the spectrum of excitations for gapless states of zero entropy density in Bethe ansatz solvable models. Our formalism is valid for an arbitrary choice of bare energy function which is relevant to situations where the Hamiltonian for time evolution differs from the Hamiltonian in a (generalized) Gibbs ensemble, i.e. out of equilibrium. The energy of particle and hole excitations, as measured with the time-evolution Hamiltonian, is shown to include additional contributions stemming from the shifts of the Fermi points that may now have finite energy. The finite-size effects are also derived and the connection with conformal field theory discussed. The critical exponents can still be obtained from the finite-size spectrum, however the velocity occurring here differs from the one in the constant Casimir term. The derivation highlights the importance of the phase shifts at the Fermi points for the critical exponents of asymptotes of correlations. We generalize certain results known for the ground state and discuss the relation to the dressed charge (matrix). Finally, we discuss the finite-size corrections in the presence of an additional particle or hole which are important for dynamical correlation functions.

We study the influence of reflective boundaries on time-dependent responses of one-dimensional quantum fluids at zero temperature beyond the low-energy approximation. Our analysis is based on an extension of effective mobile impurity models for nonlinear Luttinger liquids to the case of open boundary conditions. For integrable models, we show that boundary autocorrelations oscillate as a function of time with the same frequency as the corresponding bulk autocorrelations. This frequency can be identified as the band edge of elementary excitations. The amplitude of the oscillations decays as a power law with distinct exponents at the boundary and in the bulk, but boundary and bulk exponents are determined by the same coupling constant in the mobile impurity model. For nonintegrable models, we argue that the power-law decay of the oscillations is generic for autocorrelations in the bulk, but turns into an exponential decay at the boundary. Moreover, there is in general a nonuniversal shift of the boundary frequency in comparison with the band edge of bulk excitations. The predictions of our effective field theory are compared with numerical results obtained by time-dependent density matrix renormalization group (tDMRG) for both integrable and nonintegrable critical spin-S chains with S = 1/2, 1 and 3/2.