We use the bosonization approach to investigate quantum phases of boson-fermion mixtures (BFM) of atoms confined to one dimension by an anisotropic optical lattice. For a BFM with a single species of fermions we find a charge-density wave phase, a fermion pairing phase, and a phase separation regime. We also obtain the rich phase diagram of a BFM with two species of fermions. We demonstrate that these phase diagrams can be understood in terms of polarons, i.e., atoms "dressed" by screening clouds of the other atom species. Techniques to detect the resulting quantum phases are discussed.

We study theoretically transport of a one-dimensional single-channel interacting electron gas through barriers or constrictions. We find that electrons with repulsive interactions, incident upon a single barrier, are completely reflected at zero temperature. At finite temperature (T), the conductance is shown to vanish as a power of T, and at zero temperature, power-law current-voltage characteristics are predicted. For attractive interactions, we predict perfect transmission at zero temperature, with similar power-law corrections. We also study resonant tunneling through a double-barrier structure and related erects associated with the Coulomb blockade

We show that dark solitons in 1D Bose liquids may be created by absorption of a single quanta of an external ac field, in a close analogy with the Einstein's photoelectric effect. Similarly to the von Lenard's experiment with photoexcited electrons, the external field's photon energy $\hbar\omega$ should exceed a certain threshold. In our case the latter is given by the soliton energy $\varepsilon_s(\hbar q)$ with the momentum $\hbar q$, where $q$ is photon's wavenumber. We find the probability of soliton creation to have a power-law dependence on the frequency detuning $\omega-\varepsilon_s/\hbar$. This dependence is a signature of the quantum nature of the absorption process and the orthogonality catastrophe phenomenon associated with it.

We show how the worldline quantum Monte Carlo procedure, which usually relies on an artificial time discretization, can be formulated directly in continuous time, rendering the scheme exact. For an arbitrary system with discrete Hilbert space, none of the configuration update procedures contain small parameters. We find that the most effective update strategy involves the motion of worldline discontinuities (both in space and time), i.e., the evaluation of the Green’s function. Being based on local updates only, our method nevertheless allows one to work with the grand canonical ensemble and nonzero winding numbers, and to calculate any dynamical correlation function as easily as expectation values of, e.g., total energy. The principles found for the update in continuous time generalize to any continuous variables in the space of discrete virtual transitions, and in principle also make it possible to simulate continuous systems exactly.

We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a Luttinger liquid implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.

We review recent theoretical work on two closely related issues: excitation of an isolated quantum condensed matter system driven adiabatically across a continuous quantum phase transition or a gapless phase, and apparent relaxation of an excited system after a sudden quench of a parameter in its Hamiltonian. Accordingly, the review is divided into two parts. The first part revolves around a quantum version of the Kibble–Zurek mechanism including also phenomena that go beyond this simple paradigm. What they have in common is that excitation of a gapless many-body system scales with a power of the driving rate. The second part attempts a systematic presentation of recent results and conjectures on apparent relaxation of a pure state of an isolated quantum many-body system after its excitation by a sudden quench. This research is motivated in part by recent experimental developments in the physics of ultracold atoms with potential applications in the adiabatic quantum state preparation and quantum computation.

We investigate the zero-temperature phases of bosonic and fermionic gases confined to one dimension and interacting via a class of finite-range soft-shoulder potentials (i.e., soft-core potentials with an additional hard-core onsite interaction). Using a combination of analytical and numerical methods, we demonstrate the stabilization of critical quantum liquids with qualitatively new features with respect to the Tomonaga-Luttinger liquid paradigm. These features result from frustration and cluster formation in the corresponding classical ground state. Characteristic signatures of these liquids are accessible in state-of-the-art experimental setups with Rydberg-dressed ground-state atoms trapped in optical lattices.

We investigate the ground-state properties of a newly discovered phase of one-dimensional lattice bosons with extended interactions [E. G. Dalla Torre et al., Phys. Rev. Lett. 97, 260401 (2006)]. The new phase, termed the Haldane insulator in analogy with the gapped phase of spin-1 chains, is characterized by a nonlocal order parameter, which can only be written as an infinite string in terms of the bosonic densities. We show that the string order can nevertheless be probed with physical fields that couple locally, via the effect those fields have on the quantum phase transitions separating the exotic phase from the conventional Mott and density wave phases. Using a field theoretical analysis, we show that a perturbation that breaks lattice inversion symmetry gaps the critical point separating the Mott and Haldane phases and eliminates the sharp distinction between them. This is remarkable given that neither of these phases involves broken inversion symmetry. We also investigate the evolution of the phase diagram with the tunable coupling between parallel chains in an optical lattice setup. We find that interchain tunneling destroys the direct phase transition between the Mott and Haldane insulators by establishing an intermediate superfluid phase. On the other hand, coupling the chains only by weak repulsive interactions does not modify the structure of the phase diagram. The theoretical predictions are confirmed with numerical calculations using the density matrix renormalization group.

We discuss the superfluid (SF) to Mott-insulator transition of cold atoms in optical lattices recently observed by Greiner et al (2002 Nature 415 39). The fundamental properties of both phases and their experimental signatures ar ed iscussed carefully, including the limitations of the standard Gutzwiller approximation. It is shown that in a one-dimensional dilute Bose-gas with a strong transverse confinement (Tonks-gas), even an arbitrary weak optical lattice is able to induce a Mott-like state with crystalline order, provided the dimensionless interactio np arameter is larger than a critical value of order one. The SF–insulator transition of the Bose–Hubbard model in this case continuously evolves into a transition of the commensurate–incommensurate type with decreasing strength of the external optical lattice.

We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.

We demonstrate the application of the Metropolis–Hastings algorithm to sampling of classical thermal states of one-dimensional Bose–Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case. The presented algorithm can be easily generalized to higher dimensions and arbitrary trap geometry. For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode). The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only from canonical and grand canonical distributions, but also from the generalized Gibbs ensemble, which can help to shed new light on thermodynamics of integrable systems.

We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval (0,x). At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.

We conjecture that thermalization following a quantum quench in a strongly correlated quantum system is closely connected to many-body delocalization in the space of quasi-particles. This scenario is tested in the anisotropic Heisenberg spin chain with different types of integrability-breaking terms. We first quantify the deviations from integrability by analyzing the level spacing statistics and the inverse participation ratio of the system's eigenstates. We then focus on thermalization, by studying the dynamics after a sudden quench of the anisotropy parameter. Our numerical simulations clearly support the conjecture, as long as the integrability-breaking term acts homogeneously on the quasiparticle space, in such a way as to induce ergodicity over all the relevant Hilbert space.

We calculate the asymptotic behavior of correlators as a function of the microscopic parameters for an integrable Bose-Fermi mixture with repulsive interaction in one dimension. For two cases, namely polarized and unpolarized fermions the singularities of the momentum distribution functions are characterized as a function of the coupling constant and the relative density of bosons.

Understanding strongly correlated quantum systems is a central problem in many areas of physics. The collective behaviour of interacting particles gives rise to diverse fundamental phenomena such as confinement in quantum chromodynamics, electron fractionalization in the quantum Hall regime and phase transitions in unconventional superconductors and quantum magnets. Such systems typically involve massive particles, but optical photons can also interact with one another in a nonlinear medium. In practice, however, such interactions are often very weak. Here we describe a technique that enables the creation of a strongly correlated quantum gas of photons using one-dimensional optical systems with tight field confinement and coherent photon trapping techniques. The confinement enables the generation of large, tunable optical nonlinearities via the interaction of photons with a nearby cold atomic gas. In its extreme, we show that a quantum light field can undergo fermionization in such one-dimensional media, which can be probed via standard photon correlation measurements. Interactions between photons are typically extremely weak. But when light pulses are confined to an optical waveguide and manipulated with nearby cold atoms, strongly interacting photons can be created that may even undergo crystallization, as is now shown theoretically.

This review focuses on the field of quantum entanglement applied to condensed matter physics systems with strong correlations, a domain which has rapidly grown over the last decade. By tracing out part of the degrees of freedom of correlated quantum systems, useful and non-trivial informations can be obtained through the study of the reduced density matrix, whose eigenvalue spectrum (the entanglement spectrum) and the associated Renyi entropies are now well recognized to contains key features. In particular, the celebrated area law for the entanglement entropy of ground-states will be discussed from the perspective of its subleading corrections which encode universal details of various quantum states of matter, e.g. symmetry breaking states or topological order. Going beyond entropies, the study of the low-lying part of the entanglement spectrum also allows to diagnose topological properties or give a direct access to the excitation spectrum of the edges, and may also raise significant questions about the underlying entanglement Hamiltonian. All these powerful tools can be further applied to shed some light on disordered quantum systems where impurity/disorder can conspire with quantum fluctuations to induce non-trivial effects. Disordered quantum spin systems, the Kondo effect, or the many-body localization problem, which have all been successfully (re)visited through the prism of quantum entanglement, will be discussed in details. Finally, the issue of experimental access to entanglement measurement will be addressed, together with its most recent developments.

The study of exotic one-dimensional states, particularly those at the edges of topological materials, demand new experimental probes that can access the interplay between charge and spin degrees of freedom. One potential approach is to use a single spin probe, such as a nitrogen vacancy center in diamond, which has recently emerged as a versatile tool to probe nanoscale systems in a noninvasive fashion. Here, we present a theory describing how noise magnetometry with spin probes can directly address several questions that have emerged in experimental studies of 1D systems, including those in topological materials. We show that by controlling the spin degree of freedom of the probe, it is possible to measure locally and independently local charge and spin correlations of 1D systems. Visualization of 1D edge states, as well as sampling correlations with wave-vector resolution can be achieved by tuning the probe-to-sample distance. Furthermore, temperature-dependent measurements of magnetic noise can clearly delineate the dominant scattering mechanism (impurities versus interactions)—this is of particular relevance to quantum spin Hall measurements where conductance quantization is not perfect. The possibility to probe both charge and spin excitations in a wide range of length scales opens new pathways to bridging the large gap between atomic scale resolution of scanning probes and global transport measurements.

The spectrum of charge excitations, Im N ( q , ω), has been evaluated near the Mott transition in the one-dimensional Hubbard model with a particular value of the Coulomb interaction. The present theory based on a continuum limit describes the smooth change between the Tomonaga-Luttinger liquid state with a q -linear mode and the Mott insulator with a charge gap, by considering Umklapp scattering. The charge susceptibility is seen to diverge proportional to the inverse of the doping rate as the Mott insulator state is approached, in agreement with Bethe-ansatz result. It is shown that the q region where the renormalization group theory and conformal field theory are valid becomes vanishingly small as the transition is approached.

The quantum phase transitions in the one-dimensional asymmetric Hubbard model are investigated with the bosonization approach. The critical conditions for the transition from density wave to phase separation, the correlation functions and their exponents are obtained analytically. Our results show that the difference between the hopping integrals for upand down-spin electrons is crucial for the happening of the phase separation. When the difference is large enough, the phase separation will appear even if the on-site interaction is small.

The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the s-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences ofmore » spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.« less