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Alkaline-earth and ytterbium cold atomic gases make it possible to simulate SU( N )-symmetric fermionic systems in a very controlled fashion. Such a high symmetry is expected to give rise to a variety of novel phenomena ranging from molecular Luttinger liquids to (symmetry-protected) topological phases. We review some of the phases that can be stabilized in a one dimensional lattice. The physics of this multi-component Fermi gas turns out to be much richer and more exotic than in the standard SU(2) case. For N > 2 , the phase diagram is quite rich already in the case of the single-band model, including a molecular Luttinger liquid (with dominant superfluid instability in the N -particle channel) for incommensurate fillings, as well as various Mott-insulating phases occurring at commensurate fillings. Particular attention will be paid to the cases with additional orbital degree of freedom (which is accessible experimentally either by taking into account two atomic states or by putting atoms in the p -band levels). We introduce two microscopic models which are relevant for these cases and discuss their symmetries and strong coupling limits. More intriguing phase diagrams are then presented including, for instance, symmetry protected topological phases characterized by non-trivial edge states. [ABSTRACT FROM AUTHOR]

We investigate the infrared properties of SU ( N ) k conformal field theory perturbed by its adjoint primary field in 1 + 1 dimensions. The latter field theory is shown to govern the low-energy properties of various SU( N ) spin chain problems. In particular, using a mapping onto k -leg SU( N ) spin ladder, a massless renormalization group flow to SU ( N ) 1 criticality is predicted when N and k have no common divisor. The latter result extends the well-known massless flow between SU ( 2 ) k and SU(2) 1 Wess–Zumino–Novikov–Witten theories when k is odd in connection to the Haldane's conjecture on SU(2) Heisenberg spin chains. A direct approach is presented in the simplest N = 3 and k = 2 case to investigate the existence of this massless flow. [ABSTRACT FROM AUTHOR]

We present a field-theory analysis of a model of two SU(2n)-invariant magnetic chains coupled by a generic interaction preserving time-reversal and inversion symmetry. Contrary to the SU(2)-invariant case the zero-temperature phase diagram of such two-leg spin ladder does not contain topological phases. Only generalized valence bond solid phases are stabilized when n>1 with different wave vectors and ground-state degeneracies. In particular, we find a phase which is made of a cluster of 2n spins put in an SU(2n) singlet state. For n=3, this cluster phase is relevant to 173Yb ultracold atoms, with an emergent SU(6) symmetry, loaded in a double-well optical lattice. [ABSTRACT FROM AUTHOR]

Four-spin exchange interaction has been raising intriguing questions regarding the exotic phase transitions it induces in two-dimensional quantum spin systems. In this context, we investigate the effects of a cyclic four-spin exchange in the quasi-1D limit by considering a general N-leg spin ladder. We show by means of a low-energy approach that depending on its sign, this ring exchange interaction can engender either a staggered or a uniform dimerization from the conventional phases of spin ladders. The resulting quantum phase transition is found to be described by the SU(2)N conformal field theory. This result, as well as the fractional value of the central charge at the transition, is further confirmed by a large-scale numerical study performed by means of exact diagonalization and density matrix renormalization group approaches for N≤4. [ABSTRACT FROM AUTHOR]

We investigate the competition between different orders in the two-leg spin ladder with a ring-exchange interaction by means of a bosonic approach. The latter is defined in terms of spin-1 hard-core bosons which treat the Néel and vector-chirality order parameters on an equal footing. A semiclassical approach of the resulting model describes the phases of the two-leg spin ladder with a ring exchange. In particular, we derive the low-energy effective actions which govern the physical properties of the rung-singlet and dominant vector-chirality phases. As a by-product of our approach, we reveal the mutual induction phenomenon between spin and chirality with, for instance, the emergence of a vector-chirality phase from the application of a magnetic field in bilayer systems coupled by four-spin exchange interactions. [ABSTRACT FROM AUTHOR]

The competition between intertube hopping processes and density-density interactions is investigated in one-dimensional quantum dipolar bosons systems of A' coupled tubes at zero temperature. Using a phenomenological bosonization approach, we show that the resulting competition leads to an exotic quantum phase transition described by a U(l) x Z/v conformai field theory with a fractional central charge. The emerging Zi parafermionic critical degrees of freedom are highly nontrivial in terms of the original atoms or polar molecules of the model. We further determine the main physical properties of the quantum-critical point in a double-tube system which has central charge c = 3/2. In triple-tube systems, we show that the competition between the two antagonistic processes is related to the physics of the two-dimensional Z3 chiral Potts model. This work opens the possibility to study the exotic properties of the Z parafermions in the context of ultracold quantum Bose gases. [ABSTRACT FROM AUTHOR]

We investigate the possible formation of a molecular condensate, which might be, for instance, the analogue of the alpha condensate of nuclear physics, in the context of multicomponent cold atoms fermionic systems. A simple paradigmatic model of N-component fermions with contact interactions loaded into a one-dimensional optical lattice is studied by means of low-energy and numerical approaches. For attractive interaction, a quasi-long-range molecular superfluid phase, formed from bound-states made of N fermions, emerges at low density. We show that trionic and quartetting phases, respectively for N = 3,4, extend in a large domain of the phase diagram and are robust against small symmetry-breaking perturbations. [ABSTRACT FROM AUTHOR]

Abstract: The physical properties of arbitrary half-integer spins fermionic cold atoms loaded into a one-dimensional optical lattice are investigated by means of a conformal field theory approach. We show that for attractive interactions two different superfluid phases emerge for : A BCS pairing phase, and a molecular superfluid phase which is formed from bound-states made of 2N fermions. In the low-energy approach, the competition between these instabilities and charge-density waves is described in terms of parafermionic degrees of freedom. The quantum phase transition for is universal and shown to belong to the Ising and three-state Potts universality classes respectively. In contrast, for , the transition is non-universal. For a filling of one atom per site, a Mott transition occurs and the nature of the possible Mott-insulating phases are determined. [Copyright &y& Elsevier]

Abstract: The nature of the deconfining phase transition in the ()-dimensional SU(N) Georgi–Glashow model is investigated. Within the dimensional-reduction hypothesis, the properties of the transition are described by a two-dimensional vectorial Coulomb gas models of electric and magnetic charges. The resulting critical properties are governed by a generalized SU(N) sine-Gordon model with self-dual symmetry. We show that this model displays a massless flow to an infrared fixed point which corresponds to the Z N parafermions conformal field theory. This result, in turn, supports the conjecture of Kogan, Tekin, and Kovner that the deconfining transition in the ()-dimensional SU(N) Georgi–Glashow model belongs to the Z N universality class. [Copyright &y& Elsevier]

We discuss the nature of criticality in the β2=2πN self-dual extension of the sine-Gordon model. This field theory is related to the two-dimensional classical XY model with a N-fold degenerate symmetry-breaking field. We briefly overview the already studied cases N=2,4 and analyze in detail the case N=3 where a single phase transition in the three-state Potts universality class is expected to occur. The Z3 infrared critical properties of the β2=6π self-dual sine-Gordon model are derived using two non-perturbative approaches. On one hand, we map the model onto an integrable deformation of the Z4 parafermion theory. The latter is known to flow to a massless Z3 infrared fixed point. Another route is based on the connection with a chirally asymmetric, su(2)4⊗su(2)1 Wess–Zumino–Novikov–Witten model with anisotropic current–current interaction, where we explore the existence of a decoupling (Toulouse) point. [Copyright &y& Elsevier]

We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids--spin analogs of fractional non-Abelian quantum Hall states--with gapped bulk and gapless chiral edge excitations described by the SU(2)n Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-n/2 ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)n gapless edge excitations. [ABSTRACT FROM AUTHOR]

The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian SU(N-1)-singlet FQH states at filling fraction ν=k(N-1)/[N+k(N-1)m] labeled by integers k and m, which are potentially realized in multicomponent quantum Hall systems or SU(N) spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an (N-1)-component free-boson CFT and the SU(N)k/[U(1)]N-1 CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an SU(N) Heisenberg model. [ABSTRACT FROM AUTHOR]

Abstract: A unifying approach to competing quantum orders in frustrated two-leg spin ladders (1D) and square-lattice antiferromagnets (2D) is presented. Hidden relationship and quantum phase transitions among the competing orders are discussed. Our approach reveals that these systems have a relatively simple phase structure in spite of its complicated interactions. In particular, there is an intriguing U(1) symmetry (which exists only approximately in 2D) which mixes up antiferromagnetic order parameter with that of the p-type nematic. On top of it, we found an emergent U(1)-symmetry. On the basis of the field-theoretical- and variational analysis, we give a qualitative picture for the global structure of the phase diagram. Interesting connection to other models (e.g. bosonic t–J model) and its application to the 2D problem are also discussed. [Copyright &y& Elsevier]

The zero-temperature phase diagram of a p-orbital two-component fermionic system loaded into a one-dimensional optical lattice is mapped out by means of analytical and numerical techniques. It is shown that the p-band model away from half-filling hosts various competing superconducting phases for attractive and repulsive interactions. At quarter-filling, we analyze the possible formation of incompressible Mott phases and in particular for repulsive interactions, we find the occurrence of a Mott transition with the formation of fully gapped bond-ordering waves. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the possible classification of zero-temperature spin-gapped phases of multicomponent electronic systems in one spatial dimension. At the heart of our analysis is the existence of non-perturbative duality symmetries which emerge within a low-energy description. These dualities fall into a finite number of classes that can be listed and depend only on the algebraic properties of the symmetries of the system: its physical symmetry group and the maximal continuous symmetry group of the interaction. We further characterize possible competing orders associated to the dualities and discuss the nature of the quantum phase transitions between them. Finally, as an illustration, the duality approach is applied to the description of the phases of two-leg electronic ladders at incommensurate filling. [Copyright &y& Elsevier]

We study spin 3/2 fermionic cold atoms with attractive interactions confined in a one-dimensional optical lattice. Using numerical techniques, we determine the phase diagram for a generic density. For the chosen parameters, one-particle excitations are gapped and the phase diagram is separated into two regions: one where the two-particle excitation gap is zero, and one where it is finite. In the first region, the two-body pairing fluctuations (BCS) compete with the density ones. In the other one, a molecular superfluid (MS) phase, in which bound-states of four particles form, competes with the density fluctuations. The properties of the transition line between these two regions is studied through the behavior of the entanglement entropy. The physical features of the various phases, comprising leading correlations, Friedel oscillations, and excitation spectra, are presented. To make the connection with experiments, the effect of a harmonic trap is taken into account. In particular, we emphasize the conditions under which the appealing MS phase can be realized, and how the phases could be probed by using the density profiles and the associated structure factor. Lastly, the consequences on the flux quantization of the different nature of the pairing in the BCS and MS phases are studied in a situation where the condensate is in a ring geometry. [ABSTRACT FROM AUTHOR]

We investigate the zero-temperature phase diagram of two-orbital SU(N) fermionic models at incommensurate filling which are directly relevant to strontium and ytterbium ultracold atoms loading into a one-dimensional optical lattice. Using a low-energy approach that takes into account explicitly the SU(N) symmetry, we find that a spectral gap for the nuclear-spin degrees of freedom is formed for generic interactions. Several phases with one or two gapless modes are then stabilized which describe the competition between different density instabilities. In stark contrast to the N=2 case, no dominant pairing instabilities emerge and the leading superfluid one is rather formed from bound states of 2N fermions. [ABSTRACT FROM AUTHOR]

The zero-temperature phase diagram of a p-orbital two-component fermionic system loaded into a one-dimensional optical lattice is mapped out by means of analytical and numerical techniques. It is shown that the p-band model away from half-filling hosts various competing superconducting phases for attractive and repulsive interactions. At quarter-filling, we analyze the possible formation of incompressible Mott phases and in particular for repulsive interactions, we find the occurrence of a Mott transition with the formation of fully gapped bond-ordering waves. [ABSTRACT FROM AUTHOR]

We investigate possible realizations of exotic SU(N) symmetry-protected topological (SPT) phases with alkaline-earth cold fermionic atoms loaded into one-dimensional optical lattices. A thorough study of two-orbital generalizations of the standard SU(N) Fermi-Hubbard model, directly relevant to recent experiments, is performed. Using state-of-the-art analytical and numerical techniques, we map out the zero-temperature phase diagrams at half-filling and identify several Mott-insulating phases. While some of them are rather conventional (nondegenerate, charge-density wave, or spin-Peierls-like), we also identify, for even N, two distinct types of SPT phases: an orbital Haldane phase, analogous to a spin-N/2 Haldane phase, and a topological SU(N) phase, which we fully characterize by its entanglement properties. We also propose sets of nonlocal order parameters that characterize the SU(N) topological phases found here. [ABSTRACT FROM AUTHOR]

We study the thermodynamics of the relativistic quantum O(N) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form P(T) = P(0) + N(T³/c²)ƑN(Δ/T), where c is the velocity of the excitations at the QCP and | Δ| a characteristic zero-temperature energy scale. Using both a large-N approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function ƑN. For small values of N (N ≲ 10) we find that Ƒn(x) is nonmonotonic in the quantum critical regime (|x| ≲ 1) with a maximum near x = 0. The large-N approach--if properly interpreted--is a good approximation both in the renormalized classical (x ≲ -- 1) and quantum disordered (x ≳ 1 ) regimes, but fails to describe the nonmonotonic behavior of ƑN in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum 0(2) model and find that in the vicinity of the QCP the universal ratio TBKT/PS(0) is very close to Π/2,.... implying that the stiffness ps(TBKT) at the transition is only slightly reduced with respect to the zero-temperature stiffness ps(0). Finally, we briefly discuss the experimental determination of the universal function Ƒ2 from the pressure of a Bose gas in an optical lattice near the superfluid-Mott-insulator transition. [ABSTRACT FROM AUTHOR]

The phase diagram of one-dimensional alkaline-earth fermionic atoms and ytterbium 171 atoms is investigated by means of a low-energy approach and density-matrix renormalization group calculations. For incommensurate filling, four gapless phases with a spin gap are found and consist of two superconducting instabilities and two coexisting bond and charge density-waves instabilities. In the half-filled case, seven Mott-insulating phases arise with the emergence of four non-degenerate phases with exotic hidden orderings. [ABSTRACT FROM AUTHOR]

The zero-temperature properties of the SO(5) bilinear-biquadratic Heisenberg chain are investigated by means of a low-energy approach and large-scale numerical calculations. In sharp contrast to the spin-1 SO(3) Heisenberg chain, we show that the SO(5) Heisenberg chain is dimerized with a twofold degenerate ground state. On top of this gapful phase, we find the emergence of a nondegenerate gapped phase with hidden (Z2 × Z2)2 symmetry and spin-3/2 edge states that can be understood from a SO(5) AKLT wave function. We derive a low-energy theory describing the quantum critical point which separates these two gapped phases. It is shown and confirmed numerically that this quantum critical point belongs to the SO(5)t universality class. [ABSTRACT FROM AUTHOR]