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arXiv:1702.03263 · Physical review. B./Physical review. B

Exact ordering of energy levels for one-dimensional interacting Fermi gases with SU (N) symmetry

Lei Pan, Yanxia Liu, Haiping Hu, Yunbo Zhang, Shu Chen

Published Aug 23, 2017• 15 citations
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Abstract

Based on the exact solution of one-dimensional Fermi gas systems with $SU$(N) symmetry in a hard wall, we demonstrate that we are able to sort the ordering of the lowest-energy eigenvalues of states with all allowed permutation symmetries, which can be solely marked by certain quantum numbers in the Bethe ansatz equations. Our results give examples beyond the scope of the generalized Lieb-Mattis theorem, which can only compare the ordering of energy levels of states belonging to different symmetry classes if they are comparable according to the pouring principle. In the strongly interacting regime, we show that the ordering of energy levels can be determined by an effective spin-exchange model, and we extend our results to a nonuniform system trapped in a harmonic potential.

DOI
10.1103/physrevb.96.075149

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Based on the exact solution of one-dimensional Fermi gas systems with SUSUSU(N) symmetry in a hard wall, we demonstrate that we are able to sort the ordering of the lowest-energy eigenvalues of states with all allowed permutation symmetries, which can be solely marked by certain quantum numbers in the Bethe ansatz equations. Our results give examples beyond the scope of the generalized Lieb-Mattis theorem, which can only compare the ordering of energy levels of states belonging to different symmetry classes if they are comparable according to the pouring principle. In the strongly interacting regime, we show that the ordering of energy levels can be determined by an effective spin-exchange model, and we extend our results to a nonuniform system trapped in a harmonic potential.
The Lieb-Mattis theorem orders the lowest-energy states of total spin s of a system of P interacting fermions. We generalize these predictions to fermionic mixtures of P particles with more than N=2 spinor components or species in the thermodynamic limit P→∞. The lowest-energy state inside each permutation symmetry sector h, arising in the P-fold tensor product decomposition, is well approximated by a U(N) coherent (quasiclassical variational) state, especially in the limit P→∞. In particular, the ground state of the system belongs the most symmetric (dominant Young tableau h_{0}) configuration. We exemplify our construction with the N=3 level Lipkin-Meshkov-Glick model, with a previous motivation on pairing correlations and U(N)-invariant quantum Hall ferromagnets. In the limit P→∞, each lowest-energy state within each permutation symmetry sector h undergoes a quantum phase transition for a critical value λ_{c}(h) of the exchange coupling constant λ, depending on h. This generalizes standard quantum phase transitions and their phase diagrams corresponding to the ground state belonging to the most symmetric sector h_{0}.
The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical k−4k^{-4}k−4 laws. The weights of these tails, denoted as Tan's contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan's contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattis theorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan's contact as a function of the interaction, temperature and transverse confinement. These laws display interesting effects related to strong correlations and dimensionality.
We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum Monte Carlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.
We consider a mixture of one-dimensional strongly interacting Fermi gases with up to six components, subjected to a longitudinal harmonic confinement. In the limit of infinitely strong repulsions we provide an exact solution which generalizes the one for the two-component mixture. We show that an imbalanced mixture under harmonic confinement displays partial spatial separation among the components, with a structure which depends on the relative population of the various components. Furthermore, we provide a symmetry characterization of the ground and excited states of the mixture introducing and evaluating a suitable operator, namely the conjugacy class sum. We show that, even under external confinement, the gas has a definite symmetry which corresponds to the most symmetric one compatible with the imbalance among the components. This generalizes the predictions of the Lieb–Mattis theorem for a Fermionic mixture with more than two components.
Consider a one-dimensional system of N N N electrons subject to an external potential U U U. Let Eel(S) E_{\rm el}(S) Eel​(S) denote the ground state energy of the system with total spin S S S. The Mattis--Lieb theorem asserts that, for a broad class of potentials U U U, the inequality Eel(S)<Eel(S′) E_{\rm el}(S) < E_{\rm el}(S') Eel​(S)<Eel​(S′) holds whenever S<S′ S < S' S<S′. This result implies that the ground state of a one-dimensional many-electron system is non-ferromagnetic. In the present work, we demonstrate that the Mattis--Lieb theorem can be extended to electron-phonon interacting systems governed by the Fröhlich model. Our analysis is carried out in the setting without an ultraviolet cutoff. The cornerstone of our approach is the construction of a Feynman--Kac-type formula for the heat semigroup generated by the Fröhlich Hamiltonian.