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arXiv:2404.11030 · Physical review. B./Physical review. B

Lieb-Schultz-Mattis constraints for the insulating phases of the one-dimensional SU( N ) Kondo lattice model

P. Lecheminant, Keisuke Totsuka

Published Aug 5, 2024• 3 citations
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Abstract

The nature of the insulating phases of the $\mathrm{SU}(N$)-generalization of the one-dimensional Kondo lattice model is investigated by means of nonperturbative approaches. By extending the Lieb-Schultz-Mattis (LSM) argument to multicomponent fermion systems with translation and global $\mathrm{SU}(N$) symmetries, we derive two indices which depend on the filling and the ``$\mathrm{SU}(N$)-spin'' (representation) of the local moments. These indices strongly constrain possible insulating phases; for instance, when the local moments transform in the $N$-dimensional (defining) representation of $\mathrm{SU}(N$), a featureless Kondo insulator is possible only at filling $f=1\ensuremath{-}1/N$. To obtain further insight into the insulating phases suggested by the LSM argument, we derive low-energy effective theories by adding an antiferromagnetic Heisenberg exchange interaction among the local moments [the $\mathrm{SU}(N$) Kondo-Heisenberg model]. A conjectured global phase diagram of the $\mathrm{SU}(N$) Kondo lattice model as a function of the filling and the Kondo coupling is then obtained by a combination of different analytical approaches.

DOI
10.1103/physrevb.110.075104

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The nature of the insulating phases of the SU(N\mathrm{SU}(NSU(N)-generalization of the one-dimensional Kondo lattice model is investigated by means of nonperturbative approaches. By extending the Lieb-Schultz-Mattis (LSM) argument to multicomponent fermion systems with translation and global SU(N\mathrm{SU}(NSU(N) symmetries, we derive two indices which depend on the filling and the ``SU(N\mathrm{SU}(NSU(N)-spin'' (representation) of the local moments. These indices strongly constrain possible insulating phases; for instance, when the local moments transform in the NNN-dimensional (defining) representation of SU(N\mathrm{SU}(NSU(N), a featureless Kondo insulator is possible only at filling f=1\ensuremath−1/Nf=1\ensuremath{-}1/Nf=1\ensuremath−1/N. To obtain further insight into the insulating phases suggested by the LSM argument, we derive low-energy effective theories by adding an antiferromagnetic Heisenberg exchange interaction among the local moments [the SU(N\mathrm{SU}(NSU(N) Kondo-Heisenberg model]. A conjectured global phase diagram of the SU(N\mathrm{SU}(NSU(N) Kondo lattice model as a function of the filling and the Kondo coupling is then obtained by a combination of different analytical approaches.
We study higher-spin ($S\ensuremath{\ge}1$) generalization of the one-dimensional Kondo-Heisenberg model, in which the local spin-SSS moments of the Kondo-lattice model interact with each other via the antiferromagnetic Heisenberg interaction (JH{J}_{\text{H}}JH​), by analytical and numerical methods. The strong-coupling (i.e., large Kondo coupling) expansion maps out an insulating phase at half-filling whose magnetic correlation depends on the parity of 2S2S2S as well as a ferromagnetic metallic phase which dominates the strong-coupling region at generic fillings. Then, we carried out the density-matrix renormalization-group simulations for S=1S=1S=1 to closely investigate the phase structure at large but finite Kondo coupling. At half-filling, the Kondo coupling and JH{J}_{\text{H}}JH​ do not compete and the insulating spin-gapless phase is stable, while the competition of the two leads to a stepwise collapse of the strong-coupling ferromagnetism via an intervening dimerized insulating phase with power-law spin correlation at quarter-filling.
The Lieb-Schultz-Mattis theorem for spin chains is generalized to a wide range of models of interacting electrons and localized spins on a one dimensional lattice. The existence of a low-energy state is generally proved except for special commensurate fillings where a gap may occur. Moreover, the crystal momentum of the constructed low-energy state is 2kF{2k}_{F}2kF​, where kF{k}_{F}kF​ is the Fermi momentum of the noninteracting model, corresponding to Luttinger's theorem. For the Kondo lattice model, our result implies that kF{k}_{F}kF​ must be calculated by regarding the localized spins as additional electrons.
We map out the ground-state phase diagram of a one-dimensional SU(N)\mathrm{SU}(N)SU(N) Kondo-Heisenberg lattice model at half-filling and in the fully antisymmetric self-adjoint representation as a function of 1N\frac{1}{N}N1​ and Kondo coupling Jk/t{J}_{k}/tJk​/t. On the basis of auxiliary field quantum Monte Carlo (QMC) simulations with even NNN up to 8, we show that the enlarged $\mathrm{SU}(N\ensuremath{\ge}4$) symmetry realizes a quantum phase transition separating a valence bond solid (VBS) phase occupying a weak-coupling part of the phase diagram and the Kondo insulator (KI) state dominating in the strong Jk/t{J}_{k}/tJk​/t limit. Along the phase boundary, we always observe critical exponents that belong to a two-dimensional classical Ising universality class. We next trace the evolution of the composite fermion and spin spectral functions across the phase boundary and conclude that VBS order triggers a bond order wave state of conduction electrons, both coexisting with Kondo screening. Upon further reducing Jk/t{J}_{k}/tJk​/t, we observe that composite quasiparticles lose their spectral weight indicating that VBS order gradually liberates localized fff spins from forming Kondo singlets with conduction electron spins. We contrast the QMC results with a static large-NNN approximation, and we show that in the limit of infinite degeneracy $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the order-disorder transition becomes of first order and is accompanied by a full decoupling of conduction electrons and localized fff spins. We complete our analysis by considering the limit of a vanishing Heisenberg coupling Jh=0{J}_{h}=0Jh​=0, and we provide evidence that the enlarged SU(N)\mathrm{SU}(N)SU(N) symmetry and low dimensionality of the Ruderman-Kittel-Kasuya-Yosida exchange interaction are not sufficient to induce VBS order in the conventional SU(N)\mathrm{SU}(N)SU(N) Kondo chain, which hosts solely a KI phase. Finally, we quantify the role of increasing NNN in reducing the separation of charge and spin timescales specific to the KI state, and we show that the latter is adiabatically connected to a trivial band insulator captured by the large-NNN approximation.
We study (1 + 1)-dimensional SU(N ) spin systems in the presence of global SU(N ) rotation and lattice translation symmetries.Knowing the mixed anomaly of the two symmetries at low energy, we identify, by anomaly matching argument, a topological index for spin model-the total number of Young-tableaux boxes of spins per unit cell modulo N -characterizing the "ingappability" of the system.A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis (LSM) theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases.As an example of the latter case, we show only a class of SU(N ) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries.Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry.We also predict a general constraint on the structure factor which is measurable in experiments.To generalize to higher dimensions, we introduce a new boundary condition which renders the flux-insertion argument for the LSM-type theorems in two or higher dimensional lattices free from the specific choice of system sizes.It also enables a formulation of the LSM-type theorems in arbitrary dimensions in terms of the anomaly in field theories of bulk correspondences.Moreover, by our new boundary condition, we also show that the higher-form U(1) LSM theorem and U(1) and SU(N ) LSM theorems in arbitrary higher dimensions have unified form in the bulk viewpoint.At last, we also discuss the discrete-symmetry LSM-type ingappabilities in the lattice language and we find that projective symmetry representations per unit cell implies that the system cannot be gapped with a unique translation-symmetric ground state.I would like to express my sincere gratitude to all those having helped me during my Master and Doctoral courses at the University of Tokyo.First of all, I would like to thank my parents who have supported me financially and encouraged me to do whatever I am keen on in the past 28 years.Also, I am grateful to every one that has taught me great knowledges on life and study at various steps.More specifically, I have to appreciate the guidance and kind supports by my supervisor, Professor Masaki Oshikawa who has given me patient guidance on study and kind financial supports.Moreover, I would thank Prof. Masahiro Sato for his cooperation in our paper, "Theory of electron spin resonance in one-dimensional topological insulators with spinorbit couplings: Detection of edge states".I am also grateful for Dr. Chang-Tse Hsieh for his collaboration on our work "Anomaly Matching and Symmetry-Protected Critical Phases in SU (N ) Spin Systems in 1+1 Dimensions", on which this Thesis is largely based, and Dr. Yoshiki Fukusumi on the work "Bosonization in a background U (1) gauge field".I would also thank Prof. Yuji Tachikawa and Prof. David Tong for patient and kind help on resolving my confusion during study.I would also like to acknowledge with much appreciation the help from our group members.
The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that they can both be described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it were an internal symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, we find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly that is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gaplessness local in the parameter space.
We investigate the ground-state properties of a recently proposed model for a topological Kondo insulator in one dimension (i.e., the ppp-wave Kondo-Heisenberg lattice model) by means of the density-matrix renormalization-group method. The nonstandard Kondo interaction in this model is different from the usual (i.e., local) Kondo interaction in that the localized spins couple to the ``ppp-wave'' spin density of conduction electrons, inducing a topologically nontrivial insulating ground state. Based on the analysis of the charge- and spin-excitation gaps, the string order parameter, and the spin profile in the ground state, we show that, at half filling and low energies, the system is in the Haldane phase and hosts topologically protected spin-1/2 end states. Beyond its intrinsic interest as a useful ``toy model'' to understand the effects of strong correlations on topological insulators, we show that the ppp-wave Kondo-Heisenberg model could be experimentally implemented in $p\text{\ensuremath{-}}\mathrm{band}$ optical lattices loaded with ultracold Fermi gases.