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arXiv:2608.28898 · The European Physical Journal B

Phase stability, charge ordering, and charge-liquid formation in the Falicov–Kimball model on triangular and kagome lattices

Ammar Nejati, Marvin Lenk

Published Sep 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract We present a systematic investigation of the stability of the phase diagram of the spinless Falicov–Kimball model on two-dimensional non-bipartite lattices, the triangular and kagome lattices, using Markov-chain Monte Carlo simulations. Three filling or chemical potential conditions are examined: fixed chemical potentials ( $$\mu _f = \mu _c = U/2$$ μ f = μ c = U / 2 ), “generalized” half-filling ( $$\bar{n}_f = 1/3$$ n ¯ f = 1 / 3 , $$\bar{n}_c = 2/3$$ n ¯ c = 2 / 3 ), and conventional half-filling ( $$\bar{n}_f = \bar{n}_c = 1/2$$ n ¯ f = n ¯ c = 1 / 2 ). For each condition, the effects of a perturbatively small next-nearest-neighbor hopping are studied. On the triangular lattice, Anderson insulator, Mott insulator, and charge-density wave (CDW) phases are found in all cases, with the CDW driven by Coulomb interaction-mediated nesting. Evidence for a charge-liquid regime is found in the region between the weakly correlated regime and the CDW, characterized by competing charge-ordering wavevectors and the absence of universal scaling behavior, whose extent shrinks progressively from the grand-canonical to the conventional half-filling condition. On the kagome lattice, the CDW phase is completely absent; instead, an insulating ground state, plausibly of the Mott type, is found, with signatures of a possible quantum phase transition at zero temperature. Our results establish the crucial role of lattice geometry and particle-hole symmetry breaking in shaping the phase diagram of correlated electron systems. Graphical abstract The 2d triangular lattice (left) and the 2d kagome lattice (right) with their unit cells highlighted. Bottom: The corresponding phase diagram of the Falicov-Kimball model (FKM) on each lattice, illustrating the non-Fermi liquid (nFL), charge-density wave (CDW), Anderson insulator (AI) and Mott insulator (MI) regimes.

DOI
10.1140/epjb/s10051-026-01245-9

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•Jan 1, 2015•0 citations
Abstract We present a systematic investigation of the stability of the phase diagram of the spinless Falicov–Kimball model on two-dimensional non-bipartite lattices, the triangular and kagome lattices, using Markov-chain Monte Carlo simulations. Three filling or chemical potential conditions are examined: fixed chemical potentials ( μf=μc=U/2\mu _f = \mu _c = U/2μf​=μc​=U/2 μ f = μ c = U / 2 ), “generalized” half-filling ( nˉf=1/3\bar{n}_f = 1/3nˉf​=1/3 n ¯ f = 1 / 3 , nˉc=2/3\bar{n}_c = 2/3nˉc​=2/3 n ¯ c = 2 / 3 ), and conventional half-filling ( nˉf=nˉc=1/2\bar{n}_f = \bar{n}_c = 1/2nˉf​=nˉc​=1/2 n ¯ f = n ¯ c = 1 / 2 ). For each condition, the effects of a perturbatively small next-nearest-neighbor hopping are studied. On the triangular lattice, Anderson insulator, Mott insulator, and charge-density wave (CDW) phases are found in all cases, with the CDW driven by Coulomb interaction-mediated nesting. Evidence for a charge-liquid regime is found in the region between the weakly correlated regime and the CDW, characterized by competing charge-ordering wavevectors and the absence of universal scaling behavior, whose extent shrinks progressively from the grand-canonical to the conventional half-filling condition. On the kagome lattice, the CDW phase is completely absent; instead, an insulating ground state, plausibly of the Mott type, is found, with signatures of a possible quantum phase transition at zero temperature. Our results establish the crucial role of lattice geometry and particle-hole symmetry breaking in shaping the phase diagram of correlated electron systems. Graphical abstract The 2d triangular lattice (left) and the 2d kagome lattice (right) with their unit cells highlighted. Bottom: The corresponding phase diagram of the Falicov-Kimball model (FKM) on each lattice, illustrating the non-Fermi liquid (nFL), charge-density wave (CDW), Anderson insulator (AI) and Mott insulator (MI) regimes.
Due to the interplay between charge fluctuation and geometry frustration, the doped kagome-lattice Mott insulator is a fascinating platform to realize exotic quantum states. Through the state-of-the-art density matrix renormalization group calculation, we explore the quantum phases of the lightly doped kagome-lattice $t\text{\ensuremath{-}}J$ model in the presence of the next-nearest-neighbor electron hopping t2{t}_{2}t2​ and spin interaction J2{J}_{2}J2​. On the Ly=3{L}_{y}=3Ly​=3 cylinder (Ly{L}_{y}Ly​ is the number of unit cells along the circumference direction), we establish a quantum phase diagram with tuning t2>0{t}_{2}>0t2​>0 and J2>0{J}_{2}>0J2​>0, showing an emergent Fermi-liquid-like phase driven by increased t2{t}_{2}t2​ and J2{J}_{2}J2​, at the neighbor of the previously identified charge density wave (CDW) phase. Compared with the CDW phase, the charge order is significantly suppressed in the Fermi-liquid-like phase, and most correlation functions are greatly enhanced with a power-law decay. In particular, we find the absence of hole pairing and a strong three-sublattice magnetic correlation. On the wider Ly=4{L}_{y}=4Ly​=4 cylinder, this Fermi-liquid-like phase persists at low doping levels, strongly suggesting that this state might be stable in the two-dimensional kagome system.
Fermi-liquid-like phase driven by next-nearest-neighbor couplings in a lightly doped kagome-lattice ttt-JJJ model
Due to the interplay between charge fluctuation and geometry frustration, the doped kagome-lattice Mott insulator is a fascinating platform to realize exotic quantum states. Through the state-of-the-art density matrix renormalization group calculation, we explore the quantum phases of the lightly doped kagome-lattice ttt-JJJ model in the presence of the next-nearest-neighbor electron hopping t2t_2t2​ and spin interaction J2J_2J2​. On the Ly=3L_y = 3Ly​=3 cylinder (LyL_yLy​ is the number of unit cells along the circumference direction), we establish a quantum phase diagram with tuning t2>0t_2 > 0t2​>0 and J2>0J_2 > 0J2​>0, showing an emergent Fermi-liquid-like phase driven by increased t2t_2t2​ and J2J_2J2​, which sits at the neighbor of the previously identified charge density wave (CDW) phase. Compared with the CDW phase, the charge order is significantly suppressed in the Fermi-liquid-like phase, and most correlation functions are greatly enhanced with power-law decay. In particular, we find the absence of hole pairing and a strong three-sublattice magnetic correlation. On the wider Ly=4L_y = 4Ly​=4 cylinder, this Fermi-liquid-like phase persists at low doping levels, strongly suggesting that this state might be stable in the two-dimensional kagome system.
It has long been believed that doped quantum spin liquids (QSLs) can give rise to fascinating quantum phases, including the possibility of high-temperature superconductivity (SC) as proposed by P. W. Anderson's resonating valence bond (RVB) scenario. The Kagome lattice ttt-JJJ model is known to exhibit spin liquid behavior at half-filling, making it an ideal system for studying the properties of doped QSL. In this study, we employ the fermionic projected entangled simplex state (PESS) method to investigate the ground state properties of the Kagome lattice ttt-JJJ model with t/J=3.0t/J = 3.0t/J=3.0. Our results reveal a phase transition from charge density wave (CDW) states to uniform states around a critical doping level δc≈0.27δ_c \approx 0.27δc​≈0.27. Within the CDW phase, we observe different types of Wigner crystal (WC) formulated by doped holes that are energetically favored. As we enter the uniform phase, a non-Fermi liquid (NFL) state emerges within the doping range 0.27<δ<0.320.27 < δ< 0.320.27<δ<0.32, characterized by an exponential decay of all correlation functions. With further hole doping, we discover the appearance of a pair density wave (PDW) state within a narrow doping region 0.32<δ<1/30.32 < δ< 1/30.32<δ<1/3. We also discuss the potential experimental implications of our findings.
Two-dimensional spinless Falicov-Kimball model (FKM) with correlated hopping is studied perturbatively in the limit of large on-site Coulomb interaction U. In the half filled case (i.e., ${\ensuremath{\rho}}_{i}+{\ensuremath{\rho}}_{e}=1,$ where ${\ensuremath{\rho}}_{i},{\ensuremath{\rho}}_{e}$ are densities of ions and electrons, respectively) the effective Hamiltonian in spin variables is derived up to terms proportional to ${U}^{\ensuremath{-}3}.$ Unlike the simplest FKM case, it contains odd parity terms (resulting from the correlated hopping) in addition to even parity ones. The ground-state phase diagram of the effective Hamiltonian is examined in the (a/t,h)(a/t,h)(a/t,h) plane, where a/ta/ta/t is a parameter characterizing strength of the correlated hopping and h is a difference of chemical potentials of two sorts of particles present in the system. It appears to be asymmetric with respect to the change $\stackrel{\ensuremath{\rightarrow}}{h}\ensuremath{-}h$ and an additional ordered phase is found for a certain interval of a/t.a/t.a/t.
Abstract A recently developed numerical method, based on the modification of exact diagonalization calculations, is used to study the ground‐state properties of the spinless Falicov–Kimball model on the triangular lattice. The phase diagram as well as the valence and metal–insulator transitions are analysed and compared with results of the conventional Falicov–Kimball model on the square lattice. It is shown that the different cluster geometry leads to various changes in basic characteristics of the model. In particular, the triangular lattice destroys the symmetry of the phase diagram as well as the valence transitions, and stabilizes the new types of spatial charge ordering – the axial stripes and the phase separated distributions of localized f ‐electrons. Studying the conducting properties of the Falicov–Kimball model on the triangular lattice, the Mott‐Hubbard transition has been observed. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Strongly correlated systems on geometrically frustrated lattices can stabilize a large number of interesting phases that includes a wide array of novel Mott insulators in both bosonic and electronic systems. Charge fluctuations in a Mott insulator are suppressed due to strong mutual interaction among the particles. The presence of frustration is of particular importance as the physics it offers is often rich, unexpectedly complicated, and continues to raise many open questions. The thesis elucidates some of these issues on a kagome lattice where strong interactions among the particles in the Mott phase impose non-trivial local constraints depending on the filling fraction on the lattice. These Mott insulators, in addition to featuring unusual magnetic and/or charge ordering, can also harbor topologically ordered states of quantum matter, e.g., resonating valence bond liquids realized in certain quantum dimer models on non-bipartite lattices. The dimer models can be regarded as low-energy effective theories for different types of bosonic models in the strong-coupling limit. Exploring this connection is a central theme of this thesis with the aim of realizing novel strongly correlated ground states. Past studies of these models have revealed the existence of various ordered and disordered phases with distinct signatures. Among these low-energy phases, the presence of a stable topological liquid at a particular point, known as Rokhsar-Kivelson point, in the phase diagram is notable. The classical versions of the dimer model are also known to have garnered a vast interest in various fields ranging from problems of pure mathematical origin to ones in physical chemistry as well as statistical physics. Pioneered by Kasteleyn, several analytical works came forward to exactly calculate the partition function of the problem from which other physical observables can be derived. Classical numerical methods are extensively applied to these models to verify the analytical predictions. We introduce a new classical algorithm here to compute the correlation functions of a classical dimer model on a square (bipartite) and a triangular (non-bipartite) lattice based on a tensor network construction. The method, called tensor network renormalization group, turns out to be a powerful tool for simulating short-ranged gapped systems as inferred from our results benchmarked against the classical Monte-Carlo technique and compared with past analytical studies. One should note that the quantum dimer model at the Rokhsar-Kivelson point can also be described as an infinite temperature canonical ensemble of classical dimers because of the particular structure of the ground state which is an equal weight superposition in the configuration manifold. The geometry of the lattice plays a pivotal role in deciding the nature of the phases that arise in the dimer models. Many physical properties of the dimer liquid phase can be extracted in the simple classical setting which certainly allows for a deep understanding of the classical models to be developed. The liquid phase is gapped on non-bipartite lattices and gapless on bipartite lattices, which is reflected in the decay of correlation functions with spatial distances. In general on non-bipartite lattices, the topological nature of the dimer liquid is characterized by a Z2 topological order which survives even when the model is perturbed away from the Rokhsar-Kivelson point. Stability of this liquid phase not only depends on the lattice geometries but notably on dimer concentrations also. In this context, we focus on a particular variant of the dimer model on a triangular lattice which is known as the quantum fully packed loop model. The model is composed of nonintersecting closed loops made of dimers and governed by the same Hamiltonian as the quantum dimer model. The loop model provides an effective low-energy description of a strongly correlated bosonic system at 1/3 filling on the kagome lattice. The corresponding Bose-Hubbard Hamiltonian consists of nearest-neighbor hopping and all possible repulsive interactions within a hexagonal plaquette. Conspicuous features of the zero-temperature phase diagram for this model include (i) presence of a stable Z2 liquid even without any Rokhsar-Kivelson potential term (in distinction to the standard quantum dimer model), and (ii) an unconventional phase transition from the liquid phase to a novel crystalline phase that has nematic order (dubbed lattice nematic). For a deeper understanding of the physics, a mapping to an Ising gauge theory is presented. The gauge theoretic description provides a useful way to predict the nature of the quantum phase transition to lie in the O(3) universality class. Finally a fermionic model at the same 1/3 filling is considered in which the ground state exhibits a number of exotic local orderings resulting from the spin-charge interplay of electrons. The Hamiltonian comprises nearest-neighbor hopping, strong on-site Coulomb interaction, and repulsive interaction terms only between nearest-neighbors. In the strong correlation limit, this fermionic problem maps to a two-color fully packed loop model – a model in which the loop segments carry an additional quantum number as color on a honeycomb lattice. The effective theory is governed by coherent three-particle ring exchanges and nearest-neighbor antiferromagnetic spin exchanges. The competition between these two leads to a phase diagram composed of a novel plaquette ordered state (known as the plaquette phase) that undergoes phase transition to a new kind of charge ordered state which we call a short loop phase. From our numerical analysis, we conclude that the plaquette phase features an unusual antiferromagnetic order with gapless spin excitations while the charge-ordered state is subjugated by spin fluctuations of localized electrons arranged in small hexagonal loops on the kagome lattice.