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arXiv:2607.08862

Loop-current orderings in SU($N$) two-leg fermionic ladder through duality symmetries

M. Habchy, Félix Rose, P. Lecheminant

Published Jul 9, 2026• 0 citations
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Abstract

We investigate the formation of loop-current ordered phases in half-filled SU($N$) two-leg fermionic ladders. Using a low-energy approach, we uncover the existence of non-perturbative duality symmetries relating four competing orders. Two of these orders correspond to loop-current ordered phases that spontaneously break the time-reversal symmetry and describe charge currents circulating in a staggered pattern either around the plaquettes or along the diagonals of the ladder. These unconventional phases are shown to be dual to conventional (charge and bond) density-wave phases through an exact density-current duality symmetry existing on the lattice. From a perturbative renormalization group approach, we find that these phases for $N>2$ are stabilized in a half-filled SU($N$) two-leg Hubbard ladder with an additional SU($N$) Hund's interaction. The effect of a small doping on these phases is also discussed.

DOI
10.48550/arxiv.2607.08862

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Loop-current orderings in SU(NNN) two-leg fermionic ladder through duality symmetries
We investigate the formation of loop-current ordered phases in half-filled SU(NNN) two-leg fermionic ladders. Using a low-energy approach, we uncover the existence of non-perturbative duality symmetries relating four competing orders. Two of these orders correspond to loop-current ordered phases that spontaneously break the time-reversal symmetry and describe charge currents circulating in a staggered pattern either around the plaquettes or along the diagonals of the ladder. These unconventional phases are shown to be dual to conventional (charge and bond) density-wave phases through an exact density-current duality symmetry existing on the lattice. From a perturbative renormalization group approach, we find that these phases for N>2N>2N>2 are stabilized in a half-filled SU(NNN) two-leg Hubbard ladder with an additional SU(NNN) Hund's interaction. The effect of a small doping on these phases is also discussed.
We discuss duality relations in correlated electronic ladder systems to clarify mutual relations between various conventional and unconventional phases. For the generalized two-leg Hubbard ladder, we find two exact duality relations, and also one asymptotic relation which holds in the low-energy regime. These duality relations show that unconventional (exotic) density-wave orders such as staggered flux or circulating spin-current are directly mapped to conventional density-wave orders, which establishes the appearance of various exotic states with time-reversal and/or spin symmetry breaking. We also study duality relations in the SO(5) symmetry that was proposed to unify antiferromagnetism and d-wave superconductivity. We show that the same SO(5) symmetry also unifies circulating spin current order and s-wave superconductivity.
Using gauge transformations on electron bond operators, we derive exact duality relations between various order parameters for correlated electron systems. Applying these transformations, we find two duality relations in the generalized two-leg Hubbard ladder at arbitrary filling. The relations show that unconventional density-wave orders such as staggered flux or circulating spin current are dual to conventional density-wave orders and there are direct mappings between dual phases. Several exact results on the phase diagram are also concluded.
We provide, for the first time, in a doped strongly correlated system (two-leg ladder), a controlled theoretical demonstration of the existence of a state in which long-range ordered orbital currents are arranged in a staggered pattern, coexisting with a charge density wave. The method used is the highly accurate density-matrix renormalization group technique. This brings us closer to recent proposals that this order is realized in the enigmatic pseudogap phase of the cuprate high temperature superconductors.
The ground-state phase diagram of the half-filled two-leg Hubbard ladder with intersite Coulomb repulsions and exchange coupling is studied by using the strong-coupling perturbation theory and the weak-coupling bosonization method. Considered here as possible ground states of the ladder model are four types of density-wave states with different angular momentum (s(s(s-density-wave state, p-density-wave state, d-density-wave state, and f-density-wave state) and four types of quantum disordered states, i.e., Mott insulating states (S(S(S-Mott, D-Mott, ${S}^{\ensuremath{'}}$-Mott, and ${D}^{\ensuremath{'}}$-Mott states, where S and D stand for s- and d-wave symmetry). The s-density-wave state, the d-density-wave state, and the D-Mott state are also known as the charge-density-wave state, the staggered-flux state, and the rung-singlet state, respectively. Strong-coupling approach naturally leads to the Ising model in a transverse field as an effective theory for the quantum phase transitions between the staggered-flux state and the D-Mott state and between the charge-density-wave state and the S-Mott state, where the Ising ordered states correspond to doubly degenerate ground states in the staggered-flux or the charge-density-wave state. From the weak-coupling bosonization approach it is shown that there are three cases in the quantum phase transitions between a density-wave state and a Mott state: the Ising (Z2){(Z}_{2})(Z2​) criticality, the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ criticality, and a first-order transition. The quantum phase transitions between Mott states and between density-wave states are found to be the U(1) Gaussian criticality. The ground-state phase diagram is determined by integrating perturbative renormalization-group equations. It is shown that the S-Mott state and the staggered-flux state exist in the region sandwiched by the charge-density-wave phase and the D-Mott phase. The p-density-wave state, the ${S}^{\ensuremath{'}}$-Mott state, and the ${D}^{\ensuremath{'}}$-Mott state also appear in the phase diagram when the next-nearest-neighbor repulsion is included. The correspondence between Mott states in extended Hubbard ladders and spin-liquid states in spin ladders is also discussed.
We show using bosonization methods that extended Hubbard-Heisenberg models on two types of two leg ladders (without flux and with flux $\ensuremath{\pi}$ per plaquette) have commensurate pair-density-wave (PDW) phases. In the case of the conventional (flux-less) ladder, the PDW arises when certain filling fractions for which commensurability conditions are met. For the flux $\ensuremath{\pi}$ ladder, the PDW phase is generally present. The PDW phase is characterized by a finite spin gap and a superconducting order parameter with a finite (commensurate in this case) wave vector and power-law superconducting correlations. In this phase, the uniform superconducting order parameter, the 2kF2{k}_{F}2kF​ charge-density-wave (CDW) order parameter and the spin-density-wave N\'eel order parameter exhibit short range (exponentially decaying) correlations. We discuss in detail the case in which the bonding band of the ladder is half filled for which the PDW phase appears even at weak coupling. The PDW phase is shown to be dual to a uniform superconducting (SC) phase with quasi-long-range order. By making use of bosonization and the renormalization group, we determine the phase diagram of the spin-gapped regime and study the quantum phase transition. The phase boundary between PDW and the uniform SC ordered phases is found to be in the Ising universality class. We generalize the analysis to the case of other commensurate fillings of the bonding band, where we find higher-order commensurate PDW states for which we determine the form of the effective bosonized field theory and discuss the phase diagram. We compare our results with recent findings in the Kondo-Heisenberg chain. We show that the formation of PDW order in the ladder embodies the notion of intertwined orders.
We investigate the charge-order transition at zero temperature in a two-leg Hubbard ladder with additional nearest-neighbor Coulomb repulsion VVV using the density matrix renormalization group technique. We consider electron densities between quarter and half filling. For quarter filling and U=8t,U=8t,U=8t, we find evidence for a continuous phase transition between a homogeneous state at small VVV and a broken-symmetry state with ``checkerboard'' [wave vector $\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})$] charge order at large V.V.V. This transition to a checkerboard charge-ordered state remains present at all larger fillings, but becomes discontinuous at sufficiently large filling. We discuss the influence of U/tU/tU/t on the transition and estimate the position of the tricritical points.