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arXiv:2112.11003 · Physical review. B./Physical review. B

Driven Hubbard model on a triangular lattice: Tunable Heisenberg antiferromagnet with a chiral three-spin term

Samudra Sur, Adithi Udupa, Diptiman Sen

Published Feb 22, 2022• 11 citations
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Abstract

We study the effects of a periodically varying electric field on the Hubbard model at half filling on a triangular lattice. The electric field is incorporated through the phase of the nearest-neighbor hopping amplitude via the Peierls prescription. When the on-site interaction $U$ is much larger than the hopping, the effective Hamiltonian ${H}_{\text{eff}}$ describing the spin sector can be found using a Floquet perturbation theory. To third order in the hopping, ${H}_{\text{eff}}$ is found to have the form of a Heisenberg antiferromagnet with three different nearest-neighbor couplings $({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}})$ on bonds lying along the different directions. Remarkably, when the periodic driving does not have time-reversal symmetry, ${H}_{\text{eff}}$ can also have a chiral three-spin interaction in each triangle, with the coefficient $C$ of the interaction having opposite signs on up- and down-pointing triangles. Thus periodic driving which breaks time-reversal symmetry can simulate the effect of a perpendicular magnetic flux which is known to generate such a chiral term in the spin sector, even though our model does not have a magnetic flux. The four parameters $({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}},C)$ depend on the amplitude, frequency, and direction of the oscillating electric field. We then study the spin model as a function of these parameters using exact diagonalization and find a rich phase diagram of the ground state with seven different phases consisting of two kinds of ordered phases (collinear and coplanar) and disordered phases. Thus periodic driving of the Hubbard model on the triangular lattice can lead to an effective spin model whose couplings can be tuned over a range of values thereby producing a variety of interesting phases.

DOI
10.1103/physrevb.105.054423

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We study the effects of a periodically varying electric field on the Hubbard model at half filling on a triangular lattice. The electric field is incorporated through the phase of the nearest-neighbor hopping amplitude via the Peierls prescription. When the on-site interaction UUU is much larger than the hopping, the effective Hamiltonian Heff{H}_{\text{eff}}Heff​ describing the spin sector can be found using a Floquet perturbation theory. To third order in the hopping, Heff{H}_{\text{eff}}Heff​ is found to have the form of a Heisenberg antiferromagnet with three different nearest-neighbor couplings (J\ensuremathα,J\ensuremathβ,J\ensuremathγ)({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}})(J\ensuremathα​,J\ensuremathβ​,J\ensuremathγ​) on bonds lying along the different directions. Remarkably, when the periodic driving does not have time-reversal symmetry, Heff{H}_{\text{eff}}Heff​ can also have a chiral three-spin interaction in each triangle, with the coefficient CCC of the interaction having opposite signs on up- and down-pointing triangles. Thus periodic driving which breaks time-reversal symmetry can simulate the effect of a perpendicular magnetic flux which is known to generate such a chiral term in the spin sector, even though our model does not have a magnetic flux. The four parameters (J\ensuremathα,J\ensuremathβ,J\ensuremathγ,C)({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}},C)(J\ensuremathα​,J\ensuremathβ​,J\ensuremathγ​,C) depend on the amplitude, frequency, and direction of the oscillating electric field. We then study the spin model as a function of these parameters using exact diagonalization and find a rich phase diagram of the ground state with seven different phases consisting of two kinds of ordered phases (collinear and coplanar) and disordered phases. Thus periodic driving of the Hubbard model on the triangular lattice can lead to an effective spin model whose couplings can be tuned over a range of values thereby producing a variety of interesting phases.
Abstract We consider a half-filled system of spin-1/2 fermions on a triangular ladder with spin-dependent hopping in the presence of spin-dependent flux. Using the Schrieffer–Wolff transformation, we derive an effective spin Hamiltonian describing the infrared properties of the system in the limit of strong on-site repulsion. The expansion is performed up to the third order in the hopping amplitude over on-site interaction energy. It is shown that the low-energy spin excitations are described by an anisotropic XXZ Heisenberg model with Dzyaloshinskii–Moriya interaction and an unconventional three-spin (correlated exchange) term in the presence of an extended magnetic field. The latter couples not only to z -projection of the total spin on a triangular plaquette but also to the product of z -components of three spins located on vertices of the triangle. In the case of strong attractive interaction, the corresponding effective model is derived from the repulsive one using the one-spin-component particle-hole transformation. The effective Hamiltonian is given by the same XXZ Heisenberg model, as in the repulsive case, but in terms of pseudospin (charge) operators.
The triangular lattice Hubbard model at strong coupling, whose effective spin model contains both Heisenberg and ring exchange interactions, exhibits a rich phase diagram as the ratio of the hopping ttt to on-site Coulomb repulsion UUU is tuned. This includes a chiral spin liquid (CSL) phase. Nevertheless, this exotic phase remains challenging to realize experimentally because a given material has a fixed value of t/Ut/Ut/U, which is difficult to tune with external stimuli. One approach to address this problem is applying a dc electric field, which renormalizes the exchange interactions as electrons undergo virtual hopping processes; in addition to creating virtual doubly occupied sites, electrons must overcome electric potential energy differences. Performing a small t/Ut/Ut/U expansion to fourth order, we derive the ring exchange model in the presence of an electric field and find that it not only introduces spatial anisotropy but also tends to enhance the ring exchange term compared to the dominant nearest-neighbor Heisenberg interaction. Thus, increasing the electric field serves as a way to increase the importance of the ring exchange at constant t/Ut/Ut/U. Through density matrix renormalization group calculations, we compute the ground-state phase diagram of the ring exchange model for two different electric field directions. In both cases, we find that the electric field shifts the phase boundary of the CSL towards a smaller ratio of t/Ut/Ut/U. Therefore, the electric field can drive a magnetically ordered state into the CSL. This explicit demonstration opens the door to tuning other quantum spin systems into spin liquid phases via the application of an electric field.
The time-dependent exact-diagonalization method is used to study the light-induced phase transition of magnetic orders in the anisotropic triangular-lattice Hubbard model. Calculating the spin correlation function, we confirm that the phase transition from the ${120}^{\ensuremath{\circ}}$ order to the N\'eel order can take place due to high-frequency periodic fields. We show that the effective Heisenberg-model Hamiltonian derived from the high-frequency expansion by the Floquet theory describes the present system very well and that the ratio of the exchange interactions expressed in terms of the frequency and amplitude of the external field determines the type of the magnetic orders. Our results demonstrate the controllability of the magnetic orders by tuning the external field.
πππ-Flux Tuning of Quantum Paramagnetism in the Triangular-Lattice Hubbard Model
We compare two time-reversal-symmetric triangular-lattice Hubbard models at half filling: the zero-flux model and a staggered πππ-flux model with zero and πππ flux through oppositely oriented triangles. Although both reduce to the same Heisenberg model at leading order in the strong-coupling limit, their distinct charge fluctuation profiles yield different quantum paramagnetic regimes. Combining a strong-coupling expansion with infinite density matrix renormalization group (iDMRG) calculations, we find that the positive ring-exchange of the zero-flux model promotes spontaneous chirality, consistent with its notion of a chiral spin liquid at intermediate coupling. In the staggered-πππ-flux model, whose noninteracting limit is a Dirac semimetal, the ring exchange changes sign and disfavors chiral order and we find no resolvable scalar chirality. This renders flux-tuned ring-exchange an efficient discriminator between chiral and non-chiral quantum paramagnets at finite coupling.
We use a combination of numerical density matrix renormalization group calculations and several analytical approaches to comprehensively study a simplified model for a spatially anisotropic spin-12\frac{1}{2}21​ triangular lattice Heisenberg antiferromagnet: the three-leg triangular spin tube (TST). The model is described by three Heisenberg chains, with exchange constant JJJ, coupled antiferromagnetically with exchange constant ${J}^{\ensuremath{'}}$ along the diagonals of the ladder system, with periodic boundary conditions in the shorter direction. Here, we determine the full phase diagram of this model as a function of both spatial anisotropy (between the isotropic and decoupled chain limits) and magnetic field. We find a rich phase diagram, which is remarkably dominated by quantum states: the phase corresponding to the classical ground state appears only in an exceedingly small region. Among the dominant phases generated by quantum effects are commensurate and incommensurate coplanar quasiordered states, which appear in the vicinity of the isotropic region for most fields, and in the high-field region for most anisotropies. The coplanar states, while not classical ground states, can at least be understood semiclassically. Even more strikingly, the largest region of phase space is occupied by a spin density wave phase, which has incommensurate collinear correlations along the field. This phase has no semiclassical analog, and may be ascribed to enhanced one-dimensional fluctuations due to frustration. Cutting across the phase diagram is a magnetization plateau, with a gap to all excitations and ``up-up-down'' spin order, with a quantized magnetization equal to 13\frac{1}{3}31​ of the saturation value. In the TST, this plateau extends almost but not quite to the decoupled chains limit. Most of the above features are expected to carry over to the two-dimensional system, which we also discuss. At low field, a dimerized phase appears, which is particular to the one-dimensional nature of the TST, and which can be understood from quantum Berry phase arguments.
The antiferromagnetic phase of two-dimensional (2D) and three-dimensional (3D) Hubbard model with nearest neighbor hopping is studied on a bipartite cubic lattice by means of the quantum $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ rotor approach that yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of the model and satisfy the Mermin-Wagner theorem. The collective variables for charge and spin are isolated in the form of the space-time fluctuating U(1) phase field and rotating spin-quantization axis governed by the SU(2) symmetry, respectively. As a result interacting electrons appear as composite objects consisting of bare fermions with attached U(1) and SU(2) gauge fields. An effective action consisting of a spin-charge rotor and a fermionic field is derived as a function of the Coulomb repulsion UUU and hopping parameter ttt. At zero temperature, our theory describes the evolution from a Slater (U⪡t)(U⪡t)(U⪡t) to a Mott-Heisenberg (U⪢t)(U⪢t)(U⪢t) antiferromagnet. The results for zero-temperature sublatice magnetization (2D) and finite temperature (3D) phase diagram of the antiferromagnetic Hubbard model as a function of the crossover parameter U∕tU∕tU∕t are presented and the role of the spin Berry phase in the interaction driven crossover is analyzed.