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arXiv:2309.00037 · Physical review. B./Physical review. B

Electric field control of a quantum spin liquid in weak Mott insulators

Daniel J. Schultz, Alexandre Khoury, Félix Desrochers, Omid Tavakol, Emily Z. Zhang, Yong Baek Kim

Published Jun 17, 2024• 6 citations
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Abstract

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 $t$ to on-site Coulomb repulsion $U$ 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 $t/U$, 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 $t/U$ 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 $t/U$. 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 $t/U$. 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.

DOI
10.1103/physrevb.109.214423

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Motivated by the recent proposal of realizing an SU(4) Hubbard model on triangular moire superlattices, we present a DMRG study of an $SU(4)$ spin model obtained in the limit of large repulsion for integer filling $\nu_T=1,3$. We retain terms in the $t/U$ expansion up to $O(\frac{t^3}{U^2})$ order, that generates nearest-neighbor exchange $J$, as well as an additional three-site ring exchange term, $K$, which is absent in the SU(2) S=1/2 case. For filling $\nu_T=3$, when increasing the three-site ring exchange term $K \sim \frac{t^3}{U^2}$, we identify three different phases: a spin-symmetric crystal, an $SU(4)_1$ chiral spin liquid (CSL) and a decoupled one dimensional chain (DC) phase. The CSL phase exists at intermediate coupling: $U/t \in [11.3,\,22.9]$. The sign of $K$ is crucial to stabilizing the CSL and the DC phase. For filling $\nu_T=1$ with the opposite sign of $K$, the spin-symmetric crystal phase survives to very large $K$. We propose to search for the CSL phase in moire bilayers. For example, in twisted AB stacked transition metal dichalcogenide (TMD) bilayers, the $SU(4)$ spin is formed by layer pseudospin combined with the real spin (locked to valley). The layer pseudospin carries an electric dipole moment in $z$ direction, thus the CSL is really a dipole-spin liquid, with quantum fluctuations in both the electric moment and magnetic moment . The CSL phase spontaneously breaks the time reversal symmetry and shows a quantum anomalous Hall effect in spin transport and dipole transport. Smoking gun evidence for the CSL could be obtained through measurement of the quantized dipole Hall effect in counter-flow transport.

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Interplay between multispin and chiral spin interactions on a triangular lattice

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We investigate the spin-$\frac{1}{2}$ nearest-neighbor Heisenberg model with the four-site ring-exchange ${J}_{4}$ and chiral interaction ${J}_{\ensuremath{\chi}}$ on the triangular lattice by using the variational Monte Carlo method. The ${J}_{4}$ term induces the quadratic band touching (QBT) quantum spin liquid (QSL) with only a $d+id$ spinon pairing (without hopping term), the nodal $d$-wave QSL and $U(1)$ QSL with a finite spinon Fermi surface progressively. The effect of the chiral interaction ${J}_{\ensuremath{\chi}}$ can enrich the phase diagram with two interesting chiral QSLs (topological orders) with the same quantized Chern number $C=\frac{1}{2}$ and ground-state degeneracy GSD = 2, namely the $U(1)$ chiral spin liquid (CSL) and ${\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+id$-wave QSL. The nodal $d$-wave QSL is fragile and will turn to the ${\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+id$ QSL with any finite ${J}_{\ensuremath{\chi}}$ within our numerical calculation. However, in the process from QBT to the ${\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+id$ QSL with the increase of ${J}_{\ensuremath{\chi}}$, an exotic crossover region is found. In this region, the previous QBT state acquires a small hopping term so that it opens a small gap at the band touching points, and leads to an energy minimum which is energetically more favorable compared to another competitive local minimum from the ${\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+id$ QSL. We dub this state as the proximate QBT QSL and it gives way to the ${\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+id$ QSL eventually. Therefore, the cooperation of the ${J}_{4}$ and ${J}_{\ensuremath{\chi}}$ terms favors mostly the ${\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+id$-wave QSL, so that this phase occupies the largest region in the phase diagram.

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Hubbard Model on Triangular Lattice: Role of Charge Fluctuations

Ji-Si Xu, Zheng Zhu, Kai Wu +1 more

A chiral spin liquid (CSL) phase has been recently reported in the Hubbard model on a triangular lattice at half-filling. It emerges in an intermediate coupling regime, which is sandwiched between a $120^\circ$ antiferromagnetic (AFM) phase and a metallic phase as a function of on-site repulsion $U$. In this work, we examine how the charge fluctuation may cause the CSL and complex phase diagram. We first identify an exact sign structure of the model at arbitrary temperature, sample size, and doping, which is reduced from the original Fermi exchange signs of the electrons by finite $U$. In particular, the spin and charge degrees of freedom are generally entangled via the phase-string in the sign structure. By precisely switching off such a phase-string, the CSL phase is shown to disappear, with only the $120^\circ$ AFM order remaining down to weak $U$ by a density matrix renormalization group calculation. Here the charge fluctuation is effectively decoupled from the AFM background, which is a $120^\circ$ AFM order due to a geometric Berry phase in the sign structure of the triangular lattice. General implications for the Mott physics will also be discussed.

arXiv:2306.11096•Jun 19, 2023•1 citation
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 weak Mott insulating regime of the triangular lattice Hubbard model exhibits a rich magnetic phase diagram as a result of the ring exchange interaction in the spin Hamiltonian. These phases include the Kalmeyer-Laughlin type chiral spin liquid (CSL) and a valence bond solid (VBS). A natural question arises regarding the robustness of these phases in the presence of a weak spin-orbit coupling (SOC). In this study, we derive the effective spin model for the spin-orbit coupled triangular lattice Hubbard model in the weak Mott insulting regime, including all SOC-mediated spin-bilinear and ring-exchange interactions. We then construct a simplified spin model keeping only the most relevant SOC-mediated spin interactions. Using infinite density matrix renormalization group (iDMRG), we show that the CSL and VBS phases of the triangular lattice Hubbard model can be stabilized in the presence of a weak SOC. The stabilization results from a compensation between the Dzyaloshinskii-Moriya interaction and a SOC-mediated ring-exchange interaction. We also provide additional qualitative arguments to intuitively understand the compensation mechanism in the iDMRG quantum phase diagrams. This mechanism for stabilization can potentially be useful for the experimental realization of quantum spin liquids.
At strong repulsion, the triangular-lattice Hubbard model is described by s=1/2 spins with nearest-neighbor antiferromagnetic Heisenberg interactions and exhibits conventional 120° order. Using the infinite density matrix renormalization group and exact diagonalization, we study the effect of the additional four-spin interactions naturally generated from the underlying Mott-insulator physics of electrons as the repulsion decreases. Although these interactions have historically been connected with a gapless ground state with emergent spinon Fermi surface, we find that, at physically relevant parameters, they stabilize a chiral spin liquid (CSL) of Kalmeyer-Laughlin (KL) type, clarifying observations in recent studies of the Hubbard model. We then present a self-consistent solution based on a mean-field rewriting of the interaction to obtain a Hamiltonian with similarities to the parent Hamiltonian of the KL state, providing a physical understanding for the origin of the CSL.
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 $({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}})$ on bonds lying along the different directions. Remarkably, when the periodic driving does not have time-reversal symmetry, 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 $({J}_{\ensuremath{\alpha}},{J}_{\ensuremath{\beta}},{J}_{\ensuremath{\gamma}},C)$ 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.
An SU(4)SU(4)SU(4) chiral spin liquid and quantized dipole Hall effect in moir\'e bilayers
Motivated by the recent proposal of realizing an SU(4) Hubbard model on triangular moire superlattices, we present a DMRG study of an SU(4)SU(4)SU(4) spin model obtained in the limit of large repulsion for integer filling νT=1,3\nu_T=1,3νT​=1,3. We retain terms in the t/Ut/Ut/U expansion up to O(t3U2)O(\frac{t^3}{U^2})O(U2t3​) order, that generates nearest-neighbor exchange JJJ, as well as an additional three-site ring exchange term, KKK, which is absent in the SU(2) S=1/2 case. For filling νT=3\nu_T=3νT​=3, when increasing the three-site ring exchange term K∼t3U2K \sim \frac{t^3}{U^2}K∼U2t3​, we identify three different phases: a spin-symmetric crystal, an SU(4)1SU(4)_1SU(4)1​ chiral spin liquid (CSL) and a decoupled one dimensional chain (DC) phase. The CSL phase exists at intermediate coupling: U/t∈[11.3, 22.9]U/t \in [11.3,\,22.9]U/t∈[11.3,22.9]. The sign of KKK is crucial to stabilizing the CSL and the DC phase. For filling νT=1\nu_T=1νT​=1 with the opposite sign of KKK, the spin-symmetric crystal phase survives to very large KKK. We propose to search for the CSL phase in moire bilayers. For example, in twisted AB stacked transition metal dichalcogenide (TMD) bilayers, the SU(4)SU(4)SU(4) spin is formed by layer pseudospin combined with the real spin (locked to valley). The layer pseudospin carries an electric dipole moment in zzz direction, thus the CSL is really a dipole-spin liquid, with quantum fluctuations in both the electric moment and magnetic moment . The CSL phase spontaneously breaks the time reversal symmetry and shows a quantum anomalous Hall effect in spin transport and dipole transport. Smoking gun evidence for the CSL could be obtained through measurement of the quantized dipole Hall effect in counter-flow transport.
We investigate the spin-12\frac{1}{2}21​ nearest-neighbor Heisenberg model with the four-site ring-exchange J4{J}_{4}J4​ and chiral interaction ${J}_{\ensuremath{\chi}}$ on the triangular lattice by using the variational Monte Carlo method. The J4{J}_{4}J4​ term induces the quadratic band touching (QBT) quantum spin liquid (QSL) with only a d+idd+idd+id spinon pairing (without hopping term), the nodal ddd-wave QSL and U(1)U(1)U(1) QSL with a finite spinon Fermi surface progressively. The effect of the chiral interaction ${J}_{\ensuremath{\chi}}$ can enrich the phase diagram with two interesting chiral QSLs (topological orders) with the same quantized Chern number C=12C=\frac{1}{2}C=21​ and ground-state degeneracy GSD = 2, namely the U(1)U(1)U(1) chiral spin liquid (CSL) and Z2d+id{\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+idZ2​d+id-wave QSL. The nodal ddd-wave QSL is fragile and will turn to the Z2d+id{\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+idZ2​d+id QSL with any finite ${J}_{\ensuremath{\chi}}$ within our numerical calculation. However, in the process from QBT to the Z2d+id{\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+idZ2​d+id QSL with the increase of ${J}_{\ensuremath{\chi}}$, an exotic crossover region is found. In this region, the previous QBT state acquires a small hopping term so that it opens a small gap at the band touching points, and leads to an energy minimum which is energetically more favorable compared to another competitive local minimum from the Z2d+id{\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+idZ2​d+id QSL. We dub this state as the proximate QBT QSL and it gives way to the Z2d+id{\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+idZ2​d+id QSL eventually. Therefore, the cooperation of the J4{J}_{4}J4​ and ${J}_{\ensuremath{\chi}}$ terms favors mostly the Z2d+id{\mathrm{Z}}_{2}\phantom{\rule{4pt}{0ex}}d+idZ2​d+id-wave QSL, so that this phase occupies the largest region in the phase diagram.
A chiral spin liquid (CSL) phase has been recently reported in the Hubbard model on a triangular lattice at half-filling. It emerges in an intermediate coupling regime, which is sandwiched between a 120∘120^\circ120∘ antiferromagnetic (AFM) phase and a metallic phase as a function of on-site repulsion UUU. In this work, we examine how the charge fluctuation may cause the CSL and complex phase diagram. We first identify an exact sign structure of the model at arbitrary temperature, sample size, and doping, which is reduced from the original Fermi exchange signs of the electrons by finite UUU. In particular, the spin and charge degrees of freedom are generally entangled via the phase-string in the sign structure. By precisely switching off such a phase-string, the CSL phase is shown to disappear, with only the 120∘120^\circ120∘ AFM order remaining down to weak UUU by a density matrix renormalization group calculation. Here the charge fluctuation is effectively decoupled from the AFM background, which is a 120∘120^\circ120∘ AFM order due to a geometric Berry phase in the sign structure of the triangular lattice. General implications for the Mott physics will also be discussed.