Chiral Spin Liquid and Quantum Phase Transition in the Triangular Lattice Hofstadter-Hubbard Model

Abstract

Recent advances in moiré engineering motivate the study of lattice models of strongly-correlated electrons subjected to substantial orbital magnetic flux. We analyze the triangular lattice Hofstadter-Hubbard model at one-quarter flux quantum per plaquette and a density of one electron per site, where a chiral spin liquid phase may exist between weak-coupling integer quantum Hall and strong-coupling 120$^\circ$ antiferromagnetic phases. We use matrix product state methods and analytical arguments to investigate this model compactified to cylinders of finite circumference. We uncover a glide particle-hole symmetry operation which, we argue, is spontaneously broken at the quantum Hall to spin liquid transition on odd-circumference cylinders. We numerically verify the spontaneous symmetry breaking and further demonstrate that this transition is associated with algebraic long-range correlations of various spin-singlet, charge-neutral operators. For even-circumference cylinders, the transition becomes a crossover associated with a large correlation length that grows substantially with circumference. Our findings suggest that in the two-dimensional limit, the transition to a chiral spin liquid phase is continuous and features critical fluctuations of the current.

Figures (13)

• Figure 1: (a) Upper panel: The triangular Hofstadter-Hubbard model, with magnetic translations $T_{x,y}$, flux per triangle $\Phi_\triangle$, and Hubbard interaction $U$ indicated. Lower left panel: "butterfly" representation of non-interacting spectrum in the plane of $\Phi_\triangle$ and energy $E$. The bands at $\Phi_\triangle = \pi/2$ enjoy a doubled "magnetic" unit cell and particle-hole symmetry $\mathcal{P}$ relating opposite $C=\pm 1$ bands [lower right panel]. (b) Finite segment of the circumference-$L_y$ cylinder threaded by $\Phi_\text{ext}$ external flux, where $T_y$ translates around the cylinder and $x\in\mathbb{Z}$ indexes rings. (c) 2+1D phase diagram at a density of $n=1$ electron per site as a function of $U/t$ showing the integer quantum Hall (IQH) and chiral spin liquid (CSL) phases, with their transition marked by a star.
• Figure 2: CSL symmetry breaking at odd circumference. (a) The particle-hole operation $\mathcal{P}$ transforms the internal gauge flux through an odd-$L_y$ cylinder ring from $\phi$ to $\pi - \phi$. (b) Upper panel: In an odd-$L_y$ CSL ground state, the alternating pattern $(\pi/2,-\pi/2,\dots)$ of these internal fluxes at strong coupling spontaneously breaks $\mathcal{P} T_x$ symmetry, whose action is then equivalent to threading $\pi$ internal flux. Lower panel: the low-lying entanglement eigenvalues on the $L_y=5$ cylinder at spatial cuts between adjacent pairs of rings, which not only exhibit the approximate degeneracy expected for the CSL at each fixed $S_z$ [circled numerals], but also signal spontaneous breaking of $\mathcal{P} T_x$. (c) The internal flux configuration [upper panel] and low-lying entanglement eigenvalues [lower panel] of the IQH at $L_y=5$. The entanglement spectra at the two spatial cuts are equivalent.
• Figure 3: Evidence for the IQH-CSL continuous phase transition. (a) Upper panel: Sketch of two cylinder rings, illustrating the $\mathbb{Z}_2$ order parameter $\mathcal{O}_0$ defined as the total circumferential current operator [orange arrows]. Lower panel: $\langle \mathcal{O}_0\rangle$ versus $U/t$ for the odd-circumference cylinders, with $\langle \mathcal{O}_0\rangle\neq0$ implying $\mathcal{P} T_x$ is spontaneously broken. (b-d) MPS transfer matrix correlation length in the $(Q,S_z)=0$ charge sector for various MPS bond dimensions [increasing light to dark] and extrapolated to infinite bond dimension for $L_y=4$ [gray points]; the insets specify the external magnetic flux through the cylinder rings.
• Figure 4: Connected correlation functions on the odd $L_y$ cylinders at their critical point, $U_c(L_y=3)=11.1$ (yellow) and $U_c(L_y=5)=11.6$ (blue). The $L_y=5$ correlations are reduced by $\times 10$ for ease of viewing. Upper: Current-current correlations and negative of density-density correlations, approaching quasi-long range order as a function of MPS bond dimension $\chi$ (increasing light to dark). Lower: Spin-spin (plotted with alternating sign $(-1)^x$) and electron-hole correlations decay exponentially, demonstrating a spin gap.
• Figure S1: (a) Graphical depiction of the complex phases of the nearest-neighbor hopping amplitudes of Eq. \ref{['eq:Peierls_hoppings']} for the $(p,q)=(1,2)$ two-band Hamiltonian in a portion of the infinite 2D plane, where the orientation of the plane is consistent with the YC coordinate system specified by Eqs. (\ref{['eq:underlyingYC_bravais_vecs']},\ref{['eq:magnetic_Bravais']}). The color of each bond denotes the complex phase of $t_{\bm{r}'\leftarrow \bm{r}}$ (see Eq. \ref{['eq:hopping_ham_arrows']}) in the direction of the arrow (pictured at the bond centers). The two orange dots indicate the two sites in the magnetic unit cell. Note that the top/bottom and right/left edges are not implicitly identified. (c) Graphical depiction of the fluxes through adjacent loops of a YC-$L_y$ cylinder, defined in Sec. \ref{['sec:YCgeo']}. The outside surface of each cylindrical segment has a total area of $2L_y$ triangles, each having $\pi/2$ flux [pictured]. When $L_y$ is odd [upper scenario], this leads to a $\pi$-staggering of the flux through each cylinder ring due to Gauss' law. When $L_y$ is even [lower scenario], the flux through each ring is the same.