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arXiv:2311.01821 · Physical review. B./Physical review. B

Phase transitions in the Haldane-Hubbard model

Wan-Xiu He, Rubem Mondaini, Hong‐Gang Luo, Xiaoqun Wang, Shijie Hu

Published Jan 12, 2024• 16 citations
arXivHTMLPDF

Abstract

The Haldane-Hubbard model is a prime example of the combined effects of band topology and electronic interaction. We revisit its spinful phase diagram at half filling as a consensus on the presence of SU(2) symmetry is currently lacking. To start, we utilize the Hartree-Fock mean-field method, which offers a direct understanding of symmetry breaking through the effective mass term that can acquire spin dependence. Our results, in agreement with previous studies, provide an instructive insight into the regime where the Chern number $C=1$, with only one spin species remaining topological. Besides that, we numerically study the phase diagram of the Haldane-Hubbard model via a large-scale infinite-density matrix renormalization group (iDMRG) method. The phase boundaries are determined by the Chern number and the correlation lengths obtained from the transfer-matrix spectrum. Unlike previous studies, the iDMRG method investigates the Haldane-Hubbard model on a thin and infinitely long cylinder and examines scenarios consistent with the two-dimensional thermodynamic limit. Here, the phase diagram we obtained qualitatively goes beyond the Hartree-Fock scope, particularly in the $C=1$ region, and serves as a quantitative benchmark for further theoretical and experimental investigations.

DOI
10.1103/physrevb.109.035126

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The Haldane-Hubbard model is a prime example of the combined effects of band topology and electronic interaction. We revisit its spinful phase diagram at half filling as a consensus on the presence of SU(2) symmetry is currently lacking. To start, we utilize the Hartree-Fock mean-field method, which offers a direct understanding of symmetry breaking through the effective mass term that can acquire spin dependence. Our results, in agreement with previous studies, provide an instructive insight into the regime where the Chern number C=1C=1C=1, with only one spin species remaining topological. Besides that, we numerically study the phase diagram of the Haldane-Hubbard model via a large-scale infinite-density matrix renormalization group (iDMRG) method. The phase boundaries are determined by the Chern number and the correlation lengths obtained from the transfer-matrix spectrum. Unlike previous studies, the iDMRG method investigates the Haldane-Hubbard model on a thin and infinitely long cylinder and examines scenarios consistent with the two-dimensional thermodynamic limit. Here, the phase diagram we obtained qualitatively goes beyond the Hartree-Fock scope, particularly in the C=1C=1C=1 region, and serves as a quantitative benchmark for further theoretical and experimental investigations.
We study the ground-state phase diagram of the half-filled extended Haldane-Hubbard model on the honeycomb lattice with sublattice-dependent on-site repulsion (UA/B{U}_{\text{A/B}}UA/B​) using the exact diagonalization (ED) and mean-field (MF) methods. The resulting phase diagram shows that there is a topologically nontrivial phase with the Chern number C=1C=1C=1, emerging via the development of the imbalance between UA{U}_{\text{A}}UA​ and UB{U}_{\text{B}}UB​. In this phase, the antiferromagnetic correlations are observed in the ED calculation, in line with the finite antiferromagnetic order obtained by the MF method. The spontaneous symmetry breaking of SU(2) spin rotation in the phase is also identified in the MF level. Distinct from previous studies in which the exotic C=1C=1C=1 phase relies on the interplay between sublattice-dependent potentials and electronic interactions, our paper presents an alternative way by solely tuning the on-site interactions.
We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.
We investigate the ground-state phase diagram of the spinful extended Haldane-Hubbard model on the honeycomb lattice using an exact-diagonalization, mean-field variational approach, and further complement it with the infinite density matrix renormalization group, applied to an infinite honeycomb cylinder. This model, governed by both on-site and nearest-neighbor interactions, can result in two types of insulators with finite local order parameters, either with spin or charge ordering. Moreover, a third one, a topologically nontrivial insulator with nonlocal order, is also manifest. We test expectations of previous analyses in spinless versions asserting that once a local order parameter is formed, the topological characteristics of the ground state, associated with a finite Chern number, are no longer present, resulting in a topologically trivial wave function. Our study confirms this overall picture, and highlights how finite-size effects may result in misleading conclusions on the coexistence of finite local order parameters and nontrivial topology in this model.
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Motivated by experimental studies that have found signatures of a quantum spin liquid phase in organic crystals whose structure is well described by the two-dimensional triangular lattice, we study the Hubbard model on this lattice at half filling using the infinite-system density matrix renormalization group (iDMRG) method. On infinite cylinders with finite circumference, we identify an intermediate phase between observed metallic behavior at low interaction strength and Mott insulating spin-ordered behavior at strong interactions. Chiral ordering from spontaneous breaking of time-reversal symmetry, a fractionally quantized spin Hall response, and characteristic level statistics in the entanglement spectrum in the intermediate phase provide strong evidence for the existence of a chiral spin liquid in the full two-dimensional limit of the model.
We study the thermodynamics of the one-dimensional extended Hubbard model at half filling using a density-matrix renormalization group method applied to transfer matrices. We show that the various phase transitions in this system can be detected by measuring usual thermodynamic quantities such as the isothermal compressibility and the uniform magnetic susceptibility. For the isothermal compressibility, we show that universal crossing points exist, which allow us to accurately determine the line where the charge gap vanishes. By studying, in addition, several correlation functions, we confirm the existence of a phase with long-range dimer order (bond order), which has been a matter of debate for several years. According to our calculations, this phase is located in a narrow region between the spin-density and charge-density wave phases up to a tricritical point, which we estimate to be at ${U}_{t}=6.7\ifmmode\pm\else\textpm\fi{}0.2$, ${V}_{t}=3.5\ifmmode\pm\else\textpm\fi{}0.1$. Our results for the phase diagram are in good agreement with the most recent zero-temperature density-matrix renormalization group study; however, they disagree in some important aspects from the most recent Quantum-Monte-Carlo study.