Quantum Monte-Carlo (QMC) simulations are an important non-pertubative numeric method for studying physics of strongly correlated many-body systems, which yield unbiased results at high numeric precision in the absence of the sign-problem.

We have made significant progress in applying the sign-problem free QMC simulations to explore various fundamental problems of strong correlation problems, including interacting topological insulators, doped Mott insulators, itinerant ferromagnetism and Curie-Weiss metals, novel quantum magnetism with SU(N) symmetries.

2D topological insulators

We performed one of the two earliest QMC simulations for topological insulators and the helical edge states. It was based on the Kane-Mele Hubbard model and we proved that it is sign-problem free. See the helical edge liquid and [Ref 1.]

Curie-Weiss metal and magnetic criticality

We performed, to our knowledge, the first sign-problem free QMC simulations on itinerant ferromagnetism and the Curie-Weiss metal state. Magnetic criticality is explored. See Cuire-Weiss metal and magnetic criticality and [Ref 2.]

Doping an Antiferromagnetic Mott Insulator

The emergence of superconductivity in the vicinity of antiferromagnetism is a universal phenomenon frequently observed in strongly correlated materials. We study the transition from an antiferromagnetic insulator to a superconductor driven by hole-doping via the projector QMC simulations [Ref. 3] , which are sign-problem-free both at and away from the half-filling. It is based on a bilayer lattice model with the onsite Hubbard interaction and intra-rung charge and spin exchange interactions. An Ising anisotropic antiferromagnetic Mott insulating phase occurs at half-filling, which is weakened by hole doping. Below a critical doping value xc=0.11, the antiferromagnetism coexists with the singlet superconductivity, which is an intra-rung pairing with an extended s-wave symmetry. As further increasing doping, the antiferromagnetic order vanishes, leaving only a pure superconducting phase. These results provide important information how superconductivity appears upon doping the parent Mott insulating state.

Inter-layer current phase

The interlayer staggered current phase Strongly interacting systems have been conjectured to spontaneously develop current carrying ground states under certain conditions. However, most results in 2D are based on mean field approximations whose validity is hard to justify. In contrast, we (with S. Capponi and S. C. Zhang) performed the QMC simulation to an extended bilayer Hubbard model without the sign problem, finding a commensurate staggered interlayer current phase [Ref. 4] . To our knowledge, it is the first work to conclusively demonstrate the existence of such a phase in 2D.

Novel quantum magnetism of SU(2N) fermions -- Slater v.s. Mott, Dirac fermions, new type of quantum phase transition

There exist two basic pictures for interacting insulators -- the weak coupling Slater mechanism based on Fermi surface nesting, and the strong coupling Mott one based on local moments and super-exchange. In the SU(2) case, the evolution between these two limits is smooth without a quantum phase transition.

How about in the SU(2N) case? We have found fundamentally different features by the sign-problem free QMC based on SU(2N) Hubbard models on the square lattice [Ref. 5] . The AFM order appears in weak and intermediate coupling regions agreeing with the Slater insulator picture. However, as further increasing interaction, it becomes suppressed. In the SU(6) case, the AFM is completely suppressed in the strong interacting region, i.e., the Mott insulator, where the quantum paramagnetic columnar dimer state appears. This shows a clear quantum phase transition between the Slater and Mott insulating regions.

We investigated the Mott-insulating state of the SU(2N) Dirac fermions via the sign-problem free QMC simulations [Ref. 6 ] . It is based on the SU(2N) Hubbard model on a honeycomb lattice. Unlike the SU(2) case, the SU(4) and SU(6) Mott-insulating phases are identified with the columnar valence-bond-solid (cVBS) order in the absence of the Neel ordering. Although the Dirac semimetal-to-cVBS transitions are typically first order due to a cubic invariance possessed by the cVBS order, the coupling to gapless Dirac fermions can soften these transitions to the second order at zero temperature.

Detect edge degeneracy in interacting topological insulators using entanglement entropy

The existance of degenerate or gapless edge states is a characteristic feature of topological insulators, but is difficult to detect in the presence of interactions. We propose a new method to obtain the degeneracy of the edge states from the perspective of entanglement entropy [Ref. 7] , which is very useful to identify interacting topological states. Employing the determinant quantum Monte Carlo technique, we investigate the interaction effect on two representative models of fermionic topological insulators in one and two dimensions, respectively. In the two topologically nontrivial phases, the edge degeneracies are reduced by interactions but remain to be nontrivial.

References and Talks

1. Dong Zheng, Guang-Ming Zhang Congjun Wu, "Particle-hole symmetry and interaction effects in the Kane-Mele-Hubbard model", Phys. Rev. B 84 , 205121 (2011) , see, pdf file .

2. Shenglong Xu, Yi Li, Congjun Wu, "Sign-Problem-Free Quantum Monte Carlo Study on Thermodynamic Properties and Magnetic Phase Transitions in Orbital-Active Itinerant Ferromagnets", Phys. Rev. X 5, 021032, (2015) .

3. Tianxing Ma, Da Wang, Congjun Wu, "Doping-driven Antiferromagnetic Insulator - Superconductor Transition: a Quantum Monte Carlo Study", arXiv:1806.03652 .

4. Sylvain Capponi, Congjun Wu, and Shou-Cheng Zhang, "Current carrying ground state in a bi-layer model", Phys. Rev. B 70, 220505(R) (2004), see pdf file .

5. Da Wang, Yi Li, Zi Cai, Zhichao Zhou, Yu Wang, Congjun Wu, Competing orders in the 2D half-filled SU(2N) Hubbard model through the pinning field quantum Monte-Carlo simulations
Phys. Rev. Lett. 112, 156403 (2014), See pdf file

6. Zhichao Zhou, Da Wang, Zi Yang Meng, Yu Wang, Congjun Wu , Mott insulating states and quantum phase transitions of correlated SU(2N) Dirac fermions
Phys. Rev. B 93, 245157 (2016) . See pdf file

7. Da Wang, Shenglong Xu, Yu Wang, Congjun Wu, Detect edge degeneracy in interacting topological insulators using entanglement entropy
Phys. Rev. B 91, 115118 (2015) . See pdf file

Talk Novel magnetism with large spin fermion .

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Last modified: July 15, 2007.