Theory of phase transitions and critical phenomena plays an important role in condensed matter theory and statistical physics. The theoretical study has been carried out in two different but interactive paths. One path is constructed of various rigorous results, which include Onsager's exactly solution of the Ising model, Coulomb gas theory, renormalization theory, conformal field theory, Schramm-Loewner evolution etc. The other path makes use of the increasing power of computers, leading to the development of many effective computational methods, of which Markov chain Monte Carlo (MCMC) method is a nice example. Designing novel MCMC algorithms, especially cluster-type methods, and their applications have been hot topics for decades and will continue to show their effectiveness.
Quantum simulation is a quickly developing field in recent years. Computational results contribute to the understanding of behaviors of many-body quantum systems, as well as support experimental explorations.
We've developed Markov chain Monte Carlo algorithms that can simulate both classical and quantum models, such as percolation, Ising, Potts, O(n) loop, and quantum Ising model. Certain algorithms can simulate curved space, or effectively deal with canonical ensembles. [Phys. Rev. Lett.: 88 190602 (2002), 98 120601 (2007), 99 110601 (2007); Phys. Rev. E: 66 066110 (2002), 69 026114 (2004), 69 066129 (2004), 71 016709 (2005), 72 016126 (2005).]
We've observed the dynamical critical behaviors of some famous algorithms, e.g. Sweeny algorithm, Swendsen-Wang algorithm. Interesting new phenomena have been found, such as multi-time scales and the critical Speeding-up. [Phys. Rev. Lett.: 98 230602 (2007), 99 055701 (2007).]
Generally speaking, transfer matrix technique gives much better results than Monte Carlo method for two dimensional classical and 1+1 dimensional quantum systems, but it can hardly deal with large systems due to limitation of computational resources. We've employed transfer matrix to observe phase transitions of Potts model in two dimensions. [Phys. Rev. E 72, 056132]
Diagrammatic Monte Carlo method is a recently developed method for simulation of quantum systems. In this method, the Feynman diagrams are weighted elaborately, which may make it a candidate for alleviating the sign problem considering some Fermi systems. Employing the diagrammatic methods, we are currently working hard towards the Fermi-Hubbard model, worm-type simulation of J-Q model, multi-component Bose systems etc.
In experiments, many systems undergoing phase transitions are subject to external constraints such as the conservation of particle numbers in a mixture. Such systems are described in terms of canonical ensemble, and thus typically display a behavior different from that of unconstrained models, which are described by the grand canonical ensemble. The underline mechanism of such systems can be attributed to the so called Fisher renormalization. We've developed efficient cluster algorithms and studied the constrained systems such as the Blume-Capel model, Baxter's hard-square model, and the integer Potts model with vacancies. We have also studied the finite-size behavior of several systems near critical or tricritical point. [Phys. Rev. E: 70 046111 (2004), 71 036115 (2005), arXiv:cond-mat/0508348]
Conformal invariance relates closely to the geometrical aspects of phase transitions and critical phenomena. In two-dimensional classical systems and quantum systems in 1+1 dimensions, conformal invariance is very powerful and has lead to a large number of exact results. We've developed efficient algorithms and systematically study conformal invariance in two or higher dimensions. [Phys. Rev. Lett. 88 190602 (2002); Phys. Rev. E: 67 036107 (2003), 67 066116 (2003).]
Geometric description of fluctuations at and near criticality has a long history, which dates back to the formulation of phase transitions in terms of the droplet model. Phase transitions and critical phenomena in many statistical models can be observed from geometric aspect. For instance, Potts model can be mapped onto the random-cluster model, and the susceptibility of the former is related to the cluster-size distribution of the latter. The Mott-to-superfluid transition in the Bose-Hubbard model can be characterized by the winding number of the world lines of the particles. We've conducted extensive research in the geometric properties of critical systems, many fractal dimensions has been calculated theoretically or numerically. Models we usually studies include percolation, Potts, random cluster, O(n)loop model etc. [Phys. Rev. E: 70 056132 (2004), 70 046106 (2004), 69 026123 (2004), 72 016101 (2005); Phys. Rev. B 71 144303 (2005), J. Phys. A 40 3305 (2007).]
We've investigated critical surface phenomena of many systems. Especially, we've observed the pseudo-one-dimensional phase transition, and found that even in two-dimensional systems with short-range interactions, rich critical surface phenomena occur due to the fact that the edge spins are correlated through the critical bulk. [Phys. Rev. E: 70 035107(R)(2004),71 026109 (2004),71 016117 (2004),72 016128 (2005),73 056116 (2006).]
With the developments in the field of atomic and molecular physics, researchers are able to engineer lots of quantum systems. Real-time control over features such as lattice structure, density, level of impurities and disorder, and interactions are within our abilities. In cooperation with coworkers in the Quantum Physics and Quantum Information Division, HFNL, we are conducting theoretical and computational exploration on quantum information processing and simulation of many-body quantum systems with cold atom and optical lattice. [Phys. Rev. A 81, 043403 (2010)]
Besides the topics introduced above, we've also been working on others problems in statistical physics and related fields, for instance, defining new models and studying their properties. [J. Phys. A: Math. Theor. 42 225001 (2009), Phys. Rev. E 79, 061112 (2009).]
| Ising model | J-current Model | Fermi-Hubbard Model |