Session Y24: Focus Session: Advances in Fermionic Simulatons

Determinantal Quantum Monte Carlo simulations of fermions in optical lattices

The ability to cool fermions in optical lattices to ultra cold temperatures has led to an interdisciplinary area of research, that has attracted a lot of attention in recent years. An interesting development in this area is the possibility to realize models for strongly correlated fermions in the laboratory, such as the fermionic Hubbard Model. Determinantal Quantum Monte Carlo simulations have proven to be an important tool in the study of fermionic atoms. Nonetheless, it is important to compare the results and efficiency of different methods. Here comparisons with Numerical Linked Cluster Expansion and Dynamical Mean Field Theory data for double occupation and short range correlations, both relevant to current optical lattice experiments, will be presented and discussed. Another topic relevant in the context of optical lattice experiments is the study of metal insulator transitions. Indeed, the Mott insulating phase has been realized and observed in two-flavor mixtures of fermionic atoms loaded on optical lattices, being characterized both by the double occupation and the compressibility. An interesting point that has been addressed in the literature over the years is whether the same fermion-fermion interaction, responsible for the Mott insulating state, could drive an insulating system metallic. Here we show that, when fermions are loaded in optical lattices with spatially varying interactions a correlation induced Mott insulator to metal transition can take place. The spatial modulation of the interactions was recently demonstrated and opens the possibility for the experimental realization of such exotic phases.   

Quantum Monte Carlo Calculations of Entanglement

Spatial entanglement properties have become increasingly important in physics which includes studies in diverse fields such as condensed matter physics, astrophysics, and quantum computation. One of the important outstanding problems in the field of entanglement is to understand the effect of many body interactions. Recent advances in quantum Monte Carlo have facilitated such studies over a range of Hamiltonians that were previously inaccessible by other techniques. We apply these techniques to interacting molecular and condensed matter systems and discuss the effect interactions have on entanglement properties.   

Excited state calculations in solids by auxiliary-field quantum Monte Carlo

We present an approach for ab initio many-body calculations of excited states in solids. Using auxiliary-field quantum Monte Carlo \footnote{S.~\ Zhang and H.~\ Krakauer, Phys. Rev. Lett. {\bf 90}, 136401 (2003)}, we introduce an orthogonalization constraint with virtual orbitals to prevent collapse of the stochastic Slater determinants in the imaginary-time propagation. Trial wave functions from density-functional calculations are used for the constraints, and detailed band structures can be calculated. Results for standard semiconductors are in good agreement with GW calculations and with experiment. For the challenging ZnO, we obtain a fundamental band gap of 3.30(16) eV, consistent within the range of experimental measurements \footnote{V.~\ Srikant and D.~\ R.~\ Clarke, J. Appl. Phys. 83, 5447 (1998); S.~\ Tsoi, X.~\ Lu, A.~\ K.~\ Ramdas, H.~\ Alawadhi, M.~\ Grimsditch, M.~\ Cardona, and R.~\ Lauck, Phys. Rev. B 74, 165203 (2006); H.~\ Alawadhi, S.~\ Tsoi, X.~\ Lu, A.~\ K.~\ Ramdas, M.~\ Grimsditch, M.~\ Cardona, and R.~\ Lauck, Phys. Rev. B {\bf 75}, 205207 (2007)}. Applications to other systems are currently underway.   

Bold Diagrammatic Monte Carlo for Fermionic and Fermionized Systems

In three different fermionic cases---repulsive Hubbard model, resonant fermions, and fermionized spins-1/2 (on triangular lattice)---we observe the phenomenon of sign blessing: Feynman diagrammatic series features finite convergence radius despite factorial growth of the number of diagrams with diagram order. Bold diagrammatic Monte Carlo technique allows us to sample millions of skeleton Feynman diagrams. With the universal fermionization trick we can fermionize essentially any (bosonic, spin, mixed, etc.) lattice system. The combination of fermionization and Bold diagrammatic Monte Carlo yields a universal first-principle approach to strongly correlated lattice systems, provided the sign blessing is a generic fermionic phenomenon.   

Path Integral Quantum Monte Carlo Benchmarks for Molecules and Plasmas

Path integral quantum Monte Carlo is used to simulate hot dense plasmas and other systems where quantum and thermal fluctuations are important. The fixed node approximation---ubiquitous in ab initio ground state Quantum Monte Carlo---is more complicated at finite temperatures, with many unanswered questions. In this talk I discuss the current state of fermionic path integral quantum Monte Carlo, with an emphasis on molecular systems where good benchmark data exists. We look at two ways of formulating the fixed node constraint and strategies for constructing finite-temperature nodal surfaces. We compare different the free energies of different nodal choices by sampling an ensemble of nodal models within a Monte Carlo simulation. We also present data on imaginary-time correlation fluctuations, which can be surprisingly accurate for molecular vibrations and polarizabilty.   

Quantum Monte Carlo simulations of complex Hamiltonians

In the last two decades there have been tremendous advances in boson Quantum Monte Carlo methods, which allow for solving more and more complex Hamiltonians. In particular, it is now possible to simulate Hamiltonians that include terms that couple an arbitrary number of sites and/or particles, such as six-site ring-exchange terms. These ring-exchange interactions are crucial for the study of quantum fluctuations on highly frustrated systems. We illustrate how the Stochastic Green Function algorithm with Global Space-Time Update can easily simulate such complex systems, and present some results for a highly non-trivial model of bosons in a pyrochlore crystal with six-site ring-exchange terms.   

Quasi-adiabatic Quantum Monte Carlo algorithm for non-equilibrium quantum phase transitions

We investigate a new quantum Monte Carlo algorithm for studying static and dynamic properties of quantum phase transitions. The method, called the quasi-adiabatic quantum Monte Carlo algorithm, is based on evolution with a changing Hamiltonian to derive information pertinent to a quantum quench according to an arbitrary protocol. We demonstrate the method with results for 1D and 2D transverse-field Ising models, showing finite-size and finite-velocity scaling according to a generalization of the Kibble-Zurek mechanism. We explore ways to extract critical points and critical exponents to high precision.   

Ground state phases in the half-filled staggered $\pi$-flux Hubbard model on square lattices

Ground state phase diagram of the half-filled staggered $\pi$-flux Hubbard model on a square lattice are studied by means of constrained-path quantum Monte Carlo method. Charge and spin excitation gaps and magnetic order are calculated as a function of interaction strength $U/t$. Within our numerical scheme, it is found that the ground state phase is a semi-metal at $U/t < 5.6$, and a Mott insulator with long-range antiferromagnetic order at $U/t > 6.6$. In the window $5.6 < U/t < 6.6$, the system is an insulator in which both magnetic and dimer orders are absent. Spin excitation in the intermediate phase appears to be gapless, and the measured equal-time spin-spin correlation function shows a power-law dependence of relative distance. Our data suggests that the paramagnetic insulating intermediate phase might be a possible place to look for the putative algebraic spin liquid.   

Momentum-dependent pseudogaps in the half-filled two-dimensional Hubbard model

We compute unbiased spectral functions of the two-dimensional Hubbard model by extrapolating Green functions, obtained from determinantal quantum Monte Carlo simulations, to the thermodynamic and continuous time limits. Our results clearly resolve the pseudogap at weak to intermediate coupling, originating from a momentum selective opening of the charge gap. A characteristic pseudogap temperature $T^*$, determined consistently from the spectra and from the momentum dependence of the imaginary-time Green functions, is found to match the dynamical mean-field critical temperature, below which antiferromagnetic fluctuations become dominant. Our results identify a regime where pseudogap physics is within reach of experiments with cold fermions on optical lattices.\\[2ex] D. Rost, E. V. Gorelik, F. Assaad, N. Bl\"umer, Phys. Rev. B {\bf 86}, 155109 (2012).   

Series Expansion for the Green's Function of the Infinite-U Hubbard Model

We implement computationally a strong-coupling expansion for the dynamical single-particle Green's function of the infinite-U Hubbard model up to the eighth order in the hopping, within the formalism introduced by Metzner [1]. We obtain analytical expressions for the finite Matsubara frequency Green's functions and the Dyson self energy in the momentum space at all densities in the thermodynamic limit. The results match those obtained up to the fourth order by means of another method devised by us. Furthermore, we employ Pade approximations and various numerical re-summation techniques to extend the region of convergence to lower temperatures.\\[4pt] Ref. [1]: W. Metzner, Phys. Rev. B 43, 8549 (1991).