World-line and Determinantal Quantum Monte Carlo Methods for Spins, Phonons and Electrons

Using determinant quantum Monte Carlo (d-QMC) simulations, we demonstrate that an extended Hubbard model on a bilayer honeycomb lattice has two novel quantum phase transitions. The first is a quantum phase transition between the weakly interacting gapless Dirac fermion phase and a strongly interacting fully gapped and symmetric trivial phase, which cannot be described by the standard Gross-Neveu model. The second is a quantum critical point between a quantum spin Hall insulator with spin S z conservation and the previously mentioned strongly interacting fully gapped phase. At the latter quantum critical point the single particle excitations remain gapped, while spin and charge gap both close. We argue that the first quantum phase transition is related to the Z16 classification of the topological superconductor 3 He-B phase with interactions, while the second quantum phase transition is a topological phase transition described by a bosonic O(4) nonlinear sigma model field theory with a Θ-term.IntroductionThe interplay between topology and interactions can lead to very rich new physics. For bosonic systems, it is understood that strong interactions can lead to many symmetry protected topological (SPT) phases [1,2] that are fundamentally different from the standard Mott insulator and superfluid phases. In addition to producing various topological orders, for fermionic systems strong interactions can also reduce the classification of free fermion topological insulators and superconductors [3][4][5][6][7][8][9][10][11]. That is, interactions can drive free fermion topological superconductors to a trivial phase; namely the edge states of the free fermion topological superconductor can be gapped out without degeneracy by a symmetry preserving short range interactions without going through a bulk quantum phase transition. The most famous example is the 3 He-B topological superconductor protected by time-reversal symmetry, whose boundary is described by a (2 + 1)d Majorana fermion χ with theHe-B has a Z classification; therefore for arbitrary copies of 3 He-B, its boundary remains gapless as long as time-reversal symmetry is preserved [12][13][14]. In other words any fermion-bilinear mass term χ a σ y χ b at the boundary would break the time-reversal symmetry. However, once interactions are turned on, the classification of 3 He-B is reduced to Z 16 ; i.e., with 16 copies of 3 He-B, its boundary can be gapped out by interactions while preserving the time-reversal symmetry [9,10]. In other words, the boundary is fully gapped by interactions with χ a σ y χ b = 0, for a, b = 1 · · · 16.Although the classification of interacting 3 He-B has been understood, the following question remains: if the interactions are tuned continuously, can there be a direct second order quantum phase transition between the weakly interacting gapless boundary and the strongly interacting fully gapped nondegenerate boundary state? Even if such a second order phase transition exists, its field theory description is unknown because the standard field ...

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“…[38]. Projector QMC is a kind of determinant QMC which focuses on the zero temperature coupling constants symmetry …”

Section: Qmc Methodsmentioning

confidence: 99%

Exotic quantum phase transitions of strongly interacting topological insulators

Slagle

You

2015

Phys. Rev. B

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“…At the forefront of these methods lie the Quantum Monte-Carlo (QMC) technique 7,8 , the Numerical Renormalization Group 2,9 , the Density Matrix Renormalization Group (DMRG) [10][11][12] , cluster approaches like Cluster Perturbation Theory 13 and the Variational Cluster Approach 14 and DMFT [4][5][6] . All these methods have their strengths and weaknesses.…”

Section: Introductionmentioning

confidence: 99%

Efficient DMFT impurity solver using real-time dynamics with matrix product states

et al. 2015

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We propose to calculate spectral functions of quantum impurity models using the Time Evolving Block Decimation (TEBD) for Matrix Product States. The resolution of the spectral function is improved by a so-called linear prediction approach. We apply the method as an impurity solver within the Dynamical Mean Field Theory (DMFT) for the single-and two-band Hubbard model on the Bethe lattice. For the single-band model we observe sharp features at the inner edges of the Hubbard bands. A finite size scaling shows that they remain present in the thermodynamic limit. We analyze the real time-dependence of the double occupation after adding a single electron and observe oscillations at the same energy as the sharp feature in the Hubbard band, indicating a long-lived coherent superposition of states that correspond to the Kondo peak and the side peaks. For a two-band Hubbard model we observe an even richer structure in the Hubbard bands, which cannot be related to a multiplet structure of the impurity, in addition to sharp excitations at the band edges of a type similar to the single-band case.

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“…The ideas apply to finite-temperature determinantal QMC as well. In the standard grand-canonical algorithm [32,43], the integrand corresponding to f (X X X) takes the form det[I + l B(x l )], where B is the matrix form of the one-body propagatorB. The structure of the path integrals and how f (X X X) varies with imaginary time resemble closely [7] that of the ground-state projection, as we have invoked in the discussion above involving P l .…”

Section: Discussionmentioning

confidence: 99%

“…For any problem with f (X X X) ≥ 0, it is straightforward to sample the probability density function (PDF): f (X X X)/ f (X X X) dX X X by Metropolis [32] or branching random walks [29] and use MC to evaluate Eq. (10):…”

Section: Determinantal Qmc Formalism and The Origin Of The Problemmentioning

confidence: 99%

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Infinite variance in fermion quantum Monte Carlo calculations

Shi

Zhang

2016

Phys. Rev. E

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For important classes of many-fermion problems, quantum Monte Carlo (QMC) methods allow exact calculations of ground-state and finite-temperature properties, without the sign problem. The list spans condensed matter, nuclear physics, and high-energy physics, including the half-filled repulsive Hubbard model, the spin-balanced atomic Fermi gas, lattice QCD calculations at zero density with Wilson Fermions, and is growing rapidly as a number of problems have been discovered recently to be free of the sign problem. In these situations, QMC calculations are relied upon to provide definitive answers. Their results are instrumental to our ability to understand and compute properties in fundamental models important to multiple sub-areas in quantum physics. It is shown, however, that the most commonly employed algorithms in such situations turn out to have an infinite variance problem. A diverging variance causes the estimated Monte Carlo statistical error bar to be incorrect, which can render the results of the calculation unreliable or meaningless. We discuss how to identify the infinite variance problem. An approach is then proposed to solve the problem. The solution does not require major modifications to standard algorithms, adding a "bridge link" to the imaginary-time path-integral. The general idea is applicable to a variety of situations where the infinite variance problem may be present. Illustrative results are presented for the ground state of the Hubbard model at half-filling.

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World-line and Determinantal Quantum Monte Carlo Methods for Spins, Phonons and Electrons

Cited by 136 publications

References 93 publications

Exotic quantum phase transitions of strongly interacting topological insulators

Exotic quantum phase transitions of strongly interacting topological insulators

Efficient DMFT impurity solver using real-time dynamics with matrix product states

Infinite variance in fermion quantum Monte Carlo calculations

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