Variational Monte Carlo Techniques Research Articles

Entanglement Entropy of the ν=1/2 Composite Fermion Non-Fermi Liquid State.

The so-called "non-Fermi liquid" behavior is very common in strongly correlated systems. However, its operational definition in terms of "what it is not" is a major obstacle for the theoretical understanding of this fascinating correlated state. Recently there has been much interest in entanglement entropy as a theoretical tool to study non-Fermi liquids. So far explicit calculations have been limited to models without direct experimental realizations. Here we focus on a two-dimensional electron fluid under magnetic field and filling fraction ν=1/2, which is believed to be a non-Fermi liquid state. Using a composite fermion wave function which captures the ν=1/2 state very accurately, we compute the second Rényi entropy using the variational MonteCarlo technique. We find the entanglement entropy scales as LlogL with the length of the boundary L as it does for free fermions, but has a prefactor twice that of free fermions.

Variational Monte Carlo study of a chiral spin liquid in the extended Heisenberg model on the kagome lattice

We investigate the extended Heisenberg model on the Kagome lattice by using Gutzwiller projected fermionic states and the variational Monte Carlo technique. In particular, when both second- and third-neighbor super-exchanges are considered, we find that a gapped spin liquid described by non-trivial magnetic fluxes and long-range chiral-chiral correlations is energetically favored compared to the gapless U(1) Dirac state. Furthermore, the topological Chern number, obtained by integrating the Berry curvature, and the degeneracy of the ground state, by constructing linearly independent states, lead us to identify this flux state as the chiral spin liquid with $C=1/2$ fractionalized Chern number.

Ground States of Helium Atom and Hydrogen Negative Ion in the Presence of Magnetic Field Using Variational Monte Carlo Technique

Ground States of Helium Atom and Hydrogen Negative Ion in the Presence of Magnetic Field Using Variational Monte Carlo Technique S.B. Doma, M O. Shaker, a M. Farag and F.N. El-Gammal Mathematics Department, Faculty of Science, Alexandria University, Alexandria, Egypt Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt Mathematics Department, Faculty of Science, Meno a University, Shebin El-Kom, Egypt (Received January 29, 2014; in nal form June 15, 2014) Using variational Monte Carlo method, we calculated the 10 state energies, the derivatives of the total energy and the ionisation energies of the helium atom, and hydrogen negative ion in the presence of magnetic eld regime between 0 a.u. and 10 a.u. Our calculations are based on using two types of compact and accurate trial wave functions used before to calculate energies in the absence of magnetic eld. Our results are in good agreement with the most recent previous accurate values and also with the exact values. DOI: 10.12693/APhysPolA.126.700

Variational Monte Carlo technique

In this article, I discuss the use of the Variational Monte Carlo technique in the determination of ground state energies of quantum harmonic oscillators in one and three dimensions. In order to provide a better understanding of this technique, a pre-requisite concept of Monte Carlo integration is also discussed along with numerical examples. The technique can be easily extended to study the ground state energies of hydrogen atom, particle in a box, helium atom and other complex microscopic systems involving N particles in d-dimensions.

Theoretical investigation of the binding of a positron to vibrational excited states of hydrogen cyanide molecule

We theoretically analyzed positron affinities (PA) of hydrogen cyanide (HCN) molecule at vibrational excited states to elucidate the effect of molecular vibrations on the binding of a positron to the molecule. Using the configuration interaction method in the multi-component molecular orbital theory and anharmonic vibrational state analysis with the variational Monte Carlo technique, we found that the vibrational excitations of the CN and CH stretching modes enhance the PA value compared to that of the vibrational ground state, whereas the excitation of bending mode deenhances it. The largest PA enhancement is found at the excited states of the CH stretching mode; the PA values are 43.02 (1) and 46.34 (2) meV for the fundamental tone and overtone states, respectively. With the linear regression analysis, we confirmed that the PA variation of HCN molecule at each vibrational state arises from the variation of permanent dipole moment and dipole-polarizability due to each vibrational excitation.

Theory of a Competitive Spin Liquid State for Weak Mott Insulators on the Triangular Lattice

We propose a novel quantum spin liquid state that can explain many of the intriguing experimental properties of the low-temperature phase of the organic spin liquid candidate materials κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2. This state of paired fermionic spinons preserves all symmetries of the system, and it has a gapless excitation spectrum with quadratic bands that touch at momentum k[over →]=0. This quadratic band touching is protected by symmetries. Using variational Monte Carlo techniques, we show that this state has highly competitive energy in the triangular lattice Heisenberg model supplemented with a realistically large ring-exchange term.

Variational Monte Carlo approach to the two-dimensional Kondo lattice model

We study the phase diagram of the Kondo lattice model with nearest-neighbor hopping in the square lattice by means of the variational Monte Carlo technique. Specifically, we analyze a wide class of variational wave functions that allow magnetic and superconducting order parameters, so as to assess the possibility that superconductivity might emerge close to the magnetic instability, as is often observed in heavy-fermion systems. Indeed, we do find evidence of $d$-wave superconductivity in the paramagnetic sector, i.e., when magnetic order is not allowed in the variational wave function. However, when magnetism is allowed, it completely covers the superconducting region, which thus disappears from the phase diagram.

Paired chiral spin liquid with a Fermi surface inS=1model on the triangular lattice

Motivated by recent experiments on Ba${}_{3}$NiSb${}_{2}$O${}_{9}$, we investigate possible quantum spin liquid ground states for spin $S=1$ Heisenberg models on the triangular lattice. We use variational Monte Carlo techniques to calculate the energies of microscopic spin liquid wave functions where spin is represented by three flavors of fermionic spinon operators. These energies are compared with the energies of various competing three-sublattice ordered states. Our approach shows that the antiferromagnetic Heisenberg model with biquadratic term and single-ion anisotropy does not have a low-temperature spin liquid phase. However, for an SU(3)-invariant model with sufficiently strong ring-exchange terms, we find a paired chiral quantum spin liquid with a Fermi surface of deconfined spinons that is stable against all types of ordering patterns we considered. We discuss the physics of this exotic spin liquid state in relation to the recent experiment and suggest new ways to test this scenario.

Valence-bond crystals in the kagomé spin-1/2 Heisenberg antiferromagnet: a symmetry classification and projected wave function study

In this paper, we do a complete classification of valence-bond crystals (VBCs) on the kagomé lattice based on general arguments of symmetry only and thus identify many new VBCs for different unit cell sizes. For the spin-1/2 Heisenberg antiferromagnet, we study the relative energetics of competing gapless spin liquids (SLs) and VBC phases within the class of Gutzwiller-projected fermionic wave functions using variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. By using a state-of-the-art optimization method, we conclusively show that the U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and 36-site unit cell VBCs. This stability is also preserved on addition of a next-nearest-neighbor super-exchange coupling of both antiferromagnetic and ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is stabilized on addition of a very small next-nearest-neighbor FM super-exchange coupling, i.e. |J2| ≈ 0.045, and this VBC is the same in terms of space-group symmetry as that obtained in an effective quantum dimer model study. It breaks reflection symmetry, has a nontrivial flux pattern and is a strong dimerization of the uniform RVB SL.

Gapped spin-liquid phase in theJ1−J2Heisenberg model by a bosonic resonating valence-bond ansatz

We study the ground-state phase diagram of the spin-$1/2$${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric ${Z}_{2}$ spin liquid is stable between two antiferromagnetic phases; a continuous transition at ${J}_{2}=0.4{J}_{1}$, when the Marshall sign rule begins to be essentially violated, and a first-order transition around ${J}_{2}=0.6{J}_{1}$ are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around ${J}_{2}=0.51{J}_{1}$, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the $M$ point, i.e., $k=(\ensuremath{\pi},0)$.

Bound state form factors from knockout in10Be(d,t) neutron pickup

Existing ${}^{10}$Be($d$,$t$)${}^{9}$Be neutron pickup data are analyzed using the same bound-state form factors for the ${\ensuremath{\langle}}^{10}\mathrm{Be}{|}^{9}\mathrm{Be}+n\ensuremath{\rangle}$ overlap as in a recent analysis of single-neutron knockout [Grinyer et al., Phys. Rev. Lett. 106, 162502 (2011)]. While the knockout data were well described by a bound-state form factor calculated using the variational Monte Carlo (VMC) technique, including an appropriate neutron binding potential, the ${}^{10}$Be($d$,$t$)${}^{9}$Be pickup data are significantly overpredicted using this form factor. In addition, the no-core shell model (NCSM) and VMC form factors yield the same calculated pickup cross sections whereas the knockout results using these form factors differed by 20$%$. We explore possible sources of ambiguity in the pickup calculations that affect our ability to compare the absolute magnitudes in cross sections between these two very different reactions.

Matrix-product-based projected wave functions ansatz for quantum many-body ground states

We develop a new projected wave function approach which is based on projection operators in the form of matrix-product operators (MPOs). Our approach allows to variationally improve the short range entanglement of a given trial wave function by optimizing the matrix elements of the MPOs while the long range entanglement is contained in the initial guess of the wave function. The optimization is performed using standard variational Monte Carlo techniques. We demonstrate the efficiency of our approach by considering a one-dimension model of interacting spinless fermions. In addition, we indicate how to generalize this approach to higher dimensions using projection operators which are based on tensor products.

Scaling properties of exchange and correlation holes of the valence shell of second-row atoms

We study the exchange and correlation hole of the valence shell of second row atoms using variational Monte Carlo techniques, especially correlated estimates, and norm-conserving pseudopotentials. The well-known scaling of the valence shell provides a tool to probe the behavior of exchange and correlation as a functional of the density and thus test models of density functional theory. The exchange hole shows an interesting competition between two scaling forms -- one caused by self-interaction and another that is approximately invariant under particle number, related to the known invariance of exchange under uniform scaling to high density and constant particle number. The correlation hole shows a scaling trend that is marked by the finite size of the atom relative to the radius of the hole. Both trends are well captured in the main by the Perdew-Burke-Ernzerhof generalized-gradient approximation model for the exchange-correlation hole and energy.

Grand canonical variational approach for thet−Jmodel

Gutzwiller-projected BCS wave function or the resonating-valence-bond (RVB) state in the 2D extended t-J model is investigated by using the variational Monte Carlo technique. We show that the results of ground-state energy and excitation spectra calculated in the grand-canonical scheme allowing particle number to fluctuate are essentially the same as previous results obtained by fixing the number of particle in the canonical scheme if the grand thermodynamic potential is used for minimization. To account for the effect of Gutzwiller projection, a fugacity factor proposed by Laughlin and Anderson few years ago has to be inserted into the coherence factor of the BCS state. Chemical potential, particle number fluctuation, and phase fluctuation of the RVB state, difficult or even impossible to be calculated in the canonical ensemble, have been directly measured in the grand-canonical picture. We find that except for La-214 materials, the doping dependence of chemical potential is consistent with experimental findings on several cuprates. Similar to what has been reported by scanning tunneling spectroscopy experiments, the tunneling asymmetry becomes much stronger as doping decreases. We found a very large enhancement of phase fluctuation in the underdoped regime.

Entanglement Entropy of Critical Spin Liquids

Quantum spin liquids are phases of matter whose internal structure is not captured by a local order parameter. Particularly intriguing are critical spin liquids, where strongly interacting excitations control low energy properties. Here we calculate their bipartite entanglement entropy that characterizes their quantum structure. In particular we calculate the Renyi entropy S(2) on model wave functions obtained by Gutzwiller projection of a Fermi sea. Although the wave functions are not sign positive, S(2) can be calculated on relatively large systems (>324 spins) using the variational Monte Carlo technique. On the triangular lattice we find that entanglement entropy of the projected Fermi sea state violates the boundary law, with S(2) enhanced by a logarithmic factor. This is an unusual result for a bosonic wave function reflecting the presence of emergent fermions. These techniques can be extended to study a wide class of other phases.

Projected wave function study ofZ2spin liquids on the kagome lattice for the spin-12quantum Heisenberg antiferromagnet

Motivated by recent density-matrix renormalization group (DMRG) calculations [Yan, Huse, and White, Science 332, 1173 (2011)], which claimed that the ground state of the nearest-neighbor spin-1/2 Heisenberg antiferromagnet on the kagome lattice geometry is a fully gapped spin liquid with numerical signatures of Z2 gauge structure, and a further theoretical work [Lu, Ran, and Lee, Phys. Rev. B 83, 224413 (2011)], which gave a classification of all Schwinger-fermion mean-field fully symmetric Z2 spin liquids on the kagome lattice, we have thoroughly studied Gutzwiller-projected fermionic wave functions by using quantum variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. In particular, we investigated the energetics of all Z2 candidates (gapped and gapless) that lie in the neighborhood of the energetically competitive U(1) gapless spin liquids. By using a state-of-the-art optimization method, we were able to conclusively show that the U(1) Dirac state is remarkably stable with respect to all Z2 spin liquids in its neighborhood, and in particular for opening a gap toward the so-called Z2[0,{\pi}]{\beta} state, which was conjectured to describe the ground state obtained by the DMRG method. Finally, we also considered the addition of a small second nearest-neighbor exchange coupling of both antiferromagnetic and ferromagnetic type, and obtained similar results, namely, a U(1) Dirac spin-liquid ground state.

QSATS: MPI-driven quantum simulations of atomic solids at zero temperature

We describe QSATS, a parallel code for performing variational path integral simulations of the quantum mechanical ground state of monatomic solids. QSATS is designed to treat Boltzmann quantum solids, in which individual atoms are permanently associated with distinguishable crystal lattice sites and undergo large-amplitude zero-point motions around these sites. We demonstrate the capabilities of QSATS by using it to compute the total energy and potential energy of hexagonal close packed solid 4He at the density ρ = 4.61421 × 10 − 3 a 0 − 3 . Program summary Program title: QSATS Catalogue identifier: AEJE_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJE_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 7329 No. of bytes in distributed program, including test data, etc.: 61 685 Distribution format: tar.gz Programming language: Fortran 77. Computer: QSATS should execute on any distributed parallel computing system that has the Message Passing Interface (MPI) [1] libraries installed. Operating system: Unix or Linux. Has the code been vectorized or parallelized?: Yes, parallelized using MPI [1]. RAM: The memory requirements of QSATS depend on both the number of atoms in the crystal and the number of replicas in the variational path integral chain. For parameter sets A and C (described in the long write-up), approximately 4.5 Mbytes and 12 Mbytes, respectively, are required for data storage by QSATS (exclusive of the executable code). Classification: 7.7, 16.13. External routines: Message Passing Interface (MPI) [1] Nature of problem: QSATS simulates the quantum mechanical ground state for a monatomic crystal characterized by large-amplitude zero-point motions of individual (distinguishable) atoms around their nominal lattice sites. Solution method: QSATS employs variational path integral quantum Monte Carlo techniques to project the systemʼs ground state wave function out of a suitably-chosen trial wave function. Restrictions: QSATS neglects quantum statistical effects associated with the exchange of identical particles. As distributed, QSATS assumes that the potential energy function for the crystal is a pairwise additive sum of atom–atom interactions. Additional comments: An auxiliary program, ELOC, is provided that uses the output generated by QSATS to compute both the crystalʼs ground state energy and the expectation value of the crystalʼs potential energy. End users can modify ELOC as needed to compute the expectation value of other coordinate-space observables. Running time: QSATS requires roughly 3 hours to run a simulation using parameter set A on a cluster of 12 Xeon processors with clock speed 2.8 GHz. Roughly 15 hours are needed to run a simulation using parameter set C on the same cluster.

Magnetism and domain formation in SU(3)-symmetric multi-species Fermi mixtures

We study the phase diagram of an SU(3)-symmetric mixture of three-component ultracold fermions with attractive interactions in an optical lattice, including the additional effect on the mixture of an effective three-body constraint induced by three-body losses. We address the properties of the system in D⩾2 by using dynamical mean-field theory and variational Monte Carlo techniques. The phase diagram of the model shows a strong interplay between magnetism and superfluidity. In the absence of the three-body constraint (no losses), the system undergoes a phase transition from a color superfluid (c-SF) phase to a trionic phase, which shows additional particle density modulations at half-filling. Away from the particle–hole symmetric point the c-SF phase is always spontaneously magnetized, leading to the formation of different c-SF domains in systems where the total number of particles of each species is conserved. This can be seen as the SU(3) symmetric realization of a more general tendency for phase separation in three-component Fermi mixtures. The three-body constraint strongly disfavors the trionic phase, stabilizing a (fully magnetized) c-SF also at strong coupling. With increasing temperature we observe a transition to a non-magnetized SU(3) Fermi liquid phase.

Fermionic Shadow Wave Function Variational Calculations of the Vacancy Formation Energy inHe3

We present a novel technique well suited for studying the ground state of inhomogeneous fermionic matter in a wide range of different systems. The system is described using a fermionic shadow wave function, and the energy is computed by means of the variational Monte Carlo technique. The general form of the fermionic shadow wave function is useful for describing many-body systems with the coexistence of different phases as well in the presence of defects or impurities, but it requires overcoming a significant sign problem. As an application, we studied the energy to activate vacancies in solid 3He.

Condensate depletion in two-species Bose gases: A variational quantum Monte Carlo study

We investigate two-species Bose gases in traps with various interactions using variational quantum Monte Carlo (VMC) techniques at zero temperature. The bosons are represented by hard spheres whose diameter is equivalent to the $s$-wave scattering length in the low-energy and long-wavelength approximation. We explore the role of repulsive and attractive interspecies or intraspecies interactions on the condensate properties of the mixtures, particularly the condensate fraction of each species as compared to the case when each species is in a separate trap of its own. We model the repulsive interactions by a hard-core (HC) potential and the attractive interactions by a shallow model potential. The VMC density profiles and energies are evaluated at various interactions and two mass ratios of the species.