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arXiv:1812.05080 · Physical review. D/Physical review. D.

Few-body physics on a spacetime lattice in the worldline approach

Hersh Singh, Shailesh Chandrasekharan A

Published Apr 24, 2019• 10 citations
arXivHTMLPDF

Abstract

We formulate the physics of two species of nonrelativistic hard-core bosons with attractive or repulsive delta function interactions on a spacetime lattice in the worldline approach. We show that worm algorithms can efficiently sample the worldline configurations in any fixed particle-number sector if the chemical potential is tuned carefully. Since fermions can be treated as hard-core bosons up to a permutation sign, we apply this approach to study nonrelativistic fermions. The fermion permutation sign is an observable in this approach and can be used to extract energies in each particle-number sector. In one dimension, nonrelativistic fermions can only permute across boundaries, and so our approach does not suffer from sign problems in many cases, unlike the auxiliary field method. Using our approach, we discover limitations of the recently proposed complex Langevin calculations in one spatial dimension for some parameter regimes. In higher dimensions, our method suffers from the usual fermion sign problem. Here we provide evidence that it may be possible to alleviate this problem for few-body physics.

DOI
10.1103/physrevd.99.074511

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We formulate the physics of two species of nonrelativistic hard-core bosons with attractive or repulsive delta function interactions on a spacetime lattice in the worldline approach. We show that worm algorithms can efficiently sample the worldline configurations in any fixed particle-number sector if the chemical potential is tuned carefully. Since fermions can be treated as hard-core bosons up to a permutation sign, we apply this approach to study nonrelativistic fermions. The fermion permutation sign is an observable in this approach and can be used to extract energies in each particle-number sector. In one dimension, nonrelativistic fermions can only permute across boundaries, and so our approach does not suffer from sign problems in many cases, unlike the auxiliary field method. Using our approach, we discover limitations of the recently proposed complex Langevin calculations in one spatial dimension for some parameter regimes. In higher dimensions, our method suffers from the usual fermion sign problem. Here we provide evidence that it may be possible to alleviate this problem for few-body physics.
We study the physics of two species of non-relativistic hard-core bosons with attractive or repulsive delta function interactions on a spacetime lattice using the worldline formulation. By tuning the chemical potential carefully we show that worm algorithms can efficiently sample the worldline configurations in any fixed particle-number sector. Since fermions can be treated as hard-core bosons up to a permutation sign, we also apply this approach to non-relativistic fermions. The fermion permutation sign is treated as an observable in this approach and can be used to extract energies for each particle-number sector. Since in one dimension non-relativistic fermions can only permute due to boundary effects, unlike the auxiliary field method, in many cases our approach does not suffer from sign problems. Using our method we discover limitations of the recently proposed complex Langevin calculations in one dimension.
In this work we introduce a worldline-based fermion Monte Carlo algorithm for studying few body quantum mechanics of self-interacting fermions in the Hamiltonian lattice formulation. Our motivation to construct the method comes from our interest in studying renormalization of chiral nuclear effective-field theory with lattice regularization. In particular we wish to apply our method to compute the lattice spacing dependence of local lattice interactions as we take the continuum limit of the lattice theory. Our algorithm can compute matrix elements of the operator $exp(\ensuremath{-}\ensuremath{\beta}H)$, where HHH is the lattice Hamiltonian and $\ensuremath{\beta}$ is a free real parameter. These elements help us compute deep bound states that are well separated from scattering states even at values of $\ensuremath{\beta}$ which are not very large. Computing these bound-state energies accurately can help us study renormalization of the lattice theory. In addition to developing the algorithm, in this work we also introduce a finite-volume renormalization scheme for the lattice Hamiltonian of the leading pionless effective-field theory and show how it would work in the one- and two-body sectors.
I derive a loop representation for the canonical and grand-canonical partition functions for an interacting four-component Fermi gas in one spatial dimension and an arbitrary external potential. The representation is free of the ``sign problem'' irrespective of population imbalance, mass imbalance, and to a degree, sign of the interaction strength. This property is in sharp contrast with the analogous three-dimensional, two-component interacting Fermi gas, which exhibits a sign problem in the case of unequal masses, chemical potentials, and repulsive interactions. The one-dimensional system is believed to exhibit many phenomena in common with its three-dimensional counterpart, including an analog of the BCS-BEC crossover, and nonperturbative, universal few- and many-body physics at scattering lengths much larger than the range of interaction, making the theory an interesting candidate for numerical study. Positivity of the probability measure for the partition function allows for a mean-field treatment of the model; here, I present such an analysis for the interacting Fermi gas in the SU(4) (unpolarized, mass-symmetric) limit and demonstrate that there exists a phase in which a continuum limit may be defined.
We study lattice QED in 2+12+12+1 dimensions coupled to Wilson fermions in the Hamiltonian formulation and develop a sign-problem-free variational Monte Carlo algorithm. Using a continuous basis to describe the gauge degrees of freedom and fermionic Gaussian states for the matter fields, we construct a variational ansatz for the ground state. The parameters of the ansatz are optimized using gradient descent and/or imaginary time evolution based on the time-dependent variational principle. For a lattice with NNN matter sites, the computational complexity for sampling a gauge configuration is O(N2log⁡N)\mathcal{O}(N^2\log N)O(N2logN), and the complexity for computing the energy per gauge configuration is O(N4)\mathcal{O}(N^4)O(N4). Comparing our method to results from exact diagonalization on small lattices, we demonstrate good agreement. In addition, we present preliminary results for the ground state for two fermion flavors, Nf=2N_f=2Nf​=2, in the presence of a non-zero chemical potential, a regime in which the conventional action-based Monte Carlo methods suffer from the sign problem.
A system of three-species fermions in one spatial dimension (1D) with a contact three-body interaction is known to display a scale anomaly. This anomaly is identical to that of a two-dimensional (2D) system of two-species fermions. The exact relation between those two systems, however, is limited to the two-particle sector of the 2D case and the three-particle sector of the 1D case. Here, we implement a nonperturbative Monte Carlo approach, based on the worldline representation, to calculate the thermodynamics and static response of three-species fermions in 1D, thus tackling the many-body sector of the problem. We determine the energy, density, and pressure equations of state and the compressibility and magnetic susceptibility for a wide range of temperatures and coupling strengths. We compare our results with the third-order virial expansion and interpret the classical-quantum crossover as the onset of trimer degrees of freedom.
We analyze the pressure and density equations of state of unpolarized nonrelativistic fermions at finite temperature in one spatial dimension with contact interactions. For attractively interacting regimes, we perform a third-order lattice perturbation theory calculation, assess its convergence properties by comparing with hybrid Monte Carlo results (there is no sign problem in this regime), and demonstrate agreement with real Langevin calculations. For repulsive interactions, we present lattice perturbation theory results as well as complex Langevin calculations, with a modified action to prevent uncontrolled excursions in the complex plane. Although perturbation theory is a common tool, our implementation of it is unconventional; we use a Hubbard-Stratonovich transformation to decouple the system and automate the application of Wick's theorem, thus generating the diagrammatic expansion, including symmetry factors, at any desired order. We also present an efficient technique to tackle nested Matsubara frequency sums without relying on contour integration, which is independent of dimension and applies to both relativistic and nonrelativistic systems, as well as all energy-independent interactions. We find exceptional agreement between perturbative and nonperturbative results at weak couplings, and furnish predictions based on complex Langevin at strong couplings. We additionally present perturbative calculations of up to the fifth-order virial coefficient for repulsive and attractive couplings. Both the lattice perturbation theory and complex Langevin formalisms can easily be extended to a variety of situations including polarized systems, bosons, and higher dimension.