Simulation of constrained fermions in low-dimensional systems

We present a finite-temperature canonical-ensemble determinant quantum Monte Carlo algorithm that enforces an exact fermion number and enables numerically stable simulations of correlated lattice fermions. We propose a stabilized QR update that reduces the computational complexity from standard cubic scaling $O(βN^3)$ to linear scaling $O(βN N_e^2)$ with respect to the system size $N$ , where $N_e$ is the particle number. This yields a dramatic speedup in dilute regimes ( $N_e \ll N$ ), opening unbiased access to large-scale simulations of strongly correlated low-density phases. We validate the method on the dilute Fermi gas with onsite Hubbard interactions, observing the suppression of the fermion sign problem in the dilute limit. Furthermore, we apply this approach to a one-dimensional flat-band system, where the canonical ensemble allows for precise control over filling. We reveal a ferromagnetic instability at low temperatures in the half-filling regime. Our linear-scaling approach provides a powerful tool for investigating emergent phenomena in dilute quantum matter.We present a finite-temperature canonical-ensemble determinant quantum Monte Carlo algorithm that enforces an exact fermion number and enables numerically stable simulations of correlated lattice fermions. We propose a stabilized QR update that reduces the computational complexity from standard cubic scaling $O(βN^3)$ to linear scaling $O(βN N_e^2)$ with respect to the system size $N$, where $N_e$ is the particle number. This yields a dramatic speedup in dilute regimes ($N_e \ll N$), opening unbiased access to large-scale simulations of strongly correlated low-density phases. We validate the method on the dilute Fermi gas with onsite Hubbard interactions, observing the suppression of the fermion sign problem in the dilute limit. Furthermore, we apply this approach to a one-dimensional flat-band system, where the canonical ensemble allows for precise control over filling. We reveal a ferromagnetic instability at low temperatures in the half-filling regime. Our linear-scaling approach provides a powerful tool for investigating emergent phenomena in dilute quantum matter.

arXiv:2601.08552Physical Review LettersSep 28, 20260 citations

CONSTRAINED MOLECULAR DYNAMICS APPROACH TO FERMIONIC SYSTEMS

M. PAPA, Tomoyuki Maruyama, Aldo Bonasera A

Nuclear Physics at Border LinesJan 1, 20024 citations

Fermion ground state methods

James E. Gubernatis, Naoki Kawashima, Philipp Werner

In this chapter, we discuss the fixed-node and constrained-path Monte Carlo methods for computing the ground state properties of systems of interacting electrons. These methods are arguably the two most powerful ones presently available for doing such calculations, but they are approximate. By sacrificing exactness, they avoid the exponential scaling of the Monte Carlo errors with system size that typically accompanies the simulation of systems of interacting electrons. This exponential scaling is called the Fermion sign problem . After a few general comments about the sign problem, we outline both methods, noting points of similarity and difference, plus points of strength and weakness. We also discuss the constrained-phase method, an extension of the constrained-path method, which controls the phase problem that develops when the ground state wave function cannot be real. Sign problem The “sign problem” refers to the exponential increase of the Monte Carlo errors with increasing system size or decreasing temperature (e.g., Loh et al., 1990, 2005; Gubernatis and Zhang, 1994) that often accompanies a Markov chain simulation whose limiting distribution is not everywhere positive. Such a case generally arises in simulations of Fermion and frustrated quantum-spin systems. It seems so inherent to Monte Carlo simulations of Fermion systems that the phrase “the sign problem” to many seems almost synonymous with the phrase “the Fermion sign problem.” Explanations for the cause of the sign problem vary and are still debated. In this chapter, we choose to summarize two explanations that seem to connect the causes in ground state Fermion simulations in the continuum and on the lattice. The sign problem, of course, is not limited to ground state calculations or even to Fermion simulations. While the cause we discuss in Sections 11.2 and 11.3 focuses on the low-lying states of diffusion-like operators, several topological pictures have been proposed (Muramatsu et al., 1992; Samson, 1993; Gubernatis and Zhang, 1994; Samson, 1995). Some of these discussions are done in the context of the zero- and finite-temperature determinant methods (Muramatsu et al., 1992; Gubernatis and Zhang, 1994). Others are done more analytically from a Feynman path-integral point of view (Samson, 1993, 1995). Some are for particles with statistics other than Fermions. The presentation of the sign problem in this chapter is appropriate for the Monte Carlo methods discussed in this chapter.In this chapter, we discuss the fixed-node and constrained-path Monte Carlo methods for computing the ground state properties of systems of interacting electrons. These methods are arguably the two most powerful ones presently available for doing such calculations, but they are approximate. By sacrificing exactness, they avoid the exponential scaling of the Monte Carlo errors with system size that typically accompanies the simulation of systems of interacting electrons. This exponential scaling is called the Fermion sign problem . After a few general comments about the sign problem, we outline both methods, noting points of similarity and difference, plus points of strength and weakness. We also discuss the constrained-phase method, an extension of the constrained-path method, which controls the phase problem that develops when the ground state wave function cannot be real. Sign problem The “sign problem” refers to the exponential increase of the Monte Carlo errors with increasing system size or decreasing temperature (e.g., Loh et al., 1990, 2005; Gubernatis and Zhang, 1994) that often accompanies a Markov chain simulation whose limiting distribution is not everywhere positive. Such a case generally arises in simulations of Fermion and frustrated quantum-spin systems. It seems so inherent to Monte Carlo simulations of Fermion systems that the phrase “the sign problem” to many seems almost synonymous with the phrase “the Fermion sign problem.” Explanations for the cause of the sign problem vary and are still debated. In this chapter, we choose to summarize two explanations that seem to connect the causes in ground state Fermion simulations in the continuum and on the lattice. The sign problem, of course, is not limited to ground state calculations or even to Fermion simulations. While the cause we discuss in Sections 11.2 and 11.3 focuses on the low-lying states of diffusion-like operators, several topological pictures have been proposed (Muramatsu et al., 1992; Samson, 1993; Gubernatis and Zhang, 1994; Samson, 1995). Some of these discussions are done in the context of the zero- and finite-temperature determinant methods (Muramatsu et al., 1992; Gubernatis and Zhang, 1994). Others are done more analytically from a Feynman path-integral point of view (Samson, 1993, 1995). Some are for particles with statistics other than Fermions. The presentation of the sign problem in this chapter is appropriate for the Monte Carlo methods discussed in this chapter.

Cambridge University Press eBooksJan 31, 20160 citations

FERMION SIMULATIONS OF CORRELATED SYSTEMS

Masatoshi Imada

WORLD SCIENTIFIC eBooksDec 1, 19931 citation

Capacities of noiseless quantum channels for massive indistinguishable particles: Bosons versus fermions

Aditi Sen, Ujjwal Sen, Bartosz Gromek +2 more

We consider information transmission through a noiseless quantum channel, where the information is encoded into massive indistinguishable particles: bosons or fermions. We study the situation in which the particles are noninteracting. The encoding input states obey a set of physically motivated constraints on the mean values of the energy and particle number. In such a case, the determination of both classical and quantum capacity reduces to a constrained maximization of entropy. In the case of noninteracting bosons, signatures of Bose-Einstein condensation can be observed in the behavior of the capacity. A major motivation for these considerations is to compare the information-carrying capacities of channels that carry bosons with those that carry fermions. We show analytically that fermions generally provide higher channel capacity, i.e., they are better suited for transferring bits as well as qubits, in comparison to bosons. This holds for a large range of power-law potentials, and for moderate to high temperatures. Numerical simulations seem to indicate that the result holds for all temperatures. Also, we consider the low-temperature behavior for the three-dimensional box and harmonic trap, and again we show that the fermionic capacity is higher than the bosonic one for sufficiently low temperatures.We consider information transmission through a noiseless quantum channel, where the information is encoded into massive indistinguishable particles: bosons or fermions. We study the situation in which the particles are noninteracting. The encoding input states obey a set of physically motivated constraints on the mean values of the energy and particle number. In such a case, the determination of both classical and quantum capacity reduces to a constrained maximization of entropy. In the case of noninteracting bosons, signatures of Bose-Einstein condensation can be observed in the behavior of the capacity. A major motivation for these considerations is to compare the information-carrying capacities of channels that carry bosons with those that carry fermions. We show analytically that fermions generally provide higher channel capacity, i.e., they are better suited for transferring bits as well as qubits, in comparison to bosons. This holds for a large range of power-law potentials, and for moderate to high temperatures. Numerical simulations seem to indicate that the result holds for all temperatures. Also, we consider the low-temperature behavior for the three-dimensional box and harmonic trap, and again we show that the fermionic capacity is higher than the bosonic one for sufficiently low temperatures.

arXiv:quant-ph/0512200Physical Review AFeb 27, 20075 citations

O(βN^3)

to linear scaling

O(βN N_e^2)

with respect to the system size

N

, where

N_e

is the particle number. This yields a dramatic speedup in dilute regimes (

N_e \ll N

), opening unbiased access to large-scale simulations of strongly correlated low-density phases. We validate the method on the dilute Fermi gas with onsite Hubbard interactions, observing the suppression of the fermion sign problem in the dilute limit. Furthermore, we apply this approach to a one-dimensional flat-band system, where the canonical ensemble allows for precise control over filling. We reveal a ferromagnetic instability at low temperatures in the half-filling regime. Our linear-scaling approach provides a powerful tool for investigating emergent phenomena in dilute quantum matter.