Fermion Determinant Research Articles

Unifying view of fermionic neural network quantum states: From neural network backflow to hidden fermion determinant states

Unifying view of fermionic neural network quantum states: From neural network backflow to hidden fermion determinant states

Quantum Monte Carlo for gauge fields and matter without the fermion determinant

Monte Carlo simulations of strongly interacting fermionic systems are plagued by the fermion sign problem, making the nonperturbative study of many interesting regimes of dense quantum matter, or of theories of odd numbers of fermion flavors, challenging. Moreover, typical fermion algorithms require the computation (or sampling) of the fermion determinant. We focus instead on the meron cluster algorithm, which can solve the fermion sign problem in a class of models without involving the determinant. We develop and benchmark new meron algorithms to simulate fermions coupled to Z2 and U(1) gauge fields in the presence of appropriate four-fermi interactions. Such algorithms can be used to uncover potential exotic properties of matter, particularly relevant for quantum simulator experiments. We demonstrate the emergence of the Gauss’ law at low temperatures for a U(1) model in (1+1)D. Published by the American Physical Society 2024

A baby–Skyrme model with anisotropic DM interaction: Compact skyrmions revisited

We consider a baby–Skyrme model with Dzyaloshinskii–Moriya interaction (DMI) and two types of potential terms. The model has a close connection with the vacuum functional of fermions coupled with O(3) nonlinear n-fields and with a constant SU(2) gauge background. The energy functional is derived from the heat-kernel expansion for the fermion determinant. The model possesses normal skyrmions with topological charge Q=1. The restricted version of the model also includes both the weak-compacton case (at the boundary, not continuously differentiable) and genuine-compacton case (continuously differentiable). The model consists of only the Skyrme term, and the DMI provides soliton solutions that are known as skyrmions without any potential. The BPS equation in the supersymmetric soliton models implies that the impurity coupling is closely related to the DMI. Therefore, the effect of an exponentially localized DMI is also studied in the present model.

Spectrum of QCD with one flavor: A window for supersymmetric dynamics

We compute the spectrum of the low-lying mesonic states with vector, scalar and pseudoscalar quantum numbers in QCD with one flavour. With three colours the fundamental and the two-index anti-symmetric representations of the gauge group coincide. The latter is an orientifold theory that maps into the bosonic sector of $\mathcal{N} = 1$ super Yang-Mills theory in the large number of colours limit. We employ Wilson fermions along with tree-level improvement in the gluonic and fermionic parts of the action. In this setup the Dirac operator can develop real negative eigenvalues. We therefore perform a detailed study in order to identify configurations where the fermion determinant is negative and eventually reweight them. We finally compare results with effective field theory predictions valid in the large $N_C$ limit and find reasonably consistent values despite $N_C$ being only three. Additionally,the spin-one sector provides a novel window for supersymmetric dynamics.

On the ℤ2 topological invariant

We develop a complex fermionic field-based method to model the properties of the filled bands of topological two-dimensional (2D) matter with time reversal (TR)-symmetry. Using this fermionic representation, we give an explicit calculation of the [Formula: see text] index for 2D topological matter invariant under TR and comment on the emergence of Majorana states at the TR-fix points. Moreover, motivated by recent theoretical results on possible signatures of topological supersymmetric matter, we also give the supersymmetric generalization of our TR-invariant construction and calculate the underlying topological [Formula: see text] index. Other features such as the topological obstruction of basis sections in the fermionic determinant bundle are also investigated. Applications for the calculations of the supersymmetric charge [Formula: see text] operator and the super-Hamiltonian [Formula: see text] for the three-dimensional topological class AII are undertaken; these operators are given by Eqs. (5.48)–(5.51).

Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions

This work presents gauge-equivariant architectures for flow-based sampling in fermionic lattice field theories using pseudofermions as stochastic estimators for the fermionic determinant. This is the default approach in state-of-the-art lattice field theory calculations, making this development critical to the practical application of flow models to theories such as QCD. Methods by which flow-based sampling approaches can be improved via standard techniques such as even/odd preconditioning and the Hasenbusch factorization are also outlined. Numerical demonstrations in two-dimensional U(1) and SU(3) gauge theories with ${N}_{f}=2$ flavors of fermions are provided.

Mitigating spikes in fermion Monte Carlo methods by reshuffling measurements.

We propose a method to mitigate heavy-tailed distributions in fermion quantum Monte Carlo simulations originating from zeros of the fermion determinant. In this case, the second moment of the observables might not be well defined, and we show that by merely changing the synchronization between local updates and computation of observables, one can reduce the prefactor of the heavy-tailed distribution, thus substantially suppressing statistical fluctuations of observables. We also show that the average, or the first moment, is well defined and hence is independent on our measuring scheme. The method is especially suitable for local observables similar to, e.g., double occupancy, where the resulting speedup can reach two orders of magnitude. For observables containing spatial correlators, the speedup is more moderate but still ranges between five and ten. Our results are independent on the nature of the auxiliary field, discrete or continuous, and pave the way to improve measurement strategies for hybrid Monte Carlo simulations.

Spontaneous Lorentz symmetry breaking and one-loop effective action in the metric-affine bumblebee gravity

The metric-affine bumblebee model in the presence of fermionic matter minimally coupled to the connection is studied. We show that the model admits an Einstein frame representation in which the matter sector is described by a non-minimal Dirac action without any analogy in the literature. Such non-minimal terms involve unconventional couplings between the bumblebee and the fermion field. We then rewrite the quadratic fermion action in the Einstein frame in the basis of 16 Dirac matrices in order to identify the coefficients for Lorentz/CPT violation in all orders of the non-minimal coupling ξ. The exact result for the fermionic determinant in the Einstein frame, including all orders in ξ, is also provided. We demonstrate that the axial contributions are at least of second order in the perturbative expansion of ξ. Furthermore, we compute the one-loop effective potential within the weak field approximation.

Topological Gauge Actions on the Lattice as Overlap Fermion Determinants

Overlap fermion on the lattice has been shown to properly reproduce topological aspects of gauge fields. In this paper, we review the derivation of Overlap fermion formalism in a torus of three space-time dimensions. Using the formalism, we show how to use the Overlap fermion determinants in the massless and infinite mass limits to construct different continuum topological gauge actions, such as the level-k Chern–Simons action, “half-CS” term and the mixed Chern–Simons (BF) coupling, in a gauge-invariant lattice UV regulated manner. Taking special Abelian and non-Abelian background fields, we demonstrate numerically how the lattice formalism beautifully reproduces the continuum expectations, such as the flow of action under large gauge transformations.

Fast and scalable quantum Monte Carlo simulations of electron-phonon models.

We introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and we report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but it incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.

Four-dimensional factorization of the fermion determinant in lattice QCD

In the last few years it has been proposed a one-dimensional factorization of the fermion determinant in lattice QCD with Wilson-type fermions that leads to a block-local action of the auxiliary bosonic fields. Here we propose a four-dimensional generalization of this factorization. Possible applications are more efficient parallelizations of Monte Carlo algorithms and codes, master field simulations, and multi-level integration.

Duals of lattice Abelian models with static determinant at finite density

Dual formulations of Abelian U(1) and Z(N) LGT with a static fermion determinant are constructed at finite temperatures and non-zero chemical potential. The dual form is valid for a broad class of lattice gauge actions, for arbitrary number of fermion flavors and in any dimension. The distinguished feature of the dual formulation is that the dual Boltzmann weight is strictly positive. This allows to gain reliable results at finite density via the Monte-Carlo simulations. As a byproduct of the dual representation we outline an exact solution for the partition function of the (1+1)-dimensional theory and reveal an existence of a phase with oscillating correlations.

Mod-two APS index and domain-wall fermion

We show a mathematical relation between the mod-two Atiyah–Patodi–Singer (APS) index of a massless Dirac operator and massive domain-wall fermion determinant. The domain-wall fermion is given on a closed manifold, which is extended from the original manifold with boundary, where we instead give a fermion mass term changing its sign at the location of the original boundary. This new setup does not need the APS boundary condition, which is non-local. A mathematical proof of equivalence between the two different formulations is given by two different evaluations of the same index of a Dirac operator on a higher-dimensional manifold. The domain-wall fermion allows us to separate the edge and bulk mode contributions in a natural but not in a gauge invariant way, which offers a straightforward description of the global anomaly inflow.

Chiral symmetry breaking and confinement from an interacting ensemble of instanton dyons in two-flavor massless QCD

In this work we present the results from numerical simulations of an interacting ensemble of instanton dyons in the $SU(3)$ gauge group with ${N}_{f}=2$ flavors of massless quarks. Dynamical quarks are included via the effective interactions induced by the fermionic determinant evaluated in the subspace of topological zero modes. The eigenvalue spectrum of the Dirac operator is studied at different volumes to extract the chiral condensate and eigenvalue gap, with both observables providing consistent values of the chiral transition temperature ${T}_{c}$. We find that a sufficient density of dyons is responsible for generating the confining potential and breaking the chiral symmetry, both of which are compatible with second-order transitions.

Quantum Monte Carlo method on asymptotic Lefschetz thimbles for quantum spin systems: An application to the Kitaev model in a magnetic field

The quantum Monte Carlo method on asymptotic Lefschetz thimbles is a numerical algorithm devised specifically for alleviation of the sign problem appearing in the simulations of quantum many-body systems. In this method, the sign problem is alleviated by shifting the integration domain for the auxiliary fields, appearing for example in the conventional determinant quantum Monte Carlo method, from real space to an appropriate manifold in complex space. Here we extend this method to quantum spin models with generic two-spin interactions, by using the Hubbard-Stratonovich transformation to decouple the exchange interactions and the Popov-Fedotov transformation to map the quantum spins to complex fermions. As a demonstration, we apply the method to the Kitaev model in a magnetic field whose ground state is predicted to deliver a topological quantum spin liquid with non-Abelian anyonic excitations. To illustrate how the sign problem is alleviated in this method, we visualize the asymptotic Lefschetz thimbles in complex space, together with the saddle points and the zeros of the fermion determinant. We benchmark our method in the low-temperature region in a magnetic field and show that the sign of the action is recovered considerably and unbiased numerical results are obtained with sufficient precision.

Perturbative linearization of super-Yang-Mills theories in general gauges

Supersymmetric Yang-Mills theories can be characterized by a non-local and non-linear transformation of the bosonic fields (Nicolai map) mapping the interacting functional measure to that of a free theory, such that the Jacobi determinant of the transformation equals the product of the fermionic determinants obtained by integrating out the gauginos and ghosts at least on the gauge hypersurface. While this transformation has been known so far only for the Landau gauge and to third order in the Yang-Mills coupling, we here extend the construction to a large class of (possibly non-linear and non-local) gauges, and exhibit the conditions for all statements to remain valid off the gauge hypersurface. Finally, we present explicit results to second order in the axial gauge and to fourth order in the Landau gauge.

Deconfinement critical point of lattice QCD with Nf=2 Wilson fermions

The ${\rm SU}(3)$ pure gauge theory exhibits a first-order thermal deconfinement transition due to spontaneous breaking of its global $Z_3$ center symmetry. When heavy dynamical quarks are added, this symmetry is broken explicitly and the transition weakens with decreasing quark mass until it disappears at a critical point. We compute the critical hopping parameter and the associated pion mass for lattice QCD with $N_f=2$ degenerate standard Wilson fermions on $N_\tau\in\{6,8,10\}$ lattices, corresponding to lattice spacings $a=0.12\, {\rm fm}$, $a=0.09\, {\rm fm}$, $a=0.07\, {\rm fm}$, respectively. Significant cut-off effects are observed, with the first-order region growing as the lattice gets finer. While current lattices are still too coarse for a continuum extrapolation, we estimate $m_\pi^c\approx 4 {\rm GeV}$ with a remaining systematic error of $\sim 20\%$. Our results allow to assess the accuracy of the LO and NLO hopping expanded fermion determinant used in the literature for various purposes. We also provide a detailed investigation of the statistics required for this type of calculation, which is useful for similar investigations of the chiral transition.

Integrating out new fermions at one loop

We present the fermionic universal one-loop effective action obtained by integrating out heavy vector-like fermions at one loop using functional techniques. Even though previous approaches are able to handle integrating out heavy fermions with non-chiral interactions, i.e. vanishing γ5 interaction terms, the computations proceed in a tedious manner that obscures a physical interpretation. We show how directly tackling the fermionic functional determinant not only allows for a much simpler and transparent computation, but is also able to account for chiral interaction terms in a simple, algorithmic way. Finally, we apply the obtained results to integrate out at one loop the vector-like fermions appearing in a toy model and in a fermionic model that exhibits strong cosmological phase transitions.

Computing real time correlation functions on a hybrid classical/quantum computer

Quantum devices may overcome limitations of classical computers in studies of nuclear structure functions and parton Wigner distributions of protons and nuclei. In this talk, we discuss a worldline approach to compute nuclear structure functions in the high energy Regge limit of QCD using a hybrid quantum computer, by expressing the fermion determinant in the QCD path integral as a quantum mechanical path integral over 0 + 1-dimensional fermionic and bosonic world-lines in background gauge fields. Our simplest example of computing the well-known dipole model result for the structure function F2 in the high energy Regge limit is feasible with NISQ era technology using few qubits and shallow circuits. This example can be scaled up in complexity and extended in scope to compute structure functions, scattering amplitudes and other real-time correlation functions in QCD, relevant for example to describe non-equilibrium transport of quarks and gluons in a Quark-Gluon-Plasma.

Towards a reliable lower bound on the location of the critical endpoint

We perform the first direct determination of the position of the leading singularity of the pressure in the complex chemical potential μB plane in lattice QCD using numerical simulations with 2-stout improved rooted staggered fermions. This provides a direct determination of the radius of convergence of the Taylor expansion of the pressure that does not rely on a finite-order truncation of the expansion. The analyticity issues in the complex μB plane of the grand canonical partition function of QCD with rooted staggered fermions are solved with a careful redefinition of the fermion determinant that makes it a polynomial in the fugacity on any finite lattice, without changing the continuum limit of the observables. By performing a finite volume scaling study at a single coarse lattice spacing, we show that the limiting singularity is not on the real line in the thermodynamic limit, thus showing that the radius of convergence of the Taylor expansion gives a lower bound on the location of a possible phase transition. In the vicinity of the crossover temperature at zero chemical potential, the radius of convergence turns out to be μB/T≈2 and roughly temperature independent.