Free Fermions Research Articles

Solutions of the universal character hierarchy and BUC hierarchy by fermionic approach

In this paper, by means of the fermionic approach, we investigate the (m,n)-order τ-function of the universal character (UC) hierarchy which can be expressed as the determinants of the Schur functions. Meanwhile, we present the general τ-function for charged free fermions, from which soliton solutions of the UC hierarchy have been discussed. Furthermore, the algebraic structure and properties of the universal character hierarchy of B type (BUC hierarchy) have been developed. In terms of neutral free fermions and the generalized Q-function, the polynomial τ-functions and the extended formulas of the BUC and the 2-component BUC hierarchies have been analyzed. In addition, we derive the rational solutions of the BUC hierarchy. It is showed that soliton solutions of the BUC hierarchy can be expressed as multiplication of Pfaffians.

A relativistic discrete spacetime formulation of 3+1 QED

This work provides a relativistic, digital quantum simulation scheme for both 2+1 and 3+1 dimensional quantum electrodynamics (QED), based on a discrete spacetime formulation of theory. It takes the form of a quantum circuit, infinitely repeating across space and time, parametrised by the discretization step Δt=Δx. Strict causality at each step is ensured as circuit wires coincide with the lightlike worldlines of QED; simulation time under decoherence is optimized. The construction replays the logic that leads to the QED Lagrangian. Namely, it starts from the Dirac quantum walk, well-known to converge towards free relativistic fermions. It then extends the quantum walk into a multi-particle sector quantum cellular automata in a way which respects the fermionic anti-commutation relations and the discrete gauge invariance symmetry. Both requirements can only be achieved at cost of introducing the gauge field. Lastly the gauge field is given its own electromagnetic dynamics, which can be formulated as a quantum walk at each plaquette.

On integrability of the one-dimensional Hubbard model

We find a family of solutions to Zamolodchikov's tetrahedral algebra corresponding to the fermionic R-operator for the free fermion model of the difference type in one of the spectral parameters, construct an extension of the R-operator for a system of two spins satisfying the Yang-Baxter equation, and find the local charges. We also construct a twisted monodromy operator, which leads to the one-dimensional Hubbard model.

Fermions with SU(1, n) spacetime symmetry

We construct theories of free fermions in (2n − 1)-dimensions with SU(1, n) spacetime symmetry from the null reduction of fermions on a 2n-dimensional Ω-deformed Minkowski background for n = 2 and n = 3. These play a role in the 5d SU(1, 3)-invariant theories that are conjectured to offer a full description of certain 6d superconformal field theories. We find the (2n − 1)-dimensional manifestation of the supersymmetry of a free 2n-dimensional boson-fermion system, which we use to fix the fermion two-point functions. It is then shown that the full 2n-dimensional two-point function can be recovered through resummation. Limits of the theories are considered, and it is observed that both Galilean and Carrollian field theories appear in different regimes. We confirm that the correlation functions obey the SU(1, n) Ward identities and the representations of the fermions under this group are discussed.

Remarks on Berry connection in QFT, anomalies, and applications

Berry connection has been recently generalized to higher-dimensional QFT, where it can be thought of as a topological term in the effective action for background couplings. Via the inflow, this term corresponds to the boundary anomaly in the space of couplings, another notion recently introduced in the literature. In this note we address the question of whether the old-fashioned Berry connection (for time-dependent couplings) still makes sense in a QFT on \Sigma^{(d)}× \mathbb{R}Σ(d)×ℝ, where \Sigma^{(d)}Σ(d) is a dd-dimensional compact space and \mathbb{R}ℝ is time. Compactness of \Sigma^{(d)}Σ(d) relieves us of the IR divergences, so we only have to address the UV issues. We describe a number of cases when the Berry connection is well defined (which includes the tt^*tt* equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean \Sigma^{(d)} × \mathbb{R}_{≥ 0}Σ(d)×ℝ≥0. We then work out the examples of a free 3D Dirac fermion and a 3D \mathcal{N}=2𝒩=2 chiral multiplet. Finally, we consider 3D theories on \mathbb{T}^2× \mathbb{R}𝕋2×ℝ, where the space \mathbb{T}^2𝕋2 is a two-torus, and apply our machinery to clarify some aspects of the relation between 3D SUSY vacua and elliptic cohomology. We also comment on the generalization to higher genus.

Logarithmic negativity and spectrum in free fermionic systems for well-separated intervals

We employ a mathematical framework based on the Riemann-Hilbert approach developed by Bettelheim et al (2022 J. Phys. A: Math. Gen. 55 135001) to study logarithmic negativity of two intervals of free fermions in the case where the size of the intervals as well as the distance between them is macroscopic. We find that none of the eigenvalues of the density matrix become negative, but rather they develop a small imaginary value, leading to non-zero logarithmic negativity. As an example, we compute negativity at half-filling and for intervals of equal size we find a result of order , where N is the typical length scale in units of the lattice spacing. One may compute logarithmic negativity in further situations, but we find that the results are non-universal, depending non-smoothly on the Fermi level and the size of the intervals in units of the lattice spacing.

Discontinuities of free theories on AdS2

The partition functions of free bosons as well as fermions on AdS2 are not smooth as a function of their masses. For free bosons, the partition function on AdS2 is not smooth when the mass saturates the Breitenlohner-Freedman bound. We show that the expectation value of the scalar bilinear on AdS2 exhibits a kink at the BF bound and the change in slope of the expectation value with respect to the mass is proportional to the inverse radius of AdS2. For free fermions, when the mass vanishes the partition function exhibits a kink. We show that expectation value of the fermion bilinear is discontinuous and the jump in the expectation value is proportional to the inverse radius of AdS2. We then show the supersymmetric actions of the chiral multiplet on AdS2 × S1 and the hypermultiplet on AdS2 × S2 demonstrate these features. The supersymmetric backgrounds are such that as the ratio of the radius of AdS2 to S1 or S2 is dialled, the partition functions as well as expectation of bilinears are not smooth for each Kaluza-Klein mode on S1 or S2. Our observation is relevant for evaluating one-loop partition function in the near horizon geometry of extremal black holes.

First-order photon condensation in magnetic cavities: A two-leg ladder model

We consider a model of free fermions in a ladder geometry coupled to a non-uniform cavity mode via Peierls substitution. Since the cavity mode generates a magnetic field, no-go theorems on spontaneous photon condensation do not apply, and we indeed observe a phase transition to a photon condensed phase characterized by finite circulating currents, alternatively referred to as the equilibrium superradiant phase. We consider both square and triangular ladder geometries, and characterize the transition by studying the energy structure of the system, light-matter entanglement, the properties of the photon mode, and chiral currents. The transition is of first order and corresponds to a sudden change in the fermionic band structure as well as the number of its Fermi points. Thanks to the quasi-one dimensional geometry we scrutinize the accuracy of (mean field) cavity-matter decoupling against large scale density-matrix renormalization group simulations. We find that light-matter entanglement is essential for capturing corrections to matter properties at finite sizes and for the description of the correct photon state. The latter remains Gaussian in the the thermodynamic limit both in the normal and photon condensed phases.

Relating non-Hermitian and Hermitian quantum systems at criticality

We demonstrate three types of transformations that establish connections between Hermitian and non-Hermitian quantum systems at criticality, which can be described by conformal field theories (CFTs). For the transformation preserving both the energy and the entanglement spectra, the corresponding central charges obtained from the logarithmic scaling of the entanglement entropy are identical for both Hermitian and non-Hermitian systems. The second transformation, while preserving the energy spectrum, does not perserve the entanglement spectrum. This leads to different entanglement entropy scalings and results in different central charges for the two types of systems. We demonstrate this transformation using the dilation method applied to the free fermion case. Through this method, we show that a non-Hermitian system with central charge c = -4c=−4 can be mapped to a Hermitian system with central charge c = 2c=2. Lastly, we investigate the Galois conjugation in the Fibonacci model with the parameter \phi \to - 1/\phiϕ→−1/ϕ, in which the transformation does not preserve both energy and entanglement spectra. We demonstrate the Fibonacci model and its Galois conjugation relate the tricritical Ising model/3-state Potts model and the Lee-Yang model with negative central charges from the scaling property of the entanglement entropy.

Cosmological Bounces Induced by a Fermion Condensate.

We show that it is possible for fermion condensation of the Nambu-Jona-Lasinio type to induce a nonsingular bounce that smoothly connects a phase of slow contraction to a phase of expansion. A chiral condensate-a nonzero vacuum expectation value of the spinor bilinear ⟨Ψ[over ¯]Ψ⟩-can form spontaneously after a slow contraction phase smooths and flattens the universe and the Ricci curvature exceeds a critical value. In this approach, a high density of spin-aligned free fermions is not required, which avoids the problem of generating a large anisotropy and initiating chaotic mixmaster behavior during the bounce phase.

New conjecture on exact Dirac zero-modes of lattice fermions

Abstract We propose a new conjecture on the relation between the exact Dirac zero-modes of free and massless lattice fermions and the topology of the manifold on which the fermion action is defined. Our conjecture claims that the maximal number of exact Dirac zero-modes of fermions on finite-volume and finite-spacing lattices defined by a discretizing torus, hyperball, their direct-product space, and hypersphere is equal to the summation of the Betti numbers of their manifolds if several specific conditions on lattice formulations are satisfied. We start with reconsidering exact Dirac zero-modes of naive fermions on the lattices whose topologies are a torus, hyperball, and their direct-product space (TD × Bd). We find that the maximal number of exact zero-modes of free Dirac fermions is in exact agreement with the sum of Betti numbers $\sum ^{D}_{r=0} \beta _{r}$ for these manifolds. Indeed, the 4D lattice fermion on a torus has up to 16 zero-modes while the sum of Betti numbers of T4 is 16. This coincidence holds also for the D-dimensional hyperball and their direct-product space TD × Bd. We study several examples of lattice fermions defined on a certain discretized hypersphere (SD), and find that it has up to two exact zero-modes, which is the same number as the sum of Betti numbers of SD. From these facts, we conjecture the equivalence of the maximal number of exact Dirac zero-modes and the summation of Betti numbers under specific conditions. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory.

Ising field theory in a magnetic field: φ3 coupling at T > Tc

We study the “three particle coupling” {Gamma}_{11}^1left(xi right) , in 2d Ising Field Theory in a magnetic field, as the function of the scaling parameter ξ := h/(−m)15/8, where m ∼ Tc − T and h ∼ H are scaled deviation from the critical temperature and scaled external field, respectively. The “φ3 coupling” {Gamma}_{11}^1 is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so that m is negative. We suggest “standard analyticity”: {left({Gamma}_{11}^1right)}^2 , as the function of u := ξ2, is analytic in the whole complex u-plane except for the branch cut from – ∞ to – u0 ≈ – 0.03585, the latter branching point – u0 being associated with the Yang-Lee edge singularity. Under this assumption, the values of {Gamma}_{11}^1 at any complex u are expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of {Gamma}_{11}^1 near the Yang-Lee branching point, as well as its known asymptotic at u → +∞. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field.

Free field realization of the BMS Ising model

In this work, we study the inhomogeneous BMS free fermion theory, and show that it gives a free field realization of the BMS Ising model. We find that besides the BMS symmetry there exists an anisotropic scaling symmetry in BMS free fermion theory. As a result, the symmetry of the theory gets enhanced to an infinite dimensional symmetry generated by a new type of BMS-Kac-Moody algebra, different from the one found in the BMS free scalar model. Besides the different coupling of the u(1) Kac-Moody current to the BMS algebra, the Kac-Moody level is nonvanishing now such that the corresponding modules are further enlarged to BMS-Kac-Moody staggered modules. We show that there exists an underlying W (2, 2, 1) structure in the operator product expansion of the currents, and the BMS-Kac-Moody staggered modules can be viewed as highest-weight modules of this W-algebra. Moreover we obtain the BMS Ising model by a fermion-boson duality. This BMS Ising model is not a minimal model with respect to BMS3, since the minimal model construction based on BMS Kac determinant always leads to chiral Virasoro minimal models. Instead, the underlying algebra of the BMS Ising model is the W (2, 2, 1)-algebra, which can be understood as a quantum conformal BMS3 algebra.

Composing arbitrarily many SU(N) fundamentals

We compute the multiplicity of the irreducible representations in the decomposition of the tensor product of an arbitrary number n of fundamental representations of SU(N), and we identify a duality in the representation content of this decomposition. Our method utilizes the mapping of the representations of SU(N) to the states of free fermions on the circle, and can be viewed as a random walk on a multidimensional lattice. We also derive the large-n limit and the response of the system to an external non-abelian magnetic field. These results can be used to study the phase properties of non-abelian ferromagnets and to take various scaling limits.

New approach to KPZ models through free fermions at positive temperature

We give a short account of our new approach to study models in the Kardar–Parisi–Zhang universality class by connecting them to free fermions at positive temperature. Our ideas and methods are explained mainly for the semi-discrete directed polymer model due to O’Connell and Yor.

A scaling function for the particle entanglement entropy of fermions

Entanglement entropy under a particle bipartition generates complementary information to mode entanglement because it is sensitive to interactions and particle statistics at leading order and does not depend on any externally imposed length scale. In this paper, we investigate the particle entanglement entropy in a system of N interacting spinless lattice fermions in one spatial dimension by combining bosonization techniques with exact and approximate numerical methods. We introduce a general scaling form for the fermionic particle entanglement entropy captured by a shape function that enters as an extensive interaction-induced correction to a known free fermion result. A general asymptotic expansion in the total number of particles demonstrates that its form is robust for different values of the Rényi index and highlights how quantum correlations are encoded in the n-particle density matrix of a pure many-body quantum state.

Exceptional points in the Baxter-Fendley free parafermion model

Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a generalisation of this idea to \mathbb{Z}_NℤN-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but \mathcal{PT}𝒫𝒯-symmetric model directly generalising the Ising chain, which was later described by Fendley as a free parafermion spectrum. By extending the model’s magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general \mathcal{PT}𝒫𝒯-symmetric at these exceptional points, their proximity can have a profound impact on the model on the \mathcal{PT}𝒫𝒯-symmetric real line. Furthermore, in certain cases an exceptional point may appear on the real line (with negative field).

A hydrodynamic approach to Stark localization

When a free Fermi gas on a lattice is subject to the action of a linear potential it does not drift away, as one would naively expect, but it remains spatially localized. Here we revisit this phenomenon, known as Stark localization, within the recently proposed framework of generalized hydrodynamics. In particular, we consider the dynamics of an initial state in the form of a domain wall and we recover known results for the particle density and the particle current, while we derive analytical predictions for relevant observables such as the entanglement entropy and the full counting statistics. Then, we extend the analysis to generic potentials, highlighting the relationship between the occurrence of localization and the presence of peculiar closed orbits in phase space, arising from the lattice dispersion relation. We also compare our analytical predictions with numerical calculations and with the available results, finding perfect agreement. This approach paves the way for an exact treatment of the interacting case known as Stark many-body localization.

Boson–Fermion correspondence of the multi-component constrained mKP hierarchy

We generalize the constrained modified KP (mKP) hierarchy to the multi-component version whose tau function can be expressed as the vacuum expectation value of Clifford operators using free fermions. Meanwhile, we extend the relevant theory of the Boson–Fermion correspondence in single component constrained mKP hierarchy to the multi-component version. This leads us to obtain the rational and soliton solutions of the multi-component constrained mKP hierarchy utilizing the Boson–Fermion correspondence. Furthermore, we give five sets of differential Fay identities and two sets of difference Fay identities of the multi-component mKP hierarchy.

An optimal upper bound for the dilute Fermi gas in three dimensions

In a system of interacting fermions, the correlation energy is defined as the difference between the energy of the ground state and the one of the free Fermi gas. We consider N interacting spin 1/2 fermions in the dilute regime, i.e., ρ≪1 where ρ is the total density of the system. We rigorously derive a first order upper bound for the correlation energy with an optimal error term of the order O(ρ7/3) in the thermodynamic limit. Moreover, we improve the lower bound estimate with respect to previous results getting an error O(ρ2+1/5).