Loschmidt Chirality Amplitude in Quantum Dynamics

• Loschmidt chirality amplitude is a quantum construct that encodes handedness through complex phase structures arising from non-commuting operator dynamics and discrete path integrations.
• In practice, generalized amplitude techniques allow a joint spectral resolution in systems like Hubbard clusters, revealing energy and double occupancy distributions that signal phenomena such as impact ionization.
• The discrete checkerboard lattice approach for Weyl fermions employs phase factors (i^(±T)) to capture the net chirality of quantum paths, ensuring fidelity to continuum behavior without fermion doubling.

Loschmidt chirality amplitude denotes a class of amplitude constructions in which handedness enters the phase structure of quantum evolution. In the provided literature, chirality appears in two distinct but related ways: in a generalized Loschmidt amplitude for non-commuting operators, where the amplitude is generally complex and can encode quantum “chirality” as the handedness or asymmetry of quantum paths in operator space; and in a discrete path integral for Weyl fermions, where the path amplitude $i^{\pm T}\,3^{-B/2}\,2^{-N}$ counts right-handed minus left-handed bends through a chirality-dependent phase (Watzenböck et al., 2021, Foster et al., 2016).

1. Loschmidt amplitudes and joint spectral resolution

The standard Loschmidt amplitude for a quantum state $|\psi\rangle$ and Hamiltonian $\hat H$ is

$$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$

Its Fourier transform decomposes the state $|\psi\rangle$ with respect to the energy eigenstates of $\hat H$, yielding a “probability distribution” over energies (Watzenböck et al., 2021).

The generalized two-operator construction extends this single-observable resolution to a joint one. For two Hermitian operators $\hat A$ and $\hat B$, the generalized Loschmidt amplitude for a given state $|\psi\rangle$ is defined as

$$L^{|\psi\rangle}_{AB}(\bar\alpha,\bar\beta)\equiv \langle \psi|e^{-i\bar\alpha \hat A}e^{-i\bar\beta \hat B}|\psi\rangle.$$

Its double Fourier transform yields

$|\psi\rangle$0

where $|\psi\rangle$1 and $|\psi\rangle$2 are eigenstates of $|\psi\rangle$3 and $|\psi\rangle$4. The construction therefore resolves a state jointly over two observables rather than only over energy. In the Hubbard-model application, this joint resolution is used to identify what portion of the state overlaps with a given energy eigenvalue and a given double occupancy.

A key formal point is that the result is generally complex when $|\psi\rangle$5 and $|\psi\rangle$6 do not commute. For interpreting the result as a probability distribution, for example in the computation of expectation values, the real part is what matters. This separates the generalized amplitude from a naïve probabilistic joint distribution and places it closer to a quasidistribution-like diagnostic of operator-resolved dynamics.

2. Chirality from non-commutativity

In the generalized Loschmidt framework, chirality enters through non-commutativity. The paper on photoexcitations in the Hubbard model notes that when $|\psi\rangle$7 and $|\psi\rangle$8 do not commute, the two-operator Loschmidt amplitude is in general complex, and this complexity can encode quantum “chirality,” described there as the handedness or asymmetry of quantum paths in operator space (Watzenböck et al., 2021).

This point is interpretive rather than the main computational target of that work. The real part is used for probability-like interpretation, and the imaginary part vanishes when calculating observable averages. At the same time, Appendix A emphasizes that non-commutativity, and thus chirality, in principle leaves its mark on the detailed structure of the amplitude. The generalized construction is therefore not restricted to scalar overlap diagnostics; it retains operator-order information through its complex phase structure.

The same work explicitly states that chirality itself is not systematically analyzed there. Nevertheless, the framework is described as amenable to studying quantum evolution’s chiral aspects because it gives access to time-ordered, or “operator-path-ordered,” amplitudes. This suggests a broader conceptual role for Loschmidt-type amplitudes beyond return probabilities: they can organize out-of-equilibrium many-body dynamics in a way that preserves directional information in operator space.

3. Discrete checkerboard path amplitudes

A more explicit chirality amplitude appears in the discretization of the Weyl equation on a time-diagonal, hypercubic spacetime lattice with null faces. In that construction, the step speed is set to be three times the speed of light, so that the discrete light cone marginally encloses the continuum light cone and satisfies the Courant stability condition. The spatial step directions are given by four unit vectors $|\psi\rangle$9, the vertices of a regular tetrahedron, and the spacetime step vectors are

$\hat H$0

(Foster et al., 2016).

For right-handed Weyl spinors, the discrete one-step evolution rule is

$\hat H$1

with projection operator

$\hat H$2

For left-handed spinors, the orthogonal projector is used,

$\hat H$3

Iteration yields a path integral for the retarded propagator, with matrix path amplitude proportional to the product of projection operators. For a path of $\hat H$4 steps, the matrix amplitude is

$\hat H$5

and for fixed initial and final spinors $\hat H$6 and $\hat H$7, the scalar amplitude is

$\hat H$8

With a specific choice of phases for the spinors, successive bends in the path yield a factor of $\hat H$9, with the sign depending on the handedness of the turn. The resulting path amplitude is

$$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$0

where $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$1 is the number of steps, $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$2 is the number of bends, and $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$3 is the net number of right-handed minus left-handed bends. The $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$4 sign applies to right-handed chirality and the $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$5 sign to left-handed chirality. In the supplied account of the paper, this $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$6 factor is identified as a Loschmidt chirality amplitude: a path-dependent quantity that records the net handedness traversed in the sequence of bends.

4. Chirality flips, conjugation, and lattice-fermion structure

In the checkerboard construction, chirality is not only encoded kinematically through the use of $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$7 or $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$8; it is also dynamical when mass terms are included. A Dirac mass $$L^{|\psi\rangle}(\tau)\equiv \langle \psi|e^{-i\tau \hat H}|\psi\rangle.$$9 introduces the amplitude $|\psi\rangle$0 to flip chirality in any given time step $|\psi\rangle$1. The discrete Dirac equations couple right- and left-chiral propagators through these flip amplitudes, so the total path amplitude becomes a sum over all possible chirality-flip histories (Foster et al., 2016).

A Majorana mass is incorporated analogously, but the discrete update inserts charge conjugation rather than a simple chirality flip. The amplitude to undergo conjugation at a step is likewise proportional to $|\psi\rangle$2 in the detailed exposition. The path sum then alternates between regular propagation and its conjugate, according to where the conjugation operator acts. In this sense, the chirality amplitude is embedded in a broader combinatorics of spin projection, chirality exchange, and conjugation.

An important structural result of the same lattice scheme is that fermion doubling does not occur. The analysis reported in the paper finds no zero-frequency solutions for large wavevectors, and the only real-frequency solutions match the continuum but lie on a discrete set of null directions. The elimination of doublers is therefore tied to the time-diagonal tetrahedral lattice and the specific update rule rather than to a standard local-action discretization.

5. Hubbard-model realization: energy, double occupation, and impact ionization

The generalized Loschmidt amplitude is applied to photoexcitations in small Hubbard clusters of up to 12 sites, with the time-dependent electromagnetic field introduced through a Peierls substitution and the time evolution calculated by exact diagonalization with commutator-free Magnus integrators (Watzenböck et al., 2021). In this setting, the relevant specialization takes

$|\psi\rangle$3

where $|\psi\rangle$4 is the time-independent Hamiltonian before the pulse and

$|\psi\rangle$5

is the double-occupancy operator.

The corresponding joint amplitude is

$|\psi\rangle$6

where $|\psi\rangle$7 and $|\psi\rangle$8 are eigenstates of $|\psi\rangle$9 and $\hat H$0, respectively. This quantity acts as a dynamic map of how spectral weight is redistributed between sectors of given energy and double occupancy after a pulse.

Its main physical use in the paper is the analysis of impact ionization, defined there as the creation of additional doublon-holon pairs from a high-energy photoexcitation. The generalized amplitude resolves which many-body energy eigenstates are responsible for impact ionization and which show pronounced changes in the double occupation and spin energy. In particular, the rise of double occupation in fixed-energy slices,

$\hat H$1

directly signals impact ionization at those energies.

The analysis further reports that, for certain geometries, impact ionization arises predominantly in energy sectors corresponding to two-photon excited states, not from single-photon excitations. Conversely, in geometries or parameter regimes where impact ionization does not occur, the generalized amplitude shows no net transfer toward higher double occupancy in any energy sector. The same framework is also applied to energy and spin-correlation energy, using the Heisenberg Hamiltonian $\hat H$2, and this analysis reveals that the loss of spin energy is of little importance for impact ionization.

6. Scope, interpretation, and related results

The two source contexts delimit the scope of the notion. In the Hubbard-cluster study, a Loschmidt-type amplitude is generalized so that it can simultaneously resolve two observables; chirality enters because non-commuting operators produce a complex amplitude whose structure retains operator-order information (Watzenböck et al., 2021). In the checkerboard Weyl construction, chirality is explicit and geometric: right-handed and left-handed propagation are implemented by complementary projectors, and the phase $\hat H$3 records the net handedness of bends along a path (Foster et al., 2016).

A common misconception is to treat these as the same formal object. They are not identical constructions. One is a joint spectral decomposition of a quantum state with respect to two Hermitian operators; the other is a discrete path amplitude for a retarded propagator. Another misconception is to read the complex generalized Loschmidt amplitude directly as an ordinary probability distribution. The supplied analysis explicitly states that, when the operators do not commute, the result is generally complex, and the real part is the part used for probability-like interpretation.

At the same time, the two constructions are not unrelated. A plausible implication is that both instantiate the same broad theme: handedness can be encoded in amplitude phases, either through operator ordering in many-body dynamics or through oriented bends in lattice paths. The checkerboard formulation makes this explicit through $\hat H$4, the net number of right-handed minus left-handed bends. The Hubbard formulation leaves chirality in a more latent form, as a consequence of non-commutativity in the two-operator Loschmidt amplitude.

Within the Hubbard-model study, the generalized Loschmidt analysis is accompanied by an additional result on transport diagnostics: for one-dimensional chains, the optical conductivity has a characteristic peak structure originating solely from vertex corrections. This places the amplitude analysis within a broader nonequilibrium characterization of photoexcited Hubbard systems, where energy redistribution, double occupation, spin energy, and optical response are examined together. The paper further presents the generalized Loschmidt amplitude as useful beyond double occupation alone, since the same formalism can resolve joint distributions involving spin-correlation energy and, by comparison with kinetic and potential energy, track how photoexcitation energy is distributed among different degrees of freedom.