Carrollian Hohenberg-Kohn Mapping

• The Carrollian Hohenberg–Kohn mapping is a bijective relationship between external temporal fields and ground-state density and current trajectories, adapted for ultralocal Carrollian quantum systems.
• It introduces a framework analogous to 1D current-density functional theory, enabling effective Kohn–Sham schemes through universal energy functionals.
• The mapping distinctly captures temporal exchange–correlation effects, delineating bosonic bunching from fermionic antibunching in time-local dynamics.

The Carrollian Hohenberg-Kohn (HK) mapping is a foundational correspondence in the density-functional description of multiparticle Carroll-Schrödinger (CS) quantum systems in the regime where the Carroll limit applies ($c\rightarrow0$). Analogous to its role in conventional current-density functional theory (CDFT), the Carrollian HK mapping establishes a bijective relationship between external temporal fields and the ground-state trajectories of temporal density and current on equal-$x$ hypersurfaces. This construction, rooted in a formal isomorphism with 1D CDFT, enables the development of effective single-particle (Kohn-Sham) schemes and underpins the treatment of ultralocal quantum dynamics in Carrollian quantum mechanics (Rojas et al., 28 Nov 2025).

1. Carrollian Schrödinger Systems and Temporal Coordinates

In CS quantum systems, the evolution parameter is the spatial coordinate $x$, with physical configurations labeled by $t$. For $N$ particles, the quantum state $\Psi_0(t_1,\dots,t_N)$ is specified on equal-$x$ slices, and the generator of evolution incorporates both a temporal external “gauge” field $U(t)$, coupling to the Carroll energy operator $\widehat{E}_i = -i\hbar\partial_{t_i}$, and a scalar field $\Phi(t)$, associated with the conjugate momentum to $x$. Internal interactions $W_{\rm int}$ encode potential many-body temporal couplings, while the system ground state is assumed nondegenerate up to gauge.

On these equal-$x$ slices, observable quantities naturally defined are the one-body temporal density and current: $$n(t) = N \int dt_2\cdots dt_N \left|\Psi_0(t,t_2,...,t_N)\right|^2,$$

$$j(t) = \frac{1}{mc^2} N \int dt_2\cdots dt_N \Im\left\{\Psi_0^*\left(-i\hbar\partial_t\right)\Psi_0\right\}.$$

2. Statement of the Carrollian Hohenberg–Kohn Theorem

The Carrollian HK theorem asserts that, for a fixed internal interaction $W_{\rm int}$, the ground-state mapping

$$\{\Phi(t),\,U(t)\} \longleftrightarrow \{n(t),\,j(t)\}$$

is one-to-one, modulo an additive gauge in $\Phi$. Explicitly, knowledge of the ground-state pair $(n,j)$ uniquely determines the external field pair $(\Phi, U)$ up to gauge, and vice versa. This isomorphism follows by observing that the Carroll generator on equal-$x$,

$$\mathcal{H}^{(N)}_{[\Phi,U]} = \sum_{i=1}^N \frac{1}{2mc^2} \left(\widehat{E}_i - U(t_i)\right)^2 + \sum_{i=1}^N \Phi(t_i) + W_{\rm int}(t_1, ..., t_N),$$

is formally identical to a CDFT Hamiltonian for a 1D system.

3. Universal Carrollian Energy Functional and Its Structure

In analogy with standard DFT, the total ground-state energy decomposes into a universal part and an external-field part: $$E_{\rm gs}[\Phi, U] = F[n,j] + \int dt\, \Phi(t)\, n(t) + \int dt\, U(t)\, j(t).$$ The universal functional,

$$F[n,j] = \min_{\Psi \to (n,j)} \langle \Psi| T_{\rm carr} + W_{\rm int}|\Psi\rangle,\qquad T_{\rm carr} = \sum_{i=1}^N\frac{\widehat{E}_i^2}{2mc^2},$$

arises from contact-limit second-quantized kinetic and interaction terms, with the mean-field energy density

$$\mathcal{E}_{\rm carr}(n,j) = \int dt\, \left[ \frac{mc^2}{2}\left(\frac{j(t)}{n(t)}\right)^2 + \frac{\hbar^2}{8mc^2}\frac{(\partial_t n)^2}{n} + g\frac{n^2(t)}{2}\right]$$

plus higher-order exchange–correlation (XC) corrections. This functional is universal in that it is insensitive to the explicit form of $(\Phi, U)$.

4. Euler–Lagrange Equations and the Carrollian Kohn–Sham Construction

Variation of the total energy under fixed particle number,

$$\mathcal{E}[n,j] = F[n,j] + \int dt \Phi(t) n(t) + \int dt U(t) j(t) - \mu \int dt n(t),$$

where $\mu$ enforces $\int n = N$, yields the Euler–Lagrange conditions: $$\frac{\delta F}{\delta n(t)} + \Phi(t) - \mu = 0, \qquad \frac{\delta F}{\delta j(t)} + U(t) = 0.$$ A fictitious noninteracting Carrollian (“Kohn-Sham”) system defined by one-time orbitals $\{\varphi_k(t)\}$ reproduces the physical $(n,j)$ under

$$\mathcal{H}_s = \sum_k \int dt\, \varphi_k^*(t)\left[\frac{1}{2mc^2}(-i\hbar\partial_t - U_s(t))^2 + \Phi_s(t)\right]\varphi_k(t),$$

with occupation numbers $f_k$: $$n(t) = \sum_k f_k |\varphi_k(t)|^2,\qquad j(t) = \frac{1}{mc^2}\sum_k f_k \Im\left[\varphi_k^*(t)(-i\hbar\partial_t - U_s)\varphi_k(t)\right].$$ The KS potentials are determined by

$$\Phi_s(t) = \mu - \frac{\delta F}{\delta n(t)},\qquad U_s(t) = -\frac{\delta F}{\delta j(t)},$$

yielding the KS equations

$$\left[ \frac{1}{2mc^2} \left(-i\hbar\,\partial_t - U_s(t)\right)^2 + \Phi_s(t)\right]\varphi_k(t) = \varepsilon_k\varphi_k(t),$$

subject to self-consistency for $(n,j)$. All functionals, functional derivatives, Euler–Lagrange, and KS equations can be written in explicit LaTeX form as provided in (Rojas et al., 28 Nov 2025).

5. Ultralocality and Temporal Exchange–Correlation Structure

In the Carroll limit, spatial points decouple completely (ultralocality), eliminating any internal spatial interaction terms from $F[n,j]$. The entire many-body structure is encoded in time-domain interactions or in the momentum-space contact ($g n^2$) term. Consequences include:

• The Hartree term is strictly local in time.
• XC corrections in the contact limit are also instantaneous (time-local), though memory effects could enter in principle at higher order.
• For bosons, the temporal exchange–correlation hole is positive, enhancing $g^{(2)}(t,t)$ (“bunching”) and yielding an attractive time-local correlation potential.
• For fermions, the correlation-Pauli hole enforces $n^{(2)}(t,t) = 0$ (“antibunching”), producing a repulsive time-local XC barrier.

The direct equivalence to 1D CDFT permits importing standard XC approximations with suitable reinterpretation, although genuinely temporal XC kernels (e.g., those capturing Hanbury-Brown–Twiss-type correlations) may be developed de novo.

6. Explicit Functional Forms and Isomorphism with 1D CDFT

The Carrollian HK and KS structure is summarized by the following explicit formulas:

• Total energy functional:

$$E[n,j] = F[n,j] + \int dt\,\Phi(t)\,n(t) + \int dt\,U(t)\,j(t)$$

• Universal functional and its derivatives:

$$v(n,j;t) = \frac{\delta F}{\delta n(t)}, \qquad A(n,j;t) = \frac{\delta F}{\delta j(t)}$$

• Kohn-Sham equations:

$$\left[\frac{1}{2mc^2}(-i\hbar\partial_t - U_s(t))^2 + \Phi_s(t)\right]\varphi_k(t) = \varepsilon_k\varphi_k(t)$$

with

$\Phi_s(t)=\mu-v(n,j;t),\quad U_s(t)=-A(n,j;t)$

and

$$n(t) = \sum_k f_k |\varphi_k(t)|^2,\quad j(t) = \frac{1}{mc^2}\sum_k f_k \Im\left[\varphi_k^*(t)(-i\hbar\partial_t - U_s)\varphi_k(t)\right]$$

This formal structure follows from the isomorphism of the CS generator with a 1D CDFT Hamiltonian, with $t$ as the coordinate variable and $x$ as the evolution parameter.

7. Summary and Significance

The Carrollian Hohenberg–Kohn mapping rigorously extends the density-functional paradigm to ultralocal quantum systems evolving in “Carroll time.” The ground-state $(n(t),j(t))$ pair uniquely determines, and is uniquely determined by, the set of external temporal fields $(\Phi(t), U(t))$. The universal functional $F[n,j]$ encodes all contact-limit kinetic and temporal interaction effects, simplifying substantially due to the decoupling of spatial degrees of freedom. The self-consistent Kohn-Sham scheme provides a practical single-particle framework, with time-local Hartree and exchange–correlation structure reflecting either bunching or antibunching—contingent on bosonic or fermionic statistics—directly in the time domain. The Carrollian mapping thereby provides a powerful toolbox for the analysis and simulation of Carrollian quantum matter, leveraging parallels to and foundational results from 1D CDFT (Rojas et al., 28 Nov 2025).