---
title: Carrollian Hohenberg-Kohn Mapping
url: https://www.emergentmind.com/topics/carrollian-hohenberg-kohn-mapping
type: topic
---

# Carrollian Hohenberg-Kohn Mapping

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 [2512.00247].

## 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 [2512.00247].

## 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 [2512.00247].

Source: https://www.emergentmind.com/topics/carrollian-hohenberg-kohn-mapping