Non-Interacting Fermions with CUE Disorder

• The paper demonstrates an exact solution for the spectral form factor of non-interacting fermions under CUE disorder using combinatorial dimer methods.
• It reveals an exponential ramp in the spectral form factor, contrasting sharply with the linear ramp seen in interacting many-body chaotic systems.
• The matchgate circuit realization offers a practical experimental framework for simulating free-fermion dynamics and validating random matrix theory predictions.

Non-interacting fermions with circular unitary ensemble (CUE) disorder constitute a paradigmatic, exactly solvable model in which quantum many-body evolution is governed by random on-site phases sampled from Dyson’s unitary ensemble. The system is defined by $L$ spinless fermionic modes, evolving in discrete integer time $t$ via a many-body Floquet operator whose single-particle counterpart is Haar-random in $\mathrm{U}(L)$. The primary diagnostic for quantum chaos and random matrix universality is the spectral form factor (SFF), $K(L,t)$, whose exact solution can be obtained for arbitrary $L$ and $t$. This model captures both single-particle random matrix theory (RMT) statistics and the many-body dynamical properties of free fermions under maximal spectral disorder, revealing an exponential-in-time SFF ramp directly attributable to the structure of the many-body Hilbert space and the independence of many-body phases. The model further admits an experimental realization in quantum circuits via matchgates, leveraging the Jordan–Wigner correspondence between free fermions and number-conserving two-qubit gates.

1. System Definition and Disorder Ensemble

Consider $L$ spinless fermionic modes $c_j$, $j=1,\dots,L$. The Hilbert space is spanned by occupation basis states with $n_j=c_j^\dagger c_j\in\{0,1\}$. The dynamics at each discrete time step is specified by a single-particle unitary operator $u\in\mathrm{U}(L)$, whose eigenvalues $e^{i\theta_j}$ are drawn from the CUE joint density: $$P(\theta_1,\dots,\theta_L) = \frac{1}{(2\pi)^L\,Z_{L,2}} \prod_{1\le j<k\le L}|e^{i\theta_j}-e^{i\theta_k}|^2,$$ where $Z_{L,2}=(2\pi)^L/(L!)$. The corresponding many-body Floquet operator is diagonal,

$U = \exp\left(-i\sum_{j=1}^L \theta_j n_j\right).$

Time evolution for integer $t$ is $U^t = \exp(-i t\sum_j\theta_j n_j)$. This structure induces a correlated but tractable disorder landscape directly tied to circular-unitary statistics (Ikeda et al., 2024).

2. Spectral Form Factor: Definition and Reduction

The SFF is defined as

$$K(L,t) = \left\langle\left|\mathrm{Tr}\,[U^t]\right|^2\right\rangle_\mathrm{CUE} = \left\langle\left|\mathrm{Tr}\,\exp(-i\,tH)\right|^2\right\rangle,$$

with $H = \sum_j \theta_j n_j$. Rewriting the trace over the occupation basis gives

$$\mathrm{Tr}\,[U^t] = \prod_{j=1}^L(1+e^{-i t \theta_j}),$$

so that

$$K = 2^L\left\langle\prod_{j=1}^L[1+\cos(t\theta_j)]\right\rangle.$$

Expanding this expression yields a moment (dimer) expansion, which can be recast combinatorially: $$K(L,t) = 2^L \sum_{n=0}^{\lfloor L/2\rfloor} \left(-\frac{1}{4}\right)^n Q_n(L,t),$$ where $Q_n(L,t)$ counts configurations of $2n$ identical, hardcore particles (dimers) on a ring of $L$ sites, each paired at distance $\pm t$.

3. Exact Solution by Dimer Methods

To obtain an explicit closed form, write $L=Nt+r$ with $0\le r<t$, decomposing the ring into $t$ independent chains (congruence classes mod $t$) of lengths $N+1$ (for $r$ classes) and $N$ (for $t-r$ classes). The dimer counting formula for each chain is

$$K(M,1) = 2^M \sum_{n=0}^{\lfloor M/2\rfloor} \left(-\frac{1}{4}\right)^n \binom{M-n}{n} = M+1,$$

leading to the full SFF: $$K(L,t) = (N+1)^t \left(\frac{N+2}{N+1}\right)^{L - N t}, \qquad N = \left\lfloor\frac{L}{t}\right\rfloor,\ 0 < t \le L.$$ This formula holds for arbitrary system size $L$ and integer time $t$, encoding the detailed interplay between the structure of the many-body Hilbert space and the CUE-induced disorder (Ikeda et al., 2024).

4. Exponential Ramp and Single-Particle Universality

$K(L,t)$ exhibits exponential-in-$t$ growth. For divisors $1 = t_1 < t_2 < \cdots < t_M = L$ of $L$, for $t_j \le t < t_{j+1}$,

$$K(L,t) = K(L,t_j) \exp\left[\lambda_j (t-t_j)\right],$$

where the ramp rate is

$\lambda_j = (N+1)\ln(N+1) - N\ln(N+2).$

Whenever $t$ exactly divides $L$ ($L=Nt$), $K(L,t) = (N+1)^t$ and to leading order at early times, $\ln K \sim t\ln(L/t)$. Plotting $\ln K/L$ versus $t/L$ collapses the curves for all $L$, yielding a universal exponential ramp.

The exponential form is in sharp contrast to the linear ramp $K\sim t$ characteristic of interacting many-body RMT models. It is a consequence of the independence of many-body phases $\Theta(\{n_j\}) = \sum_j n_j \theta_j$ distributed over the $2^L$-dimensional Hilbert space: phase dephasing is uncorrelated at the many-body level for this non-interacting model. Nevertheless, single-particle eigenvalues exhibit CUE statistics, so the model realizes RMT universality at the single-particle level (Ikeda et al., 2024).

5. Matchgate Circuit Realization

Non-interacting fermions in one dimension with number-conserving nearest-neighbor gates (matchgates) are mapped to free-fermion dynamics via the Jordan–Wigner transformation. A two-qubit number-conserving matchgate on sites $i,i+1$ is represented as

$$V_{i,i+1} = \begin{pmatrix} e^{i\phi_0} & 0 & 0 & 0 \ 0 & u_{11} & u_{12} & 0 \ 0 & u_{21} & u_{22} & 0 \ 0 & 0 & 0 & e^{i\phi_3} \end{pmatrix},\quad \det u = e^{i(\phi_0+\phi_3)},$$

realized in fermionic language as $$\exp[-i(c_i^\dagger\,c_{i+1}^\dagger)\,h\,(c_i\,c_{i+1})^T]$$.

The circuit is constructed in brick-layered form:

$$U_{\mathrm{layer}} = \prod_{m=1}^{L/2} V_{2m,2m+1} \prod_{m=1}^{L/2} V_{2m-1,2m},$$

with random parameters $\phi_0\in[-\pi,\pi)$, $u\in \mathrm{U}(2)$ Haar-distributed, and $\phi_3$ set by $\det u$. A depth-$d$ circuit is $$U_{\rm CUE}^{(d)} = \prod_{\tau=1}^d U_{\mathrm{layer}}^{(\tau)}$$, converging for $d\to\infty$ to a random propagator whose SFF matches the exact result above.

Experimental realization is feasible with controlled insertion of $V$ on ion-trap or superconducting platforms. $K(t)$ can be measured, for instance, using a Hadamard test on an ancilla qubit, or randomized measurement protocols (Ikeda et al., 2024).

6. Context, Implications, and Connections

Non-interacting fermions with CUE disorder bridge two major themes: the exactly solvable dynamics of free fermions and the universal statistics of random matrix ensembles. The model provides a concrete setting in which single-particle RMT universality is realized, while the many-body SFF reflects the absence of interactions via its exponential, not linear, ramp. This stands in contrast to interacting systems, where RMT directly controls late-time spectral statistics and SFF behavior.

The matchgate circuit formulation enables scalable quantum simulation and benchmarking of RMT phenomena in platforms admitting free-fermion mappings. The combinatorial dimer solution illustrates how statistical structure in the disorder ensemble translates into tractable, universal dynamics for non-interacting many-body systems. This framework is extensible: for disorder drawn from Dyson’s COE or CSE, the disorder averaging reduces to a combinatorial problem addressable by transfer-matrix techniques, reflecting a general principle of solvability at the single-particle level in free systems with random matrix disorder (Ikeda et al., 2024).