Sign-problem-resilient singular-value probe in determinant quantum Monte Carlo

Abstract: The sign problem limits determinant quantum Monte Carlo studies of strongly correlated fermion systems. In the spin-channel Hubbard-Stratonovich decoupling, spin correlations are exactly related to auxiliary-field correlations. This relation implies that an antiferromagnetic transition reorganizes auxiliary-field configurations and thereby changes the statistical structure of the resulting fermion matrices. We use the adjacent gap ratio of low-lying singular values of the space-time fermion matrix to probe interaction-driven transitions in two half-filled honeycomb-lattice Hubbard models. In the sign-free honeycomb Hubbard model, the statistic tracks the established transition from a Dirac semimetal to an antiferromagnetic Mott insulator. In the complex-weight Haldane-Hubbard model, the transition-sensitive feature remains visible in the phase-quenched reference ensemble and occurs near previous estimates of the transition. Moreover, phase reweighting only weakly modifies the gap ratio over the regimes investigated, despite the rapid suppression of the average phase. These results establish singular-value statistics as a sign-problem-resilient probe of interaction-driven transitions in determinant quantum Monte Carlo.

• The paper introduces adjacent gap ratios of low-lying singular values from DQMC space-time fermion matrices, providing an interpretable transition diagnostic that avoids direct phase reweighting.
• Synthetic and model benchmarks show that the signal arises from auxiliary-field reorganization toward staggered order—not coupling-strength growth alone—and tracks the honeycomb AFM transition and Haldane-Hubbard crossover.
• The phase-quenched diagnostic remains reliable within accessible system sizes, including a zero-temperature Haldane-Hubbard estimate of 0.38412(71), near the Poisson value, but depends on using an HS channel matched to the targeted order.

Motivation and central question

Determinant quantum Monte Carlo (DQMC) provides an unbiased route to strongly correlated fermions, but its applicability is curtailed by the sign (or phase) problem: when the configuration weights $W[x]=\prod_\sigma\det M^\sigma[x]$ become negative or complex, sampling must proceed from the magnitude $|W[x]|$, and physical averages require reweighting by the sign or phase factor, whose average decays exponentially with inverse temperature and system size. Prior work has nevertheless shown that the phase-quenched reference ensemble retains transition information in various diagnostics, including Green's-function matrices fed to neural networks (Pataki, 2017), Hamming distances between auxiliary-field configurations (Han et al., 2022), and the average sign or phase itself. The paper by Chen and Mondaini asks whether a simpler, interpretable matrix statistic—the adjacent gap ratio of low-lying singular values of the space-time fermion matrix—can serve as a transition probe that survives reweighting.

The key structural observation motivating this probe is the exact relation between spin-channel Hubbard-Stratonovich (HS) correlations and physical spin correlations,

$$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$

which implies that an antiferromagnetic (AFM) transition reorganizes the auxiliary-field configurations and hence the statistical structure of the fermion matrices $\mathcal M^\sigma[x]$. Since the determinant magnitude equals the product of singular values while the gap ratio probes their local spacing correlations, a natural separation arises: reweighting acts through the determinant phase, whereas the gap ratio depends only on the spectrum of $\mathcal M^\dagger\mathcal M$.

Models and diagnostic

The authors study two half-filled honeycomb-lattice models with onsite repulsion $U$: the sign-problem-free Hubbard model ($t_2=0$), which transitions from a Dirac semimetal to an AFM Mott insulator at $U_c/t_1\approx 3.7$–$3.9$, and the Haldane-Hubbard model ($t_2/t_1=0.2$, flux $|W[x]|$0), whose complex NNN hopping produces complex weights and a Chern-insulator-to-Mott-insulator transition previously estimated at $|W[x]|$1. Rather than forming the ill-conditioned long product $|W[x]|$2, they analyze the sparse block-cyclic space-time matrix $|W[x]|$3, whose determinant coincides with that of $|W[x]|$4, computing its lowest singular values via distributed PETSc/SLEPc routines at cost well below a full SVD.

The statistic is the adjacent gap ratio

$|W[x]|$5

averaged over the lowest $|W[x]|$6 singular values and over the ensemble. Because the real kinetic matrix of the honeycomb model places $|W[x]|$7 in non-Hermitian class AI (BDI under Hermitization, benchmark $|W[x]|$8) while complex hopping places the Haldane case in class AIII (benchmark $|W[x]|$9), random-matrix theory supplies sharp baselines; the Poisson value $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$0 marks uncorrelated levels.

Controlled synthetic ensembles isolate the mechanism

To disentangle two effects—increasing HS coupling amplitude $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$1 versus reorganization of the field distribution toward staggered order—the authors construct synthetic ensembles with the same block-cyclic structure, where binary fields are drawn relative to a staggered template with bias $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$2, independently of $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$3. Two trajectories yield an unambiguous conclusion: along the amplitude-only path ($$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$4 fixed), the gap ratio stays near its random-matrix benchmark for all $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$5; along the coupled path ($$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$6), it departs strongly in both symmetry classes. Coupling-strength growth alone does not produce the spectral response; reorganization of the field distribution toward staggered order is essential.

Benchmark: honeycomb and Haldane-Hubbard models

Applying the same comparison within DQMC confirms this interpretation. With independently randomized fields (same $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$7 and $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$8), the gap ratio approaches its benchmark and saturates; with importance-sampled fields it departs precisely across the literature critical interval and decreases on the strong-coupling side, tracking the semimetal-to-AFM-Mott transition. For the Haldane-Hubbard model sampled entirely from the phase-quenched measure $$\bigl\langle x_{i,\tau}x_{j,\tau'}\bigr\rangle_{\rm phys} = \tanh^2\lambda \,\bigl\langle \hat m_i(\tau)\hat m_j(\tau')\bigr\rangle_{\rm phys},$$9, the downward crossover occurs with its steepest region overlapping the reported transition range $\mathcal M^\sigma[x]$0, and an imaginary-time extrapolation at $\mathcal M^\sigma[x]$1—inside the same-cluster exact-diagonalization level-crossing interval $\mathcal M^\sigma[x]$2 on the 18-site cluster—yields $\mathcal M^\sigma[x]$3, essentially the Poisson value. The transition-sensitive signal therefore persists in the $\mathcal M^\sigma[x]$4 limit and is not a finite-temperature artifact, even though no phase reweighting is performed.

Two analytic limits in the Supplemental Material explain the mechanism: (i) for a perfectly staggered, time-uniform field on a bipartite lattice, the lowest singular values scale as $\mathcal M^\sigma[x]$5, controlled by single-particle states nearest the band center and the smallest antiperiodic momentum $\mathcal M^\sigma[x]$6, so field organization enters most directly at the hard edge; and (ii) time-uniformity generates exact $\mathcal M^\sigma[x]$7 doublets whose weak splitting suppresses adjacent gap ratios as $\mathcal M^\sigma[x]$8. Increasing temporal coherence of the HS field (quantified by zero-frequency autocorrelation $\mathcal M^\sigma[x]$9 and DEFLATE-based compressibility $\mathcal M^\dagger\mathcal M$0) strengthens this near-doublet structure. Window-size audits confirm that the choice $\mathcal M^\dagger\mathcal M$1 is conservative and robust.

Phase-reweighting bias

Whether the reference-ensemble average faithfully proxies the physical one is governed by the covariance identity

$\mathcal M^\dagger\mathcal M$2

Although $\mathcal M^\dagger\mathcal M$3 of a fixed configuration contains no phase information, the two quantities can be statistically correlated since both derive from the same auxiliary-field configuration. Numerically, the average phase decreases rapidly with $\mathcal M^\dagger\mathcal M$4, yet the covariance remains smaller than the average phase by roughly an order of magnitude, so the reweighting correction shows no resolved systematic growth over the accessible sizes. Within statistical resolution, the phase-quenched gap ratio is thus a quantitatively reliable proxy for locating the Haldane-Hubbard crossover. This claim carries caveats: corrections are shown only for regimes where the denominator is statistically resolved, and the resolution-limited nature of the result means coincidence at larger sizes or lower temperatures is not established.

Representation dependence and limitations

A substantive limitation is acknowledged directly: the diagnostic is not independent of the HS channel. A negative control using the SU(2)-invariant decomposition on the same honeycomb transition shows no transition-tracking response—consistent with that channel encoding charge rather than Ising-spin correlations, so no staggered, $\mathcal M^\dagger\mathcal M$5-uniform field organization develops and the doublet mechanism does not apply. The probe's sensitivity therefore requires matching the decoupling channel to the ordering channel, which the authors elevate to a design principle via channel-matched HS decompositions relating auxiliary-field correlators directly to targeted operator correlations. Further open questions include whether the reference and physically reweighted averages coincide asymptotically in the thermodynamic limit, and how the hard-edge nonuniformity of randomized fields affects small-window statistics.

Conclusion

This work establishes low-lying singular-value gap ratios of DQMC space-time fermion matrices as an interpretable, sign-problem-resilient probe of interaction-driven transitions. Synthetic ensembles demonstrate that spectral response requires field-distribution reorganization rather than coupling-amplitude growth alone; benchmarks confirm the statistic tracks the honeycomb AFM transition and locates the Haldane-Hubbard crossover without phase reweighting, persisting to zero temperature; and a covariance identity bounds the reweighting bias explicitly. The principal qualifications—HS-channel specificity and finite-size-limited evidence for reweighting robustness—are stated plainly by the authors, framing a concrete program of channel-tailored matrix diagnostics for ordered phases beyond the reach of conventional reweighted observables.