Quantum Critical Points and the Sign Problem

Over the last several decades, quantum Monte Carlo (QMC) simulations have provided great insight into challenging strong correlation problems in chemistry [1, 2], condensed matter [3, 4], nuclear [5], and high energy physics [6]. In all these areas, however, the sign problem (SP), which occurs when the probability for specific quantum configurations in the importance sampling becomes negative, significantly constrains their application. Solving, or at least mitigating, the SP is one of the central endeavors of computational physics. The extent and importance of the effort is indicated by the many proposed solutions, and their continued development over the last three decades — see the Supplemental Materials (SM) for a review [7].

Despite enormous effort, the SP remains unsolved. In fact, the lack of progress is one of the main driving forces behind a number of large-scale efforts, including the quest for quantum emulators [8, 9, 10] as well as quantum computing itself [11, 12]. One of the most fundamental mysteries concerns the possible link between the sign problem and the underlying physics of the Hamiltonian being investigated.

Here, instead of challenging this NP-hard problem [13], or proposing solutions that can partially ameliorate its behavior [14, 15] we show that there is a clear quantitative connection between the behavior of the average sign $\langle{\cal S}\rangle$ in the widely used Determinant Quantum Monte Carlo (DQMC) method and several quantum phase transitions: that of the semimetal to antiferromagnetic Mott insulator (AFMI) of Dirac fermions in the spinful [SU(2)] honeycomb-Hubbard Hamiltonian [16, 17], the band to correlated insulator transition  [18, 19, 20], and charge density wave transitions of spinless [U(1)] fermions on a honeycomb lattice [21, 22]. In the first case, simulations at half-filling, where the quantum critical point (QCP) occurs, are SP free. We introduce a small doping $\mu$ and show, in the limit $\mu\to 0^{+}$ at temperature $T\to 0$, that $\langle{\cal S}\rangle$ evolves rapidly as we tune through the QCP.

Our second illustration, the ionic Hubbard model, has a SP even at half-filling. Here the average sign undergoes an abrupt drop at the band-insulator (BI) to correlated metal (CM) transition. The third case, spinless fermions on a honeycomb lattice, also has a semimetal to (charge) insulator transition. Its interest relies on the existence of a SP-free approach. Studying it with a method which contains an ‘unnecessary’ SP lends insight into the key question of the algorithm dependence of links between the SP and the physics of model Hamiltonians.

These three discussions establish a link between known physics of the models and the fermion sign. Having made that connection, we then turn to the iconic square lattice Hubbard model whose physics has not been conclusively established. We find that the onset of the SP occurs in a dome-shaped region of the filling-temperature phase space underneath that of the pseudogap (PG) physics. The SP is well-enough controlled in the PG phase to obtain reliable results for various observables, including the pairing correlations in various channels, exhibiting dominant enhancement for $d$-wave symmetry. Because it onsets exponentially in inverse temperature, the SP provides a rather sharp demarcation of the regime, which mimics the superconducting dome of the cuprates [23]. Although the SP prevents DQMC from resolving a signal of a $d$-wave transition, the ground work established for the honeycomb lattice and BI-CM models suggests that this sign problem dome might be linked to the onset of a superconducting phase. In the conclusions we will make connections which will further corroborate the generality of our results to other models and QMC methods.

The Sign Problem; Model and Methodology

The origin of the sign problem can be understood in two related classes of algorithms, ‘world-line’ Quantum Monte Carlo (WLQMC) [24] and Green’s function Quantum Monte Carlo (GFQMC) [25, 26], by considering Feynman’s path integral approach, which provides a mapping of quantum statistical mechanics in $D$ dimensions to classical statistical mechanics in $D+1$ dimensions. Paralleling Feynman’s original exposition for the operator $e^{-{\rm i}\hat{H}t/\hbar}$, the imaginary time evolution operator $e^{-\beta\hat{H}}$ is subdivided into $L_{\tau}$ incremental pieces $\hat{U}_{\Delta\tau}=e^{-\Delta\tau\hat{H}}$, where $L_{\tau}\,\Delta\tau=\beta$, the inverse temperature. Complete sets of states $\hat{\cal I}_{\tau}=\sum_{S_{\tau}}|S_{\tau}\rangle\langle S_{\tau}|$ are introduced between each $\hat{U}_{\Delta\tau}$ so that the partition function ${\cal Z}={\rm Tr}\,e^{-\beta\hat{H}}$ becomes a sum over the classical degrees of freedom associated with the spatial labels of each $\hat{\cal I}_{\tau}$, and also an additional imaginary time index denoting the location $\tau=1,2,\ldots,L_{\tau}$ of $\hat{\cal I}_{\tau}$ in the string of operators $\hat{U}_{\Delta\tau}$. The quantity being summed in the calculation of ${\cal Z}$ is the product of matrix elements $\langle S_{\tau}|\hat{U}_{\Delta\tau}|S_{\tau+1}\rangle$.

In such WLQMC/GFMC methods, the SP arises when $\prod_{\tau}\langle S_{\tau}|\hat{U}_{\Delta\tau}|S_{\tau+1}\rangle<0$. Negative matrix elements are unavoidable for itinerant fermionic models in $D>1$ because their sign depends on the number of fermions intervening between two particles undergoing exchange, and hence changes as the particle positions are updated. The basis dependence of the sign problem is apparent by considering intermediate states $|S_{\tau}\rangle$ chosen to be eigenstates $|\phi_{\alpha}\rangle$ of $\hat{H}$. In that case the matrix elements are just $e^{-\Delta\tau E_{\alpha}}$ and hence are trivially positive definite. Of course, since the eigenstates of $\hat{H}$ are unknown, this is not a practical choice in any non-trivial situation. Moreover, the SP can generally be avoided for bosonic or spin models as long as the lattice is bipartite. Nonetheless, even bosonic and spin Hamiltonians can have negative matrix elements on frustrated geometries [27], especially for antiferromagnetic models, emphasizing the SP is not solely a consequence of Fermi statistics.

‘Auxiliary field’ Quantum Monte Carlo (AFQMC) algorithms [28, 29, 30] typically have a much less severe SP than WLQMC [31, 7]. They are based on the observation that the trace of an exponential of a quadratic form of fermionic operators can be done analytically, resulting in the determinant of a matrix of dimension set by the cardinality of fermionic operators. The determinant is the product $\prod_{j}\big{(}1+e^{-\beta\varepsilon_{j}}\big{)}$, where $\varepsilon_{j}$ are the non-interacting energy levels, and is always positive.

If interactions are present, quartic terms in $\hat{H}$ are reduced to quadratic ones with a Hubbard-Stratonovich (HS) transformation. The trace of the resulting product of exponentials of quadratic forms can be performed, but now they each depend on a different, i.e. imaginary-time dependent, auxiliary field. The resulting determinant is no longer guaranteed to be positive; the consequence is the SP since the HS field needs to be sampled stochastically in order to compute operator expectation values.

In AFQMC, the trace over fermionic degrees of freedom is done for all species (i.e. all spin and orbital indices $\alpha$). If there is no hybridization between different $\alpha$, each trace gives an individual determinant. In some situations, particle-hole, time-reversal, or other symmetries [32, 33, 34] impose a relation between the determinants for different $\alpha$’s, and as a consequence the negative determinants always come in pairs. Low temperature (ground-state) properties can be accessed in such ‘sign problem free’ cases, and a host of interesting quantum phase transitions has been explored [35, 36, 37, 38]. If such a partnering does not occur, a reasonable rule of thumb is that the average sign $\langle{\cal S}\rangle$ is sufficiently bounded away from zero with measurements that exhibit sufficiently small error bars for $T\gtrsim W/20\,$-$\,W/40$, at intermediate interaction strengths (of the same order as the bandwidth $W$) ¹¹1This is just a rough guideline; the precise onset of the SP is determined by lattice geometry, doping (chemical potential), and interaction strength. A catalog of the SP in DQMC for the single band Hubbard model in different situations is given in Ref. [59]..

The DQMC methodology [28, 29] we use is a specific implementation of AFQMC. We employ the discrete HS transformation introduced by Hirsch [40] and choose the Trotter discretization $\Delta\tau$ such that systematic errors in $\langle{\cal S}\rangle$ and other observables are of the same order as statistical sampling errors. (See [7] for additional details.)

We mostly consider models in which two (spin) species of itinerant electrons hop on a lattice with an on-site repulsion, i.e., variants of the Hubbard Hamiltonian,

Here $\hat{c}^{\dagger}_{j\sigma}(\hat{c}^{\phantom{\dagger}}_{j\sigma})$ are creation (destruction) operators at site $i$ with spin $\sigma$ and $\hat{n}_{i\sigma}=\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\phantom{\dagger}}_{i\sigma}$ is the number operator. In Sec. I, $i,j$ are near-neighbor sites on a honeycomb lattice, with $t_{ij}=t$. As a consequence of particle-hole symmetry (PHS), $\mu_{i}=0$ corresponds to half-filling, and $\rho=\langle\hat{n}_{i\sigma}\rangle=1/2$, for arbitrary $U$ and temperature $T$. In Sec. II, we consider a $t_{ij}=t$ square lattice with $\mu_{i}=+\Delta$ on one sublattice and $\mu_{i}=-\Delta$ on the other, a situation which has a SP even at half-filling, but which is mild enough to allow its phase diagram to be established with reasonable reliability. Sec. III concerns a single species model with interactions between fermions on neighboring sites, notable because a SP-free QMC formulation exists [22, 41].

All these models have QCPs which have been located to fairly high precision and so serve as testbeds for demonstrating that the average sign can be used as an alternate means to study the onset of quantum criticality. In our final investigation, Sec. IV, we consider the doped, spinful, square lattice Hubbard model, much of whose low temperature physics remains shrouded in mystery. We correlate the behavior of the SP with some of its properties at intermediate temperature, and then describe what might be inferred concerning one the most elusive puzzles–the presence of a low temperature superconducting dome.

I. Semimetal to AFMI on a Honeycomb Lattice

On a honeycomb lattice (Fig. 1A), the $U=0$ Hubbard Hamiltonian has a semi-metallic density of states which vanishes linearly at $E=0$. Its dispersion relation $E({\bf k})$ has Dirac points in the vicinity of which the kinetic energy varies linearly with momentum. Unlike the square lattice that displays AF order for all $U\neq 0$, the honeycomb Hubbard model at $T\to 0$ remains a semimetal for small nonzero $U$, turning to an AF insulator only for $U$ exceeding a critical $U_{c}$. Early DQMC and series expansion calculations estimated $U_{c}\sim 4\,t$ [42], with subsequent studies [16, 17] yielding the more precise value $U_{c}/t=3.869$.

The upper panel of Fig. 1B gives $\langle{\cal S}\rangle$ in the $U$-$T$ plane. By introducing a small non-zero $\mu=0.1$, we can induce a SP which begins to develop at $T/t\sim 0.1$. As $T$ is lowered further the average sign deviates from $\langle{\cal S}\rangle=1$ in a relatively narrow window of $U/t$ close to the known $U_{c}$. In turn, we show the $\langle{\cal S}\rangle$ on the $U-\mu$ plane at fixed $T/t=0.05$ in the lower panel of Fig. 1B. For large $\mu$ the sign is small for a broad swath of interaction values. As $\mu$ decreases this region pinches down until it terminates close to $U_{c}$; the dashed white line displays the minimum $\langle{\cal S}\rangle$ in the relevant range. In both panels, the behavior of the average sign outlines the quantum critical fan that extends above the QCP.

Figure 1C shows a finite size extrapolation of $\langle{\cal S}\rangle$ in the $1/L-U$ plane, where $L$ is the linear lattice size. Just as $\langle{\cal S}\rangle$ worsens with increasing $\beta$, it is also known to deviate increasingly from $\langle{\cal S}\rangle=1$ with growing $L$ [29]. What these data further indicate is that the extrapolation $L\rightarrow\infty$ clearly reveals $U_{c}$ in the presence of a small chemical potential. So far, we have exclusively used $\langle{\cal S}\rangle$ in locating $U_{c}$. Original investigations employed more ‘traditional’ (and more physical) correlation functions such as the AF structure factor and conductivity. For comparison to the evolution of $\langle{\cal S}\rangle$, Fig. 1D shows one example, the rate of change of the double occupancy $D$, again in the $1/L-U$ plane. A peak in $-dD/dU$ indicates where local moments $\langle m^{2}\rangle$ are growing most rapidly. The similarity between Figs. 1C and 1D emphasizes how $\langle{\cal S}\rangle$ is tracking the physics of the model in a way remarkably similar to $\langle m^{2}\rangle$. The combination of the three limits, $\mu,\beta,L$, unequivocally points out the QCP location; the SM [7] contains further discussion, and other observables.

II. Ionic Hubbard BI to AF Transition

Among the different types of non-conducting states are ‘band insulators’ (BI), in which the chemical potential lies in a gap in the non-interacting density of states (DOS), and ‘Mott insulators’ (MI) in which strong repulsive interactions prevent hopping at commensurate filling. The evolution from BI to MI is a fascinating issue in condensed matter physics [18, 44, 45, 46, 19, 20]. In the ionic Hubbard model we investigate here, a staggered site energy $\mu_{i}=\pm\Delta$ on the two sublattices of a square lattice (Fig. 2A) leads to a dispersion relation $E(k)=\pm\sqrt{\varepsilon(k)^{2}+\Delta^{2}}$ with $\varepsilon(k)=-2t\,({\rm cos}k_{x}+{\rm cos}k_{y}\,)$. The resulting DOS vanishes in the range $-\Delta<E<+\Delta$ in which the lattice is half-filled, resulting in a BI. The occupation of the ‘low energy’ sites $\mu_{i}=-\Delta$ is greater than that of the ‘high energy’ sites $\mu_{i}=+\Delta$, so that there is a trivial charge density wave (CDW) order associated with an explicit breaking of the sublattice symmetry in the Hamiltonian.

An onsite repulsion $U$ disfavors this density modulation: The potential energy $Un_{i\uparrow}n_{i\downarrow}$ is higher than for a uniform occupation. Thus the driving physics of the BI, the staggered site energy $\Delta$ and that of the MI, the repulsion $U$, are in competition. Although the simplest scenario is a direct BI to MI transition with increasing $U$, one of the more exotic possible outcomes is the emergence of a metallic phase when these two energy scales are in balance and neither type of insulator can dominate the behavior. Past DQMC simulations suggest this less trivial case occurs, and have used the temperature dependence of the dc conductivity to bound the metallic phase [46, 47].

Here we investigate how this physics might be reflected in the average sign. Figure  2 B shows $\langle{\cal S}\rangle$ in the $U/t-T/t$ plane at $\Delta=0.5\,t$. As $T$ is lowered, $\langle{\cal S}\rangle$ deviates from unity for a range of intermediate $U$ values. Figure 2 C gives the behavior in the $U/t-\Delta/t$ plane at fixed low $T=t/24$. The central result is that $\langle{\cal S}\rangle$ is small in a region which maps well with the previously determined boundaries of the metallic phase [46, 47]. This is emphasized by a comparison to Fig. 2 D, which uses one of the ‘traditional’ methods for phase boundary location, namely the behavior of the double occupancy. The BI has a low occupancy, and hence very low double occupancy on the $+\Delta$-sites. Increasing $U$ smooths out the density, so that the double occupancy on the $+\Delta$-sites increases: $d\langle n_{\uparrow\downarrow,+\Delta}\rangle/dU>0$. In contrast, in the MI region, $U\gtrsim\Delta$, the physics is that of the usual Hubbard Hamiltonian and double occupancy decreases as $U$ grows: $d\langle n_{\uparrow\downarrow,+\Delta}\rangle/dU<0$.

In the CM region between BI and MI, however, obtaining a relevant signal-to-noise ratio for the traditional observables is exponentially challenging precisely because the average sign vanishes in this region. The ‘phase diagram’ obtained by $\langle{\cal S}\rangle$ (Fig. 2 C) is remarkably similar to that given by the physical observable, the rate of change of double occupancy with $U$ (Fig. 2 D)

As in the determination of the QCP for the spinful Hubbard model on a honeycomb lattice, $\langle{\cal S}\rangle$ emerges as more than a mere nuisance, but as a harbinger of the physics. An in-depth similarity between these two situations is discussed in the SM [7], where we show that the BI-metal QCP is again uniquely identified by the $1/L$ scaling of $\langle{\cal S}\rangle$, in precise analogy with the honeycomb case. These results suggest the existence of a quantum critical region associated with the CM phase and the vanishing $\langle{\cal S}\rangle$.

III. An ‘Unnecessary’ Sign Problem

We now consider spinless fermions, where the on-site Hubbard interaction $U$, made irrelevant by the Pauli principle, is replaced by an intersite repulsion $V$,

Equation 2 provides an example of a model where the SP can be completely solved by utilizing special techniques such as the fermion bag in the Continuous Time QMC approach [35], or by going to a different basis employing a Majorana representation of the fermions in the AFQMC method [41], as long as the system is on a bipartite lattice and $V>0$. The standard Blankenbecler, Scalapino, and Sugar (BSS) approach  [28], on the other hand, manifestly displays a SP in the low temperature regime. Nevertheless, in order to study the sign and its connection with the underlying physics, we use a BSS based algorithm to investigate the system on a honeycomb lattice (Fig. 3 A). Consideration of this ‘unnecessary’ SP allows us to address fundamental issues related to the algorithm dependence of links between the SP and the physics of model Hamiltonians.

At $T=0$, the model displays a QPT between a Dirac semimetal and an insulating staggered CDW state as the interaction is tuned through a critical value $V_{c}$ [22]. At large $V$, the repulsive interaction favours a CDW state, distinguished from that of the ionic Hubbard model by the fact that there is no staggered external field here – the CDW phase is a result of spontaneous symmetry breaking. As $V$ is reduced, increasing quantum fluctuations due to hopping finally destroy the CDW state, resulting in a Dirac semimetal for $V<V_{c}$. Accurate estimates based on SP free methods yield $V_{c}\sim 1.35t$ [41].

In Fig. 3 B, we show a map of the temperature extrapolation of $\langle{\cal S}\rangle$ as a function of $V$. The sign shows a clear reduction around the known $V_{c}$ (denoted by the star).

Figure 3 D shows the spatial lattice size dependence of the sign, and Fig. 3 C, once again, a more ‘traditional’ local variable, the derivative of the nearest-neighbor (NN) density-density correlation, $\langle n_{i}n_{j}\rangle_{\rm NN}$, with respect to $V$. In the CDW phase, increasing $V$ strengthens the staggered order, reducing the NN density correlations, and hence $-d\langle n_{i}n_{j}\rangle_{\rm NN}/dV$ is positive. Conversely, the effect is much smaller in the semimetal state, where the derivative is close to zero. The transition $V_{c}$ is characterized by a clear downturn in this quantity, which becomes progressively sharper as $L$ increases, as Figure 3 C shows. This variable, thus, serves as a physical indicator of the QPT, allowing a comparison of Figs. 3 C,D to demonstrate the connection between the QCP and the behavior of $\langle{\cal S}\rangle$. In this model, $\langle{\cal S}\rangle$ is well enough behaved that a study of the finite-temperature CDW transition with DQMC is feasible [7], without having to resort to SP-free approaches [48].

IV. Square Lattice Hubbard Model

The essential physics of the cuprate superconductors consists of antiferromagnetic order at and near one hole per CuO₂ cell, a superconducting dome upon doping, which typically extends to densities $0.6\lesssim\rho\lesssim 0.9$ and a ‘pseudogap’/‘strange metal’ phase above the dome [49, 23]. There are many quantitative, experimentally-based, phase diagrams of different materials which determine the regions occupied by these phases [50]. Likewise, there are computational studies of individual $(\rho,T,U)$ points establishing magnetic/charge order [51], linear resistivity [52], a reduction in the spectral weight for spin excitations [53, 54], and $d$-wave pairing [55, 56].

Here we reveal a ‘sign problem phase diagram’ which bears significant resemblance to that of experiments. As is well known, the severity of the sign problem itself precludes determination of $d$-wave order in DQMC via ‘traditional’ observables such as the associated correlation functions. However, Figure 4, based on the behavior of the sign itself, is suggestive. We report the average sign (Fig. 4 A), the enhancement of the $d$-wave pairing susceptibility over its value in the absence of the pairing vertex [57] (Fig. 4 B), and the uniform, static spin susceptibility $\chi(\textbf{q}=0)$ in both the $T/t-\mu/t$ (A–C) and $T/t-\rho$ (D–F) planes – see SM [7].

The most salient features of this ‘sign phase diagram’ are (i) the ‘dome’ of vanishing $\langle{\cal S}\rangle$ which occurs in a range of densities $0.4\lesssim\rho\lesssim 1$ as $T$ is lowered (Fig. 4 D); (ii) the enhancement of $d$-wave pairing (Fig. 4 E) surrounding the sign dome; and (iii), that the magnetic properties are also linked to the $\langle{\cal S}\rangle$-dome: the trajectory tracing the peak value of $\chi(q=0)$ as $T$ is decreased terminates precisely at the top of the dome (Fig. 4 F). In isolation, the comparisons of the behavior of the sign and the pairing and magnetic responses in the square lattice Hubbard model appear likely to be merely coincidental. Indeed, the fact that the sign is worst precisely for optimal dopings has been previously remarked, but thought to be just ‘bad luck’ [57, 32, 58, 59]. However, that the known QCP of the three models discussed in Secs. I-III can be quantitatively linked to the behavior of $\langle{\cal S}\rangle$ suggest that the sign dome might actually be indicative of the presence of $d$-wave superconductivity.

Discussion and Outlook

Early in the history of the study of the sign problem, a simple connection was noted between the fermionic physics and negative weights in auxiliary field QMC: If one artificially constructs two Hubbard-Stratonovich field configurations, one associated with two particle exchanging as they propagate in imaginary time and another with no exchange, one finds that the associated fermion determinants are negative in the former case, and positive in the latter [60]. This interesting observation, however, pertains to low density, that is, to the propagation of just two electrons. Another key observation is that the SP can be viewed as being proportional to the exponential of the difference of free energy densities of the original fermionic problem, and the one used with the weights in the Monte Carlo sampling taken to be positive, akin to a bosonic formulation of the problem [32, 13]. It highlights how intrinsic the SP is in QMC methods. A last important remark is that ordered phases are often associated with a reduction in the importance of configurations which scramble the sign. This is graphically illustrated in the snapshots of [24]. Although less crisp, similar effects are seen in auxiliary field QMC, for example in considering the evolution from the attractive Hubbard model to the Holstein model with decreasing phonon frequency $\omega_{0}$. Reducing $\omega_{0}$ acts to increase the effect of the phonon potential energy term $\hat{P}^{2}$ in $\hat{H}$, thereby straightening the auxiliary field in imaginary time.

Here we have shown that the behavior of the average sign $\langle{\cal S}\rangle$ in DQMC simulations holds information concerning finite density thermodynamic phases and transitions between them: the quantum critical points in the semimetal to antiferromagnetic Mott insulator transition of Dirac fermions, the band insulator to metal to correlated insulator evolution of the ionic Hubbard Hamiltonian, and the QCP of spinless fermions (even though a sign-problem free formulation exists). Specifically, a rapid evolution of $\langle{\cal S}\rangle$ marks the positions of quantum critical points. We have chosen these models as representative examples of QCP physics of itinerant electrons which have been extensively studied in the condensed matter physics community, but believe the result to be general. In fact, in a model for frustrated spins in a ladder, using a completely different QMC method (stochastic series expansion), similar conclusions can be inferred [61], further corroborating this generality.

Likewise, in the square lattice version of the $U(1)$ Hubbard model that we studied here, with an added $\pi$-flux, it can be shown that in the sign-problem free formulation, the QMC weights, when expressed in terms of the square of Pfaffians (Pf), holds similar information, namely, that $\langle{\rm sgn(Pf)}\rangle$ departs from 1 close to the QCP for this model [62]. These results provide further evidence that the average sign of the QMC weights is inherently connected to the physics of the model in many unrelated models and methods, but an even broader study may be necessary to establish this conclusively.

Having established this connection in Hamiltonians with known physics, we have also presented a careful study of the sign problem for the Hubbard model on a 2D square lattice, which is of central interest to cuprate $d$-wave superconductivity. The intriguing ‘coincidence’ that the sign problem is worst at a density $\rho\sim 0.87$, which corresponds to the highest values of the superconducting transition temperature, has previously been noted [57, 32, 58, 59]. It is worth emphasizing that we have not here presented any solution to the sign problem. However, our work does establish the surprising fact that $\langle{\cal S}\rangle$ can be used as an ‘observable’ which can quite accurately locate quantum critical points in models like the spinful and spinless Hubbard Hamiltonians on a honeycomb lattice, and the ionic Hubbard model, and also provides a clearer connection between the evolution of the fermion sign and the strange metal/pseudogap and superconducting phases of the iconic square lattice Hubbard model.

In these Supplementary Materials we provide some additional context and history of the sign problem. We also show the behavior of other observables across the transitions described in the main body of the paper, as well as exploring more carefully finite spatial size and Trotter effects in the evolution of $\langle{\cal S}\rangle$.