Abstract
Quantum Monte Carlo (QMC) simulations of correlated electron systems provide unbiased information about system behavior at a quantum critical point (QCP) and can verify or disprove the existing theories of nonFermi liquid (NFL) behavior at a QCP. However, simulations are carried out at a finite temperature, where quantum critical features are masked by finitetemperature effects. Here, we present a theoretical framework within which it is possible to separate thermal and quantum effects and extract the information about NFL physics at T = 0. We demonstrate our method for a specific example of 2D fermions near an Ising ferromagnetic QCP. We show that one can extract from QMC data the zerotemperature form of fermionic selfenergy Σ(ω) even though the leading contribution to the selfenergy comes from thermal effects. We find that the frequency dependence of Σ(ω) agrees well with the analytic form obtained within the Eliashberg theory of dynamical quantum criticality, and obeys ω^{2/3} scaling at low frequencies. Our results open up an avenue for QMC studies of quantum critical metals.
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Introduction
Understanding nonFermi liquid (NFL) behavior near a metallic quantum critical point (QCP) remains one of the most ambitious goals of the studies of interacting electrons. Examples of systems evincing metallic quantum criticality include fermions in spatial dimensions D ≤ 3 at the verge of either spin densitywave, charge densitywave, or nematicorder, 2D fermions at a halffilled Landau level, quarks at the verge of an instability to color superconductivity, and several Sachdev–Ye–Kitaev (SYK)type models with either electron–electron or electron–phonon interaction^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43}. At a QCP, fluctuations of the corresponding bosonic order parameter become soft. The fermion–fermion interaction, mediated by these soft fluctuations, yields a fermionic selfenergy Σ(ω) ∝ ∣ω∣^{a} with a < 1. The real and imaginary parts of this selfenergy are comparable in magnitude and both are larger than ω at low frequencies. This implies that the damping of quasiparticles remains comparable to their energy even infinitesimally close to the Fermi surface, in variance with the central paradigm of Landau’s theory of a Fermi liquid (FL). Studies of NFL became the mainstream of research on correlated electrons after a series of discoveries of hightemperature superconductors, which display unconventional metallic properties in the normal state^{15,24,44,45}. In most of these materials, superconductivity borders other ordered phases with either spin or charge order. There are also multiple overlaps between the behavior of fermions at a QCP and highenergy physics and string theory^{44,46}.
In recent years, several analytical approaches have been developed to study NFL behavior at a QCP. These approaches are based on effective fermion–boson models, in which soft fluctuations of a specific order parameter serve as the source of NFL behavior. The longstanding goal of these studies is to find the functional form of Σ(ω) at a QCP and extract the exponent a < 1 from its small ω behavior. Oneloop calculations show that Σ(ω) does become singular at a QCP, for example, in 2D at a transition to nematic or Ising ferromagnetic (FM) order with momentum Q = 0, it scales at the lowest frequencies as ω^{2/3} (a = 2/3). Whether this behavior extends beyond one loop is a more tricky issue. Power counting arguments indicate that higherorder terms in the loop expansion for the selfenergy reproduce the ω^{2/3} scaling form^{7}. However, detailed calculations reveal that additional \({(\mathrm{log}\,\omega )}^{n}\) factors appear, and that n increases with the loop order^{25,27,29,31,32,35}. Such logarithms imply that at low enough frequencies, \(\omega \ll {\omega }_{{\rm{mod}}}\), Σ(ω) gets modified from its oneloop form. As a further complication, the same interaction that gives rise to NFL behavior also gives rise to superconductivity at a nonzero T_{c}, so normal state selfenergy holds only at ω > T_{c}. It is difficult to extract from analytical studies whether \({\omega }_{{\rm{mod}}}\) is larger or smaller than T_{c}, that is, whether the modification of Σ(ω) from higherorder processes is relevant for a metal, which displays superconductivity near a QCP, or only for a putative normal state at T = 0. This uncertainty has triggered an interest in independent numerical studies of the behavior of fermions at a metallic QCP.
Numerical methods for itinerant fermions near a QCP have witnessed great progress in recent years, and at present one can analyze quantum criticality via reliable largescale numerical simulations^{47,48}. In particular, it has been found that designer models of fermion–boson models offer a pathway to access fermionic QCPs while avoiding the notorious sign problem in largescale quantum Monte Carlo (QMC) simulations. Such models have been implemented in several simulations, studying nematic^{49,50}, ferromagnetic^{51}, antiferromagnetic^{52,53,54,55,56}, gauge field^{57,58,59,60,61,62}, and YukawaSYKtype^{41} QCPs. The focusing on a particular soft boson offers an unbiased numerical test for either a Q = 0 or a finite Q analytical theory of metallic quantum criticality. The mutual inspiration and dialog between numerical and theoretical communities, arising from these studies, has also stimulated progress along the numerical front (SLMC^{63} and EMUS^{64} are successful examples of this).
Signproblemfree QMC has its own limitations as well. To avoid superconductivity and finite size effects, simulations are done at a finite T, which is not the smallest energy scale in the system, such that on a Matsubara axis the fermionic selfenergy Σ(ω_{n}) is a function of a discrete Matsubara frequency ω_{n} = (2n + 1)πT. (The selfenergy also has a momentum dependence Σ = Σ(ω_{n}, k), but here and henceforth we suppress this notation for clarity, except where needed.) At nonzero T, it can generally be expressed as Σ(ω_{n}) = Σ_{T}(ω_{n}) + Σ_{Q}(ω_{n}), where the “thermal” part Σ_{T}(ω_{n}) is the contribution from static thermal fluctuations and the “quantum” part Σ_{Q}(ω_{n}) is the contribution from dynamical bosonic fluctuations. At T = 0, ω_{n} is a continuous variable, Σ_{T} = 0, and Σ = Σ_{Q}(ω_{n}) is an NFL selfenergy at a QCP. However, at a finite T, the selfenergy differs from its T = 0 form, and the presence of Σ_{T}(ω_{n}) can mask the behavior associated with Σ_{Q}(ω_{n}). Besides, at a finite T, Σ_{Q}(ω_{n}) also generally differs from its T = 0 form. We note that in the YukawaSYK model these finitetemperature effects have recently been analyzed using an emergent conformal (reparametrization) symmetry of the lowenergy theory, which automatically incorporates thermal and quantum effects^{41}. However, the treatment of the present critial FS model without conformal symmetry requires separate analyses of Σ_{Q} and Σ_{T}.
The main purpose of this paper is to provide the method to disentangle Σ_{T} and Σ_{Q} from QMC data for the selfenergy. Our approach is based on three observations:

First, to study QC behavior one should avoid the effect of fluctuations from fermions with energies of order of the bandwidth, such as would lead to, or example, Mott physics. For this, the effective femion–boson coupling (labeled \(\bar{g}\) in the text) should be much smaller than the bandwidth W. In systems with a large Fermi surface, W is comparable to the Fermi energy, so the necessary condition is \(\bar{g}\ll {E}_{{\rm{F}}}\).

Second, at small \(\bar{g}\), there is a wide range of frequencies ω_{n} ≪ E_{F}, for which ω_{n} is much larger than Σ(ω_{n}). In this range, the thermal selfenergy has a simple form, valid for finite temperatures and frequencies ω_{n} ≫ Σ, Σ_{T}(ω_{n}) ≈ α(T)/ω_{n} up to logarithmic corrections, that is, ω_{n}Σ_{T}(ω_{n}) = α(T) is approximately independent of ω_{n}.

Third, in the same range, Σ_{Q}(ω_{n}) still has NFL form and is well approximated by the oneloop, T = 0, expression, modulo that ω_{n} is discrete.
By considering the above points, one arrives at the following conclusion: if a QMC study is performed at \(\bar{g}\ll {E}_{{\rm{F}}}\) and provides data for Σ(ω_{n}) for a substantial number of Matsubara points in the range ω_{n} ≫ Σ(ω_{n}), it is possible to extract Σ_{Q} from the data by the following simple procedure. First, extract (the approximately constant) α(T) from the data by fitting ω_{n}Σ(ω_{n}) by a continuous function of frequency and extrapolating to zero frequency, where it is equal to α(T) because ω_{n}Σ_{Q}(ω_{n}) extrapolates to zero. Once α_{T} is known, subtract Σ_{T}(ω_{n}) = α(T)/ω_{n} from the full Σ(ω_{n}) and obtain Σ_{Q}(ω_{n}), which, as we said, should have the same form as T = 0 selfenergy. For a more accurate separation of Σ_{Q} from Σ_{T} include the slow frequency dependence of α(T) in the fitting procedure, which is still quite straightforward to do, as we will show later.
We apply our strategy to a metal near an Ising FMQCP. We show the schematic phase diagram in Fig. 1a. It contains regions of a paramagnetic metal (PM) and an ordered Ising FM, separated by a QCP. Right above the QCP, there is a region of small T, where the system displays truly NFL behavior, that is, Σ(ω_{n}) is nonanalytic and larger than ω_{n}. At higher T, Σ(ω_{n}) becomes smaller than ω_{n}, yet the selfenergy still has nonFL form, and, by our reasoning, its quantum part, Σ_{Q}(ω_{n}) should be almost the same as at T = 0. In Fig. 1b we show the full selfenergy, obtained in QMC simulation, and in Fig. 1c we show Σ_{Q}(ω_{n}), extracted using the approximate procedure outlined above. The black line in Fig. 1c is the analytical oneloop result for the selfenergy at a QCP at T = 0. We see that the data for all ω_{n} nicely fall onto this curve. At small ω_{n}, the analytic oneloop selfenergy behaves as \({\omega }_{n}^{2/3}\), and the fact that QMC data fall onto the T = 0 curve implies that the QMC data are consistent with \({\omega }_{n}^{2/3}\) scaling at the lowest ω_{n} at a QCP. The deviation from \({\omega }_{n}^{2/3}\) scaling in the analytical formula (Eq. (16) in the text) is due to two reasons. First, for the model used for QMC simulations, the bosonic propagator D(q, Ω) contains a regular Ω^{2} term along with the Landau damping term, Ω/q. When this term becomes relevant, Σ_{Q}(ω_{n}) tends to saturate. Second, even when the Landau damping term dominates, the ω^{2/3} form is the lowfrequency limit of a more complicated function \({\Sigma }_{{\rm{Q}}}({\omega }_{n})\propto {\omega }_{n}^{2/3}{\mathcal{U}}\left({\omega }_{n}/{\omega }_{b}\right)\), and ω^{2/3} behavior holds only when ω_{n} ≪ ω_{b}, that is, \({\mathcal{U}}(z)\approx {\mathcal{U}}(0)\). The crossover frequency \({\omega }_{{b}} \sim {(\bar{g}{E}_{{\rm{F}}})}^{1/2}\) (see Eq. (12) below). In our simulations, this ω_{b} is much larger than the upper boundary of NFL behavior, \({\omega }_{{\rm{F}}} \sim {\bar{g}}^{2}/{E}_{{\rm{F}}}\), but is still much smaller than E_{F}. Accordingly, most of our ω_{n} fall into ω_{n} > ω_{b}, where Σ_{Q}(ω_{n}) differs from \({\omega }_{n}^{2/3}\). We emphasize that Σ_{Q}(ω_{n}) has an NFL form regardless of the ratio ω_{n}/ω_{b}. Figure 2 presents a summary of the relevant energy scales in our QMC study.
It is instructive to compare our results with recent analysis of QMC data for similar models. Reference^{65} demonstrated that a rather flat dispersion of Σ(ω_{n}), obtained in QMC simulations, is reasonably well reproduced by Σ(ω_{n}) = Σ_{T}(ω_{n}) + Σ_{Q}(ω_{n}), where both are computed analytically within a metallic QC theory. For that study, a larger coupling \(\bar{g} \sim {E}_{{\rm{F}}}\) was used to increase the magnitude of the selfenergy. The discrepancy between the analytic and QMC selfenergies in ref. ^{65} was ~20%. This was small enough to see that analytic and QMC selfenergies have similar dispersion, but still too high to reliably extract Σ_{Q}(ω_{n}) from the QMC data. For the current study, \(\bar{g}\) is smaller, and typical Σ(ω_{n})/ω_{n} is roughly five times smaller than in that work. In this situation, we argue that the QC form of Σ_{Q}(ω_{n}) can be extracted from the data.
The structure of the paper is the following. In section “The lattice model, phase diagram and QMC selfenergy”, we describe the lattice model for which the QMC simulations have been performed, and present the numerical results for the selfenergy. In section “Analytic selfenergy at Ising FMQCP”, we present the analytical results for the selfenergy within the selfconsistent oneloop analysis. In section “Analysis of QMC data”, we extract Σ_{Q}(ω_{n}) from QMC data and show that for all n > 0 it falls onto the analytic, T = 0 form of Σ_{Q}(ω_{n}). In section “Discussion”, we summarize the results and discuss the implication of this work to other QC cases studied in QMC simulations. We argue that the computational scheme that we proposed can be used as a generic method to extract NFL selfenergy at a QCP and can be further extended to study more subtle effects, for example, the flow of the dynamical exponent z.
Results
The lattice model, phase diagram, and QMC selfenergy
As shown in Fig. 3, we consider a model describing Ising FM fluctuations coupled to a Fermi surface^{51}. The model is implemented on a square lattice with Hamiltonian \(\hat{H}={\hat{H}}_{{\rm{f}}}+{\hat{H}}_{{\rm{s}}}+{\hat{H}}_{{\rm{sf}}}\) and each part reads
where \({\hat{H}}_{{\rm{f}}}\) describes two layers (or orbitals, λ = 1, 2) of spinful (σ = ↑, ↓) fermions with nearestneighbor hopping on a square lattice, and the chemical potential μ tunes the size of bare Fermi surface. The bare fermion dispersion dictated by \({\hat{H}}_{{\rm{f}}}\) is \(\epsilon ({\bf{k}})=2t(\cos ({k}_{x})+\cos ({k}_{y}))\mu\) and the bandwidth is W = 8t. \({\hat{H}}_{{\rm{s}}}\) represents a transverse field Ising model on the same lattice, where by tuning T and h/J an Ising FM to PM transition can be obtained. The onsite coupling term \({\hat{H}}_{{\rm{sf}}}\) between the fermions and Ising spins mediates a fermion–fermion interaction, establishing a metallic system with ferromagnetic fluctuations. We present the schematic phase diagram in Fig. 1a. In the analysis that follows, we focus on the model parameters {t = 1, μ = − 0.5t, J = 1, ξ/t = 1}, for which we find an FMQCP at h_{c}/J ≈ 3.270(6). The parameters associated with the fermiology for these parameters are listed in Table 1.
As shown in ref. ^{51}, our model gives rise to an FMQCP. However, the bare numerical fermionic selfenergy data from QCP, as shown in Fig. 1b, shows a behavior distinctively different from the expected NFL, \(\Sigma ({\omega }_{n})\propto {\omega }_{n}^{2/3}\). At low frequency, the selfenergy shows an unusual upturn instead of going to zero. Such a upturn in the imaginary part of fermionic selfenergy, in the usual numeric setting, implies a gap opening on the Fermi surface. However, our data of the fermionic Green’s function does not show a wellformed gap on the FS. Similar behavior of the numerical NFL selfenergies has also been observed in other cases, including nematic and AFMQCPs^{49,53}. As discussed in the “Introduction”, the rest of this paper is devoted to an analysis of the selfenergy data in Fig. 1b, and to understand how to disentangle the thermal and quantum parts of the selfenergy, as shown in Fig. 1c.
Analytic selfenergy at Ising FMQCP
We begin with a brief review of the diagrammatic theory for interacting fermions near the ferromagnetic QCP. As the derivations of the electron–boson models and their relationship to itinerant QCP and NFL physics, as well as superconductivity, are scattered over numerous research papers and reviews encompassing decades of work, assiduous readers are suggested to directly consult these refs ^{2,3,5,6,7,12,14,16,17,19,21,22,24,25,27,28,29,33,34,35,43,66,67,68,69,70,71,72}. Here, we will keep our derivation concise and try to be selfcontained.
To understand the situation described in Eq. (1) of itinerant electrons coupled to critical bosonic fluctuations, we can encode the dynamics of bosons and fermions in their propagators,
and
where k = (ω_{n}, k), q = (Ω_{m}, q) are threevectors with ω_{n} = (2n + 1)πT and Ω_{m} = 2mπT, the fermionic and bosonic Matsubara frequencies, respectively, ϵ(k) is the dispersion from section “The lattice model, phase diagram and QMC selfenergy”, \({M}_{0}^{2}\) represents the bare distance to the QCP before the interaction is turned on (in the QMC it is controlled by the transverse magnetic field, M_{0} = M_{0}(h)), and Σ, Π are, respectively, the fermionic and bosonic selfenergies. Both selfenergies are represented by a diagrammatic series in \(\bar{g}={(\frac{\xi }{2})}^{2}{D}_{0}\). The series is depicted pictorially in Fig. 4, where solid and wiggly lines are the full propagators G(k), D(q) and the triangles are fully dressed vertices. In general, it is not justified to neglect the vertex corrections. However, it is customary to split the corrections into two types: those coming from fermions away from the Fermi surface (“highenergy” fermions on the scale of the bandwidth W), and those coming from near the Fermi surface. The highenergy contributions just give some static corrections to an effective lowenergy theory, which can be absorbed into an effective renormalized coupling \(\bar{g}\). The condition for the smallness of these corrections is weak coupling,
This condition is valid away from the QCP, that is, when \({M}_{0}^{2} \sim {k}_{{\rm{F}}}^{2}\). In the lowenergy theory, at low enough temperatures and frequencies, it is not justified to neglect vertex corrections. However, those vertex corrections that contribute to Σ(k) can be neglected if we are in a regime where ∣Σ(ω_{n})∣ ≪ ω_{n}. As shown in Fig. 1b, the lowest fermionic frequency in our QMC simulation is ω_{0} = πT = 0.157 with T = t/20 and the corresponding fermionic selfenergy ∣Σ(k_{F}, ω_{0})∣ = 0.058, so this condition is satisfied. A longer discussion on this is presented in another work by some of us^{65}.
In our QMC study, we are always in the regime ∣Σ(ω_{n})∣ ≪ ω_{n}, and Eq. (4) is obeyed, so without further discussion we will assume that vertex corrections are negligible. Then, Π, Σ are described by the coupled selfconsistent equations,
Here N_{f} is the number of fermion flavors (N_{f} = 2 in the model of section “The lattice model, phase diagram and QMC selfenergy”) and the factor 2 in Π comes from spin summation.
In principle, Eqs. (5) and (6) have momentum integrals over the entire Brillouin zone, which means that they still include contributions to the selfenergies that come from high energies. One of these is a static contribution to Π. This contribution just renormalizes the mass towards the QCP, that is, \({M}_{0}^{2}\) in Eq. (3) is replaced by
Thus, M^{2} can be tuned to a QCP by varying \(\bar{g}\), or alternatively by varying \({M}_{0}^{2}\) (this is what is done in the QMC simulations). An additional static contribution renormalizes D_{0} and we absorb it into \(\bar{g}\). There are also static contributions to Σ, but they do not change the critical dynamics so we absorb them into the fermionic dispersion. Then, there are dynamical contributions that we will now compute.
Beyond neglecting vertex corrections, we further assume that the fermionic dispersion can be linearized near the FS, which means that the theory describes a lowenergy effective theory near the FS. Then, integrating over linearized fermionic dispersion we obtain,
where Θ(x) is the step function, the density of states \({{\mathcal{V}}}_{{\rm{F}}}(\theta )={k}_{{\rm{F}}}(\theta )/{\upsilon }_{{\rm{F}}}(\theta )\) and k_{F}, \({\upsilon}_{\mathrm{F}}\) are the Fermi vector and velocity at an angle θ on the FS, as given in Table 1. In Eq. (8), as ∣Σ(ω_{n})∣ ≪ ω_{n}, we neglected contributions from selfenergy and assumed that the k_{F} and υ_{F} vector are approximately parallel. For the fermionic selfenergy we get,
where σ(x) is the sign function and \(\hat{n}({\theta }_{k})={\left.\frac{({{\mathbf{\upsilon}}}_{{{\rm{F}}}_{y}},{{\mathbf{\upsilon}}}_{{{\rm{F}}}_{x}})}{{\upsilon }_{{\rm{F}}}}\right}_{\theta = {\theta }_{k}}\) is an unit vector pointing parallel to the FS at the angle θ_{k}. In a C_{4} symmetric system, we can replace \(\hat{n}\) by \({\boldsymbol{\upsilon}}_{\mathrm{F}}/{\upsilon}_{\mathrm{F}}\), since the unit vector only determines the value of \(\upsilon_{\mathrm{F}}\)(θ) in Eq. (8).
We first evaluate the bosonic selfenergy which to leading order is,
where the C_{4} symmetry of the lattice is used to replace \(\upsilon_{\mathrm{F}}\)(θ ± π/2) = \(\upsilon_{\mathrm{F}}\)(θ) and similarly for \({{\mathcal{V}}}_{{\rm{F}}}\). Next, we turn to the fermionic selfenergy. Plugging Eq. (10) into Eq. (9) yields,
In Eq. (11) we rescaled momentum to frequency ω = υ_{F}p, and hid the explicit angular dependence υ_{F} = υ_{F}(θ_{k}) for conciseness. The frequency scale introduced by Π is
From Eq. (11) we can read the relevant frequency scales for Σ(ω_{n}). The typical scale of the ω_{l} sum is ω_{l} ~ ω_{n} due to the sign function, that is, typical internal frequencies are constrained to be on order of the external frequency.
We now show that at finitetemperature, but as long as ∣Σ(ω_{n})∣ ≪ ω_{n} the fermionic selfenergy in Eq. (5) splits into two parts: thermal and quantum (for detailed derivations and discussions see, e.g., refs ^{16,21,34,43,65}). The quantum part recovers the zerotemperature fermionic selfenergy, while the thermal part takes on a very simple form and scales as 1/ω_{n}. Thus, after simply deducting this 1/ω_{n} term, the finitetemperature selfenergy directly provides the zerotemperature behavior of fermions, although the measurement is done at finite temperature, at which thermal fluctuations has a significant contribution. This is one of the key conclusions of this work. We separate the summation in Eq. (11) into two parts
where Σ_{T} is the ω_{l} = ω_{n} piece of the sum in Eq. (11), namely
where
As \({\mathcal{S}}(x)\) vanishes rapidly at large x, it predicts that Σ_{T} only contributes significantly at finite temperature and close enough to the QCP (πT ≳ \(\upsilon_{\mathrm{F}}\)υ_{F}M). In that regime, as noted in the introduction, α(T, ω_{n}) = ω_{n}Σ_{T}(ω_{n}) depends at most logarithmically on frequency at the smallest ω_{n}, α(T, ω_{n}) ≈ α(T).
The quantum part includes all other terms in the Matsubara sum. This sum can be approximately replaced by an integral, which immediately recovers the T = 0 form of the fermionic selfenergy, that is,
with
The scaling function \({\mathcal{U}}(z)\) has the following asymptotics,
where \({z}_{c}^{1}={({\upsilon }_{{\rm{F}}}/c)}^{3/2}\) and u_{0} is a constant which depends on \(\upsilon_{\mathrm{F}}/c\). For \(\upsilon_{\mathrm{F}}/c\) ≪ 1, u_{0} ≈ 1/8; while for parameters of section “The lattice model, phase diagram and QMC selfenergy” (\(\upsilon_{\mathrm{F}}/c\) ≈ 0.42), u_{0} ≈ 0.1. Note in the case of our QMC study z_{c} ≈ 3.7, so that the intermediate regime cannot really be seen. Equation (17) is exactly the formula we used to generate the black line in the Fig. 1c, and is the quantum NFL selfenergy Σ_{Q} of an FMQCP. It saturates in the large frequency region as shown in the figure, as predicted by Eq. (18) in the z_{c} ≪ z limit. Combining Eqs. (14) and (16), we indeed see that the selfenergy has a thermal 1/ω_{n} term plus the zerotemperature quantum selfenergy.
Let us briefly elaborate on the physics behind the scaling function \({\mathcal{U}}(z)\). In Eq. (17), the part in the square brackets correspond to the fermionic propagator and the other part in the integral corresponds to the bosonic propagator. Consider the limit ω ≪ ω_{b} corresponding to z ≪ 1. Expanding for z ≪ 1 we find that to leading order the terms in the square brackets are a constant, and the dx integral is limited to 0 < x < 1. Physically this is the statement that the momentum integration (∫dy) is only on bosonic momentum parallel to the FS. In addition, the \({({\upsilon }_{{\rm{F}}}/c)}^{2}{x}^{2}y\) term in the boson propagator is also negligible, which corresponds to the fact that the bare Ω^{2} part of the boson dynamics is irrelevant at low frequency. Evaluating Eq. (16) for ω_{n} ≪ ω_{b} we find,
where
Equation (19) is the formula used to generate the red dashed line in Fig. 1c, as an asymptotic line of the quantum part of the selfenergy predicted by Eq. (17). The analysis of Σ leading to Eqs. (19) and (20), as well as analogous analysis for superconducting selfenergy, is conventionally termed “Eliashberg theory” (ET), due to its similarity to ET of superconductivity from electron–phonon interactions^{73}.
Now consider the opposite limit, ω ≫ ω_{b}, corresponding to z ≫ 1. For simplicity let us assume υ_{F}/c ≪ 1. In that case the term in the square brackets, corresponding to the fermionic propagator, is not constant, and the bulk of the contribution to u_{0} is given by the range 1 < x < ∞. Physically this means that scattering is not confined to be parallel to the FS and is two dimensional, although it is still confined to be near the FS. It is instructive to compute the subleading term for small z. After some algebra, one finds that this contribution is also given by 2D scattering, and gives \((2\pi /\sqrt{3}){\mathcal{U}}(z)\approx 10.73{z}^{1/3}\). This means that for z ~ 1, Σ_{0} is reduced by a factor of almost 4 from the expected value if one considers only the leading contribution. This is the reason that the deviation from the asymptotic red line in Fig. 1c is so large. At even larger z, the Ω^{2} term in the bosonic propagator begins to contribute, which just modifies the highfrequency behavior of the selfenergy. However, the deviations from ω^{2/3} scaling occur already at z ~ 1. We term the theory which accounts for both highfrequency modifications and the finitetemperature corrections of Eq. (14) a modified Eliashberg theory (MET).
Analysis of QMC data
Now we turn to the QMC data analysis. We study the fermionic selfenergy from the FMQCP model described in section “The lattice model, phase diagram and QMC selfenergy” and compare the QMC data with the MET in section “Analytic selfenergy at Ising FMQCP”.
Let us begin by going through the relevant physical parameters in the QMC data. We normalize all quantities by the hopping energy t = 1, see section “The lattice model, phase diagram and QMC selfenergy” for details. By tuning the transverse field h, ref. ^{51} was able to extract QMC data for different T above the QCP, and also deep in the disordered phase, where the selfenergy should have an FL form Σ(ω_{n}) ∝ ω_{n} at low frequencies. We concentrate on the data at the QCP. The parameters from section “The lattice model, phase diagram and QMC selfenergy” imply a bare \(\bar{g}=1/4\), which is much smaller than E_{F} ≈ 1.6, implying the QMC is in the weak coupling regime (we remind that all energies are quoted in units of the hopping). The bosonic propagator in QMC was found to agree well (as shown in ref. ^{51}, it turns out the bosonic propagator is dominated by the bosonic selfenergy part, with a small finite anomalous dimension in q^{2} and Ω^{2} terms, it will not change the main results of this paper) with Eqs. (3), (7), and (10), that is,
with D_{0} = 1, M^{2}(T) = 0.13T^{1.48}, c = 3.16, that is, the measured D_{0} agrees with the bare one, all obtained from the bosonic propagator data in ref. ^{51}. In addition, it was found that Σ(ω_{n}) ≪ ω_{n} for all temperatures and Matsubara frequencies that were obtained. Thus, we may expect that corrections to the bare \(\bar{g}\) are small, and the renormalized \(\bar{g}\), which is an input to MET is at the order of the bare one. Under this condition, the relevant scales for Σ are
The temperatures we analyze are T = 0.05, …, 0.1, which implies the first Matsubara frequency is πT = 0.16, …, 0.31, see the schematics of energy scale in Fig. 2. Thus, ω_{F} is completely irrelevant as is verified by the fact that the selfenergy is always small. As we discussed in section “Introduction”, the QMC selfenergy appears to have a leading term of the form
as shown in Fig. 1b. This is consistent with the prediction of MET, see Eqs. (14) and (15).
We analyze the data in two ways. First, we extract the quantum selfenergy and compare it to the T = 0 prediction. To do this, we need to remove the thermal part. This is most conveniently done simply by studying the product ω_{n}Σ(ω_{n}). As discussed in the “Introduction” and the previous section, according to Eqs. (13) and (16) we have,
providing we treat α(T) as constant, neglecting its slow frequency dependence, see Eq. (15). In Eq. (24), α(T) includes both the contribution from Σ_{T}(ω_{n}, T), and corrections from finite size effect (such as a possible small gap due to the mismatch of finite size h_{c} and the thermodynamic h_{c}). The second part, that is, ω_{n}Σ_{Q}(ω_{n}), comes from the MET prediction for Σ_{Q}(ω_{n}), Eq. (16), which recovers ET prediction Eq. (19) in the lowfrequency limit (ω_{n} ≪ ω_{b}).
We fit Eq. (24) to the data for all T simultaneously. Importantly, in Eq. (24), the fitting parameters are only the constants α(T) and \(\bar{g}\). This is because Σ_{Q} is a function only of \(\bar{g}\) and system parameters, see Eq. (17). Figure 1c from the beginning of our paper depicts the result of our fit. We obtain a fitting of \(\bar{g}=0.245\pm 0.023\) for 95% confidence intervals, in excellent agreement with the theory. Regarding α(T), we find that α(T) ≈ 8 × 10^{−3} is almost a constant, in disagreement with the expected ∝ T behavior of ω_{n}Σ_{T}(ω_{n}, T). Clearly, part of this discrepancy is due to our neglecting the frequency dependence of α. We therefore repeat the analysis using the following fitting procedure,
which takes the full frequency behavior of Σ_{T} into account. Guided by the previous fit, we set \(\bar{g}=0.25\) to be the bare one to reduce the number of fitting parameters. We show the result of this fit in Fig. 5 and the extracted \(\alpha ^{\prime} (T)\) in Fig. 6. The agreement is very good, and we checked that the data collapse can be made even better by allowing \(\bar{g}\) to vary somewhat (equivalent to about 13% change in the bare vertex ξ). The extracted \(\alpha ^{\prime} (T)\) indicates the formation of a small gap forming at around T = 0.1, which is expected to yield a selfenergy contribution of the form α'(T)/ω_{n} = Δ^{2}(T)/ω_{n}. The gap size Δ corresponding to \(\alpha ^{\prime} (T)\) is much less than the numerical inverse reciprocal lattice spacing, so the appearance of this gap is actually an expected effect, which however is beyond the resolution of the standard methods for verifying the appearance of longrange order. Thus, our analysis of the selfenergy yields a method for more accurately finding the QCP in our system.
Here we add a word of caution. Previous work has shown^{33,38,74} that the first Matsubara frequency does not obey the quantum critical scaling Σ_{Q}(πT) ∝ (πT)^{2/3}, and therefore should not be included in the fitting procedure. We verified that dropping the first Matsubara point does not change our results. Also, note that within the error range in Fig. 5, it is possible that Σ_{Q}(πT) < 0. In fact, it can be verified that Σ_{Q}(πT) is always negative^{21,65}.
To avoid this issue, we also numerically computed Σ_{T}(ω_{n}) and Σ_{Q}(ω_{n}) by performing the Matsubara sum in Eq. (9), using \(\bar{g}=0.25\). This procedure takes into account the full frequency dependence of Σ_{T}(ω_{n}) as well as finite mass effects and the first Matsubara frequency issues. Figure 7 depicts a comparison of the QMC selfenergy with the numerical summation. There is an excellent agreement between the two, except for a Tdependent constant offset between the MET and QMC results. The result is consistent with the first analysis we performed above. For completeness, we also performed a comparison between the MET and QMC data for the data in the disordered phase (the FL regime). Figure 8 shows this comparison, again with very good agreement.
We therefore conclude that we have extracted the quantum selfenergy from the QMC data, and that it shows excellent agreement with the expected QC behavior.
Discussion
NFLs play a crucial role in a wide range of quantum manybody phenomena, such as quantum criticality, hightemperature superconductivity in correlated materials, unconventional transport in strange metals, and have been a key focus in the study of modern condensed matter physics^{1,2,3,5,6,7,12,14,15,16,17,18,19,22,24,25,27,28,29,33,35,36,44,45,47,48,53,66,67,68,69,70,71,72,75}. Despite of the intensive research efforts, key questions remained open and the problem of NFLs is still one of the most challenging topics in manybody physics, even with the most sophisticated field theoretical treatments^{16,25,27,29,35}, powerful numerical manybody algorithms and highperformance supercomputers^{47,48,63,64}.
Our work provides a pathway to address a key challenge in the study of NFLs, that is, the fact that the smokinggun signature of NFLs (the predicted unconventional lowtemperature fermion selfenergy), has never been directly observed or verified in largescale unbiased numerical methods. Through combined numerical and theoretical efforts, we proved that this key signature of NFLs can be accessed through QMC simulations, by simply deducting a ∝ 1/ω_{n} thermalfluctuation background. This technique enabled us to directly compare numerical results with theoretical predictions, providing a bridge between theoretical, numerical, and experimental studies.
Although this paper mainly focuses on the itinerant ferromagnetism QCP as an example to demonstrate the physics, the technique is universal and can be easily generalized to other itinerant QCPs, such as nematic and AFMQCPs^{49,52,53,56}. Furthermore, this technique can also be used to explore the predicted nontrivial effects from higherorder corrections^{16,25,27,29,35,76}, and thus open up a pathway towards a full understanding about this challenging subject of NFLs.
Methods
Numerical calculations
The numerical results for fermionic and bosonic selfenergies have been obtained using stateofart determinantal QMC simulations as reported in ref. ^{51}.
Analytical calculations
Analytical calculations have been carried out diagrammatically within ET and MET, by solving the set of selfconsistent equations for fermionic and bosonic selfenergies.
Data availability
The data that support the findings of this study are available from the first author upon reasonable request.
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Acknowledgements
We thank Subir Sachdev, Max Metlitski, Yuxuan Wang, Yoni Schattner, Erez Berg, and Dmitrii Maslov for insightful discussions on fermionic QCPs and NFL. X.Y.X. also thank Tarun Grover for helpful discussion on related projects. We acknowledge the support from RGC of Hong Kong SAR China through 17303019 and 17301420, MOST through the National Key Research and Development Program (2016YFA0300502). The work by A.K. and A.V.C. was supported by the Office of Basic Energy Sciences, U.S. Department of Energy, under award DESC0014402. We also thank the Center for Quantum Simulation Sciences in the Institute of Physics, Chinese Academy of Sciences, the Computational Initiative at the Faculty of Science at the University of Hong Kong and the Tianhe platforms at the National Supercomputer Centers in Tianjin and Guangzhou for their technical support and generous allocation of CPU time. This research was initiated at the Aspen Center for Physics, supported by NSF PHY1066293.
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All authors discussed and designed the study together. X.Y.X. and A.K. analyzed the data. All authors together discussed the theory and drafted the article.
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Xu, X.Y., Klein, A., Sun, K. et al. Identification of nonFermi liquid fermionic selfenergy from quantum Monte Carlo data. npj Quantum Mater. 5, 65 (2020). https://doi.org/10.1038/s41535020002666
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DOI: https://doi.org/10.1038/s41535020002666
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