Introduction

The high-temperature superconductivity (HTSC) in copper-oxide-based perovskites was discovered almost 40 years ago1, but a clear and complete theoretical view on this remarkable effect is still missing. Regarding the generic electronic structure of HTSC cuprates, there is a consensus on the paramagnetic Fermi surface: the angle resolved photoemission spectrum (ARPES)2 agrees well with the results of the Density Functional Theory (DFT)3. The main feature, related to a single Cu3d-O2p band per plane crossing the Fermi energy, is described by a simple tight-binding model with a large next-nearest-neighbor (NNN) hopping \({t}^{{\prime} }\). The typical ratio of \({t}^{{\prime} }\) to the nearest-neighbor (NN) hopping amplitude t is \({t}^{{\prime} }/t\simeq -0.3\) for bismuth strontium calcium copper oxide “BSCCO” or yttrium barium copper oxide “YBCO” with a critical temperature on the order of Tc ≈ 100 K2. The sign and large values of \({t}^{{\prime} }\) originate from the DFT band structure, which accurately describes the general chemical bonds in YBCO-like systems and is related to the strong hybridization of the valence band with the Cu − 4s orbital3. Furthermore, the estimation of \({t}^{{\prime} }/t\) for different HTSC cuprates4 shows strong empirical correlations between this parameter and the Tc values. Additionally, the negative sign of \({t}^{{\prime} }/t\) leads to a shift of the van Hove singularity towards the chemical potential for optimal hole doping, which efficiently increases the critical transition temperature. According to DFT calculations5, the Fermi energy goes closer to van Hove singularity simultaneously to the growth of Tc in mercury barium copper superconductor under pressure, the system with the highest Tc among cuprates.

The energy spectrum observed in ARPES compared to the DFT bands reveals a strong many-body renormalization2. The most unusual features of ARPES experiments6,7,8,9,10,11,12,13, are related to the structure of the superconducting gap. The key findings of these experiments, particularly the emergence of two distinct gaps, are shown schematically in Fig. 1, together with the generic phase diagram of cuprates in the temperature T versus hole-doping δ plane. A growing body of experimental evidences suggests that the pseudogap phase terminates at the same doping as the superconducting dome7,11,13,14,15. It is therefore natural to propose that the driving mechanism for d-wave superconductivity in cuprates is connected to the formation of “bad fermions” near the antinodal regions in the normal phase, which are responsible for the pseudogap. Furthermore, this region of the Fermi surface is consistently associated with the pseudogap (Fig. 1), as evidenced by spin susceptibility measurements16 and scanning tunneling microscopy (STM) results14,17. Note also appearance of the second line in Ref. 11, coherence temperature growing with the hole concentration which means that we probably deal with the coexistence of “bad” and “good” (coherent) fermions, as was also discussed based on transport properties18. Our theoretical consideration seems to confirm the experimental picture11,18; note that at the qualitative level a similar view was discussed in the last work of P. W. Anderson19.

Fig. 1: A generic phase diagram of cuprates.
Fig. 1: A generic phase diagram of cuprates.
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a Schematic borders for antiferromagnetic (AF), superconducting (SC) and pseudogap (PG) phases of HTSC cuprates as a function of temperature T and hole doping δ. The insert shows the corresponding Fermi surface (FS) with two distinct gap structures, where \({d}_{w}^{\pm }\) market the positive and negative parts of d-wave order parameter. b Normal G(k) and c anomalous F(k) (right) parts of the Nambu-Gor'kov Green’s function obtained at the lowest Matsubara frequency ν0 = πT using the Dual Fermion (DF) perturbation within the determinant Quantum Monte Carlo (DQMC) scheme. The Green’s functions are calculated for δ = 12% hole doping of the t–\({t}^{{\prime} }\) Hubbard model. The non-interacting Fermi-surface is depicted by a white line.

From a theoretical perspective, it has been shown using numerically exact density matrix renormalization group (DMRG) methods and the constrained-path determinant Quantum Monte Carlo (DQMC) scheme that the single-band Hubbard model with \({t}^{{\prime} }=0\) does not exhibit superconductivity in its ground state for any parameter values20. Instead, it displays a stripe phase at low temperatures. However, introducing NNN hopping \({t}^{{\prime} }\) leads to the emergence of d-wave superconductivity along with a partially filled stripe order for moderate values of \({t}^{{\prime} }=-0.2t\)21, which correspond to LSCO-like systems4. For larger values of \({t}^{{\prime} }=-0.25t\), d-wave superconductivity becomes the dominant instability in the single-band Hubbard model22. The realistic calculations of effective Coulomb parameters in the broad range of HTSC matertials23 gives the following value of effective Hubbard-like interaction Ueff ≈ U − V ≃ 6t, where V is nearest-neighbor Coulomb interaction24.

In turn, variational QMC calculations provide conflicting results regarding the effect of \({t}^{{\prime} }\) on superconductivity in HTSC cuprates. While recent studies of realistic one-band model of different compounds23 find no correlations with \({t}^{{\prime} }\) and a monotonic increase of pairing with U, a similar calculations for the simple t–\({t}^{{\prime} }\) model25 find strong evidence for a two-gap structure with strong pairing, for \({t}^{{\prime} }\le 0\) and an optimal value of U ≃ 6t. The latter calculations also reveal “bad fermion” behavior near the antinodal point for a certain doping range, meaning that the occupation number n(k) deviates significantly from the Fermi step function, and this effect turns out to be very sensitive to \({t}^{{\prime} }/t\). However, this approach cannot access frequency-dependent properties, providing limited insight into electron dynamics.

The first successful application of cluster extension of the Dynamical Mean Field Theory (C-DMFT) to study the interplay between antiferromagnetism (AFM) and superconductivity in cuprates26, along with numerous subsequent efforts in the same direction, faced challenges related to a poor momentum-space resolution and artificial breaking of translational symmetry unavoidable after separation of a cluster from the full crystal lattice. However, cluster approaches remain valuable for semi-quantitative analysis of peculiarities in the electronic structure by connecting them to the quantum states of individual plaquettes and complicated interplay of the pseudogap and d-wave gap27,28. This connection, in particular, explains high sensitivity of the results to the \({t}^{{\prime} }/t\) ratio and uncovers a useful relations to Kondo lattice systems due to nontrivial degeneracy in plaquettes29,30,31. The first attempt to investigate k-dependent pseudogap phenomena was made within 8-site Dynamical Cluster Approximations (DCA)32 and coins the term “sector selective Mott transition.” Such an effect may exist due to the fact that different 8 k-point are not mixed within DCA scheme and can be view nowadays as a nodal-antinodal dichotomy of the pseudpgap.

An important finding from the DCA calculations33 is that more than 50% of superconducting pairing in the single-band Hubbard model with optimal values of parameters arises from strong-coupling effects beyond the standard spin-fluctuation mechanism within BCS-like theories34. At the same time, it is crucial to emphasize that the large interaction limit, where the Hubbard interaction U ≫ W (with the bandwidth W = 8t), is equivalent to the t–J model and fails to accurately describe cuprate physics. In this limit, d-wave superconductivity in the t–J model emerges mainly in the “unphysical” regime of \({t}^{{\prime} }/t > 0\)35,36,37 and only very weak part for \({t}^{{\prime} }/t < 0\)38. Furthermore, the large-U limit is unfavorable for two-hole pairing energy, as demonstrated by exact diagonalization of a small 4 × 4 periodic system31.

We would like to point out an important synergy effect with standard spin-fluctuation mechanism of d-wave superconductivity34,39 and effects of van-Hove singularity which provides band flattening and nontrivial interplay of instabilities in particle-particle and particle-hole channels40,41. Different investigations shows that for optimal doping, Coulomb correlation parameter U/t ≃ 5.6 and \({t}^{{\prime} }/t\simeq -0.3\) the system is very close to Lifshitz transition with coincidence of van-Hove singularity with chemical potential30,42 which drastically boosts spin-fluctuations and together with strong-coupling effects favors the high-temperature superconductivity. An important connection between topologically ordered Higgs phase related to SU(2) local moment theory and pseudogap phenomena in cuprate put forward43 and confirmed by eight-site momentum space C-DMFT calculations42. More elaborated investigation of pseudogap states using recently developed diagammatic Monte-Carlo method44 also proposes a connection to the ground-state stripe phase calculated at zero temperature.

What is still missing in the HTSC puzzle is the understanding of nontrivial relation of pseudogap phenomena to d-wave superconductivity. From the one hand, both effects are related to antiferromagnetic Mott phase of a parent undoped compound. From the other hand, there is a competition between them for antinodal region. All this makes theoretical description of a generic HTSC phase diagram (Fig. 1) largely unsolved problem.

In this paper, we employ a Green’s function strong-coupling lattice perturbation theory45 to investigate the spectral properties of the doped single-band t–\({t}^{{\prime} }\) Hubbard model in the parameter regime relevant for HTSC cuprates. We find that at optimal doping, the electronic spectral function exhibits a pseudogap at the antinodal point in the Brillouin zone (BZ). This pseudogap coexists with a superconducting response to an external d-wave field, which displays \({d}_{{x}^{2}-{y}^{2}}\) symmetry in momentum space. Since the pseudogap is present already at temperature higher than the magnetic ordering one, extrapolated for δ → 0, we propose that the emergence of this “bad fermion” behavior at the antinodal point is directly linked to the appearance of local magnetic moments (LMMs) and their AFM correlations (a local resonating valence bond - RVB46 physics), constituting the key microscopic mechanism of high-temperature superconductivity. The LMMs are mainly formed by the quasi-localized heavy “bad fermions”, which is a dynamical process, while the region near the nodal point remains metallic, with well-defined quasiparticle states near the Fermi surface. This dichotomy between the nodal and antonidal points in the spectral function is a characteristic feature of the intermediate regime of electronic correlations, where the strong-coupling physics associated with LMMs coexists with itinerant electronic behavior47. We further support this claim by explicitly disentangling the contributions of spin fluctuations of purely itinerant order (paramagnons) and the LMMs, introduced via an effective Higgs field, to the electronic self-energy within a single-site-based strong-coupling approach. Our analysis reveals that while the presence of the pseudogap in the electronic spectrum suppresses superconductivity, the AFM correlations between the LMMs that give rise to this pseudogap simultaneously play a crucial role in enhancing the superconducting response.

Results

Perturbation expansion based on a lattice reference system

To investigate the two-dimensional one-band model for the Cu-O plane, we consider the t–\({t}^{{\prime} }\) Hubbard model on a 16 × 16 × 64 space-time grid (periodic in space, antiperiodic in imaginary time) with the NN t = 1 (which defines the unit of energy) and NNN \({t}^{{\prime} }=-0.3\) hopping amplitudes, and a local (on-site) Coulomb interaction U = 5.6 t. The choice of the interaction strength is based on a highly degenerate point in energy spectrum observed in small cluster calculations, which favors superconductivity29,30,31. The same value of U, related to the optimal nodal-antinodal dichotomy near the Lifshitz transition, was reported in Ref. 48. Results for other lattice sizes further support the main conclusions of this work and are available in the Supplementary Fig. 2.

To solve the considered model, we employ a strong-coupling perturbative scheme that performs a diagrammatic expansion based on the interacting lattice problem45. Importantly, the reference problem is introduced on the same 16 × 16 × 64 space-time grid as the original model and accounts for the same value of the Coulomb interaction U. However, the reference problem is considered particle-hole symmetric, i.e., at half-filling with \({t}^{{\prime} }=0\), which enables its efficient numerically exact solution using the DQMC scheme, as it turns out to be free of fermionic sign problem thanks to the particle-hole symmetry. The diagrammatic expansion beyond the reference problem is carried out in the spirit of the Dual Fermion (DF) approach49,50. The next-nearest-neighbor hopping \({t}^{{\prime} }\) and the difference between the chemical potentials of the reference and original systems μ serve as perturbative parameters for this expansion. This approach allows us to start from a correlated antiferromagnetic Mott insulator as the reference state and investigate doped pseudo-gap metals with strong d-wave-like superconducting fluctuations. Since the difference in chemical potential is of the order of μ = − 1.5 for 15% hole doping, and the maximum contribution of the NNN hopping \(4{t}^{{\prime} }=-1.2\) is small compared to the bandwidth W = 8t, while the simulation temperature is not very low, we consider the first-order DF diagrammatic correction to be sufficient for reasonably large lattice systems (see Materials and Methods). Note that the diagrammatic expansion is formulated in momentum k, Matsubara frequency ν51, and 2 × 2 Nambu space with normal (diagonal) G(k, ν) and anomalous (off-diagonal) F(k, ν) contributions to the Nambu-Gor’kov Green’s function.

In Fig. 2 we show the transformation of the electronic spectral function from the half-filled (reference) case with μ0 = 0 and \({t}^{{\prime} }=0\) (left panel) to the doped system with μ = − 1.45 and \({t}^{{\prime} }=-0.3\) (right panel). The results are calculated at the inverse temperature β = 1/T = 5 using maximum entropy analytical continuation from Matsubara space to real energy52. In the Mott insulator phase, corresponding to δ = 0% doping, one can see the formation of broad Hubbard bands around the energy E = ± 6, and shadow antiferromagnetic bands at E ≃ − 4 in the vicinity of the M = (π, π) point. Upon δ = 15% hole doping, the spectral function changes dramatically. One can clearly see a strong effect of \({t}^{{\prime} }\) on the van Hove singularity that results in the formation of a narrow, almost flat band in the Γ-X direction and the appearance of a pseudogap near the X point, which signals the quasi-localized behavior of electrons related to the formation of LMMs. On the other hand, the spectrum remains metallic near the nodal point (Γ − M)/2. In addition, one can see the formation of a narrow in-gap band first discussed in the framework of dynamical variational Monte Carlo scheme53,54 for the 12 × 12 cluster and also presented in our previous normal state DF-QMC calculations of 8 × 8 system in a bath45. Such new states are located just above the Fermi level from k = X to k = (Γ − M)/2 and match the quasipartical states bellow EF. Flattening of the bands and enhancement of van Hove singularities near the Fermi energy due to correlation effects were earlier found and studied by weak-coupling renormalization group41 and strong-coupling DF approach55.

Fig. 2: The electronic spectral function.
Fig. 2: The electronic spectral function.
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Results for the Hubbard model with ithe nearest-neighbor (t) and the next-nearest-neighbor (\({t}^{{\prime} }\)) hopping amplitudes on a square lattice. calculated for the interaction parameter U = 5.6 at inverse temperature β = 5 relative to the chemical potential together with the non-interacting dispersions (black line) are obtained for (a) the half-filled particle-hole symmetric case (\({t}^{{\prime} }=0\)), and (b) for δ = 15% hole doping with \({t}^{{\prime} }=-0.3\). The colour bar proportional to imaginary part of the Green’s function at the lowest Matsubara frequency.

In Fig. 3 we plot the imaginary part of the normal Green’s function G(k) (top panel), as well as the real part of the anomalous Green’s function F(k) (bottom panel). The results are obtained at the lowest Matsubara frequency ν0 = πT for the first BZ in the presence of a small external superconducting d-wave field with the amplitude hdw = 0.05. The 2D projection of these Green’s functions on the BZ plane is shown in bottom panels of Fig. 1. These calculations clearly capture the formation of a large pseudogap in the electronic spectral function \(-\frac{1}{\pi }\,{{\rm{Im}}}\,\,G({{\bf{k}}})\) at the antinodal X = (π, 0) point, which exists at already relatively high temperatures (β = 5 corresponds to T ≃ 700 K for the realistic hopping amplitude t = 0.3 eV). Additionally, we find that the anomalous Green’s function F(k) is relatively large and has a very unusual shape. Indeed, ReF(k) features a suppressed spectral weight at the X points, related to the pseudogap formation in the spectral function, which shifts its extrema in the direction toward the nodal point. We attribute such a strong deviation of the anomalous Green’s function from a usual \((\cos {k}_{x}-\cos {k}_{y})\) form of an applied external \({d}_{{x}^{2}-{y}^{2}}\) field to a fingerprint of a strongly-correlated superconductivity.

Fig. 3: The normal and anomalous Green’s function.
Fig. 3: The normal and anomalous Green’s function.
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a The imaginary part of the normal Green function G and b the real part of the anomalous Green function F (bottom panel) calculated at the lowest Matsubara frequency ν0 = π/β for 12% hole doping and inverse temperature β = 8.

Single-site reference problem: Effects of Higgs field

To identify the mechanism of pseudogap formation, we perform additional calculations using the triply irreducible local expansion (D-TRILEX) method56,57,58. This method is similar to the previously introduced DF lattice DQMC scheme, as it also employs a strong-coupling diagrammatic expansion based on an interacting reference problem. Compared to the DF lattice DQMC scheme, D-TRILEX accounts for more diagrammatic contributions to the self-energy by including an infinite set of ladder-type diagrams, enabling an accurate treatment of collective charge and paramagnetic spin fluctuations (paramagnons) with arbitrary spatial range. The trade-off, however, is that this approach cannot handle large lattice reference problems as efficiently as the DF lattice DQMC scheme. In this work, we use a single-site DMFT impurity problem as the reference system for D-TRILEX calculations, which are performed for 64 × 64 k-points in the Brillouin zone and 128 Matsubara frequencies.

In this work, the D-TRILEX method serves as a complementary tool to provide deeper insights into the obtained results, as it enables the separation of different contributions to the self-energy. In Fig. 4, we present the results calculated using D-TRILEX for the same model parameters as those in the DF lattice DQMC scheme, obtained at β = 10 for δ = 16% hole doping and the hdw = 0.01 value of the superconducting d-wave field. The left column shows the momentum-resolved normal G(k) (top panel) and anomalous F(k) (middle panel) contributions to the Nambu-Gor’kov lattice Green’s function, as well as the anomalous lattice self-energy S(k) (bottom panel), all obtained for the lowest Matsubara frequency. At this relatively large temperature, we find that the spatial magnetic fluctuations are still relatively weak and do not lead to the development of a pseudogap in the normal Green’s function G. The form of the anomalous Green’s function F in momentum space resembles the normal Green’s function G, but with an additional d-wave-like symmetry, displaying zero response at the nodal point. The anomalous self-energy also exhibits a d-wave-like symmetry in momentum space, but in contrast to the Green’s function F, it reveals a \((\cos {k}_{x}-\cos {k}_{y})\) profile that closely resembles the form of the applied external d-wave superconducting field (see Supplementary Fig. 9).

Fig. 4: The Green’s function and the self-energy.
Fig. 4: The Green’s function and the self-energy.
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a The imaginary part of the normal Green’s function G. b The real part of the anomalous Green’s function F. c The real part of the anomalous self-energy S. d The imaginary part of the normal Green’s function G calculated with the effective Higgs field hAFM = 0.485. e The real part of the anomalous Green’s function F with the Higgs field. f The real part of the anomalous self-energy S with the Higgs field. The results are obtained using D-TRILEX scheme at β = 10 for δ = 16% of hole doping at the lowest Matsubara frequency ν0 = π/β as a function of momentum k.

We note that at the considered temperature T = 0.1 and doping level of 16%, our D-TRILEX calculations do not reveal any divergence in the spin channel, indicating that the system does not develop local magnetic moments (LMM) through the formation of magnetic order. Nevertheless, in this parameter regime the spin fluctuations are already rather strong, which may suggest that the system could still form LMM in close proximity to the ordered state. However, applying the Landau-like criterion formulated in Ref. 59 indicates the absence of LMM. A remaining possible source of LMM formation is the emergence of singlet states (local RVB46 physics), which cannot be captured by the single-site impurity problem and are very difficult to describe within a perturbative diagrammatic expansion. The switching from LMM regime, with well-defined magnetic moments on each site, to formation of inter-site quantum singlet state was discussed in detail in the framework of exactly solvable double-Bethe model60. Additionally, the cluster D-TRILEX61 results obtained for the staggered dimer reference system indicate the presence of pseudogap at 15% hole doping. These observations allow us to assume, that in our case we are on the “singlet” (similar to the RVB) side of this transition, which explains the difference between the cluster DF-DQMC and single-site D-TRILEX results.

The effect of the LMM formation can be incorporated into the D-TRILEX framework by introducing an effective Higgs condensate of paramagnons, corresponding to a classical (static) AFM field hAFM, as detailed in Materials and Methods. The Higgs field plays a crucial role in the formation of a pseudogap, as discussed in Refs. 42,43. In the right column of Fig. 4, we show the D-TRILEX results calculated in the presence of the Higgs condensate. We find, that by setting the value of the field to hAFM = 0.485 the obtained results qualitatively align with those of the DF lattice DQMC scheme.

Discussion

The obtained results call for identifying the microscopic processes behind the formation of the two-gap structure in the superconducting spectral function and understanding the implications of these processes for the mechanism of the strong-coupling theory of d-wave superconductivity. Our general answer to this question is that this mechanism is similar to Anderson RVB46,62 and kinetic t⊥ mechanism of multilayer pair-hole hopping46,63,64 (see Suplemental Materials for simple double-plaquette results). The main idea is that, informally speaking, it is not energetically favorable for fermions in the normal phase to be “bad”, that is, non-quasiparticle, so that they prefer to form, instead, a superconducting condensate. Anderson connected this energy balance with the interlayer hopping which is more difficult for non-Fermi-liquid state than for regular Fermi liquid. This seems to be not supported by further experimental development, and recent discovery of monolayer high-temperature cuprate superconductors65 probably closes the discussion in this particular point. Contrary, we believe that this is in-plane NNN kinetic energy, which plays the role that Anderson attributed to the interlayer hopping. Despite we show that the pseudogap formation by itself competes with superconductivity, this is not the main effect of the NNN hopping. As was have demonstrated by exact diagonalization for a small, 4 × 4 periodic system31, \({t}^{{\prime} }=-0.3\) leads to a huge enhancement of the binding energy of two holes, and this is related to the peculiarities of energy spectrum of isolated plaquette. We believe that this essentially strong-coupling physics provides the necessary missing component of superconducting glue. Indeed, in our point of view, the preformed superconducting pairs already exist in the small clusters of the order of 4 × 4 due to strong AFM correlations, and it is \({t}^{{\prime} }\) that makes these hole pairs “coherent” even in the single Cu-O plane.

Interestingly, at the theoretical level the role of interlayer hopping is very similar to the role of NNN in-plane hopping. To check this, a bit unexpected, statement we performed calculations for the simple model of double plaquette (see Suplemental Materials), with including of t⊥ but with \({t}^{{\prime} }=0\) between plaquettes. The results show that t⊥ also leads to a huge increase of hole-hole binding energy. In this sense, we believe that the key idea of P. W. Anderson connecting the origin of superconductivty in cuprates with non-Fermi-liquid behavior in normal phase is basically correct, and the only (but important!) modification is that the role which he initially attributed to t⊥ is actually played by \({t}^{{\prime} }\).

Another important effect comes from the large density of state at high-temperatures near the Fermi level for hole doping around δ ≃ 15% and \({t}^{{\prime} }=-0.3\), which is related to a “highly degenerate” ground state of a small cluster31. In this case, the formation of a pseudogap we can attribute to the physics of periodic Kondo problem or “destructive interference phenomena”66. It would be very interesting to connect our microscopic approach to the phenomenological two-fluid model of cuprate superconductors18, which assumes the coexistence of two fermionic subsystems in the normal phase, a Fermi-liquid and a non-Fermi-liquid ones, where only the latter becomes superconducting at low temperatures. For the large doping δ≥25% our DF lattice DQMC scheme shows that the “bad electrons” become much more coherent correlated quasiparticles (see Supplementary Fig. 7).

To identify the precise mechanism by which “bad” electrons become superconducting, let us compare the results obtained within the lattice DF-DQMC and the single-site D-TRILEX approaches. The results of the DF lattice DQMC scheme (Fig. 1), which clearly exhibit the pseudogap feature in G and a substantial reduction in the intensity of F at the antinodal point, are therefore not supported by the single-site approximation. The observed difference indicates that accounting for the effect of paramagnons alone is insufficient to reproduce the two-gap structure in the electronic spectra. To identify the missing ingredient, we note the key difference between the two methods. The DF lattice DQMC scheme utilizes an exactly solved lattice reference system, which efficiently captures the “bad fermion” behavior that emerges already at rather high temperatures. This behavior manifests itself as the simultaneous appearance of the pseudogap at the antinodal point in the electronic spectrum and the formation of LMMs in the system, composed of these “bad” electrons. These effects are entirely absent in the single-site impurity problem and cannot be fully accounted for by paramagnetic spin fluctuations within ladder-type diagrammatic approximations. In the DF lattice DQMC scheme this effect exists already in the non-perturbative solution of the reference AFM Mott insulator system due to strong space-time correlations in the Monte-Carlo auxiliary spin-fields.

We note that the LMM in the single-band model for HTSC cuprates forms differently from that in multi-orbital systems such as iron pnictides67. In the latter case, LMM formation is driven by strong Hund’s exchange coupling68, making the static approximation for the LMM well justified. In contrast, in the single-band Hubbard model, the LMM, associated with a single lattice site, does not exist in the static limit, and its emergence is not a true physical transition but rather a crossover59. Nevertheless, LMM formation in the single-band case can be identified through spontaneous symmetry breaking associated with the formation of a short-range AFM ordering, which gives rise to a nonzero average value of the static AFM component of the Higgs field linked to the LMM. Here, we further reveal its impact on the superconducting response. It turns out that what plays the role in superconductivity of cuprates (assuming applicability of \(t-{t}^{{\prime} }\) single-band Hubbard model) is not the formation of single-site LMM but, rather, a RVB-style singlet state at the bounds60.

After accounting for the effect of the Higgs field within the D-TRILEX scheme, the momentum dependence of the Green’s function closely matches the DF lattice DQMC results shown in the bottom panels of Fig. 1. In particular, now the normal Green’s function G exhibits a pseudogap at the antinodal point, which further confirms that the formation of LMMs and the emergent “bad fermion” behavior are intrinsically connected processes. The appearance of the pseudogap leads to a reduction in spectral weight in the anomalous Green’s function F, causing the extrema of F to shift from the antinodal point towards the nodal point. The inclusion of the Higgs field also leads to another significant modification. The momentum dependence of the anomalous self-energy S changes from the cosine form of the applied superconducting field, resulting from the scattering on the spatial magnetic fluctuations (bottom left panel), to a pattern resembling the Green’s function with a momentum shift of k → k + Q, where Q = {π, π}. This behavior arises from the AFM correlations between the LMMs, which generate an effective classical AFM mode that drives electronic scattering in the self-energy, as detailed in the Materials and Methods section. We also observe that incorporating the effect of the Higgs field significantly enhances the superconducting response, leading to a substantial increase in the maximum values of the anomalous part of both the Green’s function F and self-energy S.

This result might look surprising since the formation of a pseudogap typically suppresses superconductivity by reducing the spectral weight at the Fermi surface. To investigate the role of the pseudogap in the superconducting response, we perform an additional D-TRILEX calculation in which the contribution of the Higgs field is included only in the normal part of the self-energy Σ. As shown in the Supplementary Note 4, this results in a reduced superconducting response, consistent with expectations. Specifically, the maximum value of the anomalous self-energy S decreases from ≃ 0.03 (bottom left panel of Fig. 4) to ≃ 0.02, confirming that the formation of the pseudogap suppresses superconductivity. However, when the Higgs condensate is included in both the normal (Σ) and anomalous (S) self-energies (bottom right panel of Fig. 4), the maximum value of the anomalous self-energy increases to ≃ 0.05. This indicates that the AFM correlations of LMMs, which are responsible for the formation of the pseudogap, simultaneously enhance superconductivity. By comparing the two cases with the pseudogap present, we conclude that the Higgs condensate contributes a bit more than 50% to the superconducting response, in addition to the standard spin-fluctuation mechanism of electronic scattering on paramagnons. This estimation is consistent with the amount of mysterious contributions to the superconducting pairing found in DCA calculations33, establishing the LMMs formation and their AFM correlations as the missing strong-coupling mechanism of d-wave superconductivity.

Note that bifacial effect of AFM correlations of LMMs on superconductivity can be considered as an example of a more general tendency. The effect of bosonic degrees of freedom on superconductivity is frequently twofold. On the one hand, they can produce additional glue between electrons (that is, additional attractive interaction) which favors superconductivity. On the other hand, they can lead to redistribution of spectral density decreasing the density of states at the Fermi energy which may be harmful for superconductivity. In the weak-coupling regime, the example of this twofold behaviour is provided by plasmonic superconductivity69. We consider here strong coupling situation which means that direct analogies are not possible but this example may be make the bifacial character of the effects under consideration less surprising.

Conclusion

We have presented a strong coupling perturbation scheme within the lattice Quantum Monte-Carlo framework, which is allows to study the reconstruction of electronic spectrum from the Mott-Slater insulating state for the half-field particle-hole symmetric Hubbard model for finite temperature to the correlated metal for about 15% hole doping and large \({t}^{{\prime} }/t\approx -0.3\) which dictate by the chemical bonding in the High-temperature cuprate superconductors. We investigate formation of the two different gap structures, one is related with the pseudogap in the normal part of Green’s function and located near antinode, while other consists of d-wave like superconducting gap in the anomalous part of the Green’s function between nodal and antinodal point. Using additional comparison with single-site D-TRILEX scheme we demonstrate that enhancement of superconductivity in the strong coupling regime is related to short-range antiferromagnetic fluctuations of local moments produced by effective Higgs fields.

In simple words, we present argument supporting Anderson basic idea on bad fermions in the normal phase as a driving force of superconducting instability but with an important correction that the role which he initially attributed to the interlayer hopping can be played also by in-plane next-nearest-neighbor hopping.

Methods

The t–\({t}^{{\prime} }\) Hubbard model for single correlated cuprate band can be written as

$$\widehat{{H}_{\alpha }}=-{\sum }_{i,j,\sigma }{t}_{ij}^{(\alpha )}{c}_{i\sigma }^{{\dagger} }{c}_{j\sigma }+{\sum }_{i}U({n}_{i\uparrow }-\frac{1}{2})({n}_{i\downarrow }-\frac{1}{2})\,,$$
(1)

where the operator \({c}_{i\sigma }^{({\dagger} )}\) creates (destroys) an electron on a lattice site i with the spin projection σ ∈ {↑, ↓}. \({n}_{i\sigma }={c}_{i\sigma }^{{\dagger} }{c}_{i\sigma }\) is the density operator, \({t}_{ij}^{\alpha }\) is the hopping amplitude between the i and j sites, and U is the on-site Coulomb repulsion.

We solve the introduced model using the so-called dual diagrammatic techniques pioneered by the Dual Fermion (DF) theory49. This approach enables the treatment of essential correlation effects non-perturbatively by exactly solving a suitably chosen reference system. The remaining correlation effects, which extend beyond the reference problem, are then systematically included through a perturbative diagrammatic expansion. The diagrammatic expansion is performed in the effective (dual) space for several key reasons. First, it prevents double counting of correlations between the reference system and the remaining parts of the problem. Second, it transforms a non-perturbative at large U expansion in terms of the original degrees of freedom into a perturbative one in the dual space. Consequently, the dual diagrammatic approach simultaneously combines weak- and strong-coupling expansions, making it exact in both these limits. Furthermore, the dual diagrammatic expansion can be constructed based on arbitrary interacting reference systems, ranging from single49 or multiple58,70,71 impurity problems of dynamical mean-field theory72 to clusters of lattice sites31,45,50.

Dual Fermion lattice DQMC scheme

In this work, we explore dual fermion schemes that are based on two different reference systems. The first approach considers the 16 × 16 half-filled particle-hole-symmetric lattice as a reference. We introduce an α parameter to distinguish hopping amplitudes of the reference (α = 0) and original (α = 1) system. The hoppings can then be written in the following form:

$${t}_{ij}^{(\alpha )}=\left\{\begin{array}{ll}t, \hfill & {{\rm{for}}}\,(i,j)\,\in \,{{\rm{NN}}} \hfill\\ \alpha {t}^{{\prime} }, & {{\rm{for}}}\,(i,j)\,\in {{\rm{NNN}}}\\ \alpha \mu , & {{\rm{for}}}\,i=j \hfill\\ 0, \hfill & \,{{\rm{otherwise}}} \hfill\end{array}\right.$$
(2)

where t is the nearest-neighbor (NN) and \({t}^{{\prime} }\) is the next-nearest-neighbor (NNN) hopping amplitudes on a square lattice. Note, that the chemical potential for the half-filled particle-hole symmetric (\({t}^{{\prime} }=0\)) reference system is equal to μ0 = 0 in agreement with Eq. (2) and our definition of the Hubbard interaction term in Eq. (1).

The advantage of considering the specified reference problem is that it can be solved numerically exactly in the strongly correlated regime (U ≈ 8t) using the Green’s function auxiliary-field determinant quantum Monte Carlo (DQMC) method73. This approach allows simulations on relatively large N × N × L space-time lattices, with N ≃ 20 sites and L ≃ 100 imaginary time slices, due to the absence of the fermionic sign problem74 in the particle-hole-symmetric case. It is crucial for the present theoretical scheme that the reference system already incorporates a significant portion of the correlation effects in the system. If we look at the density of states proportional to local Green’s function obtained by stohastic analitical continuation75 for the half-filled Hubbard model for U = 5.6 and \({t}^{{\prime} }=0\) in Fig. 5, one can see the existence of high-energy Hubbard bands as well as lower-energy antiferromagnetic Slater peaks at the Mott gap. Such a “four-peak” structure is the characteristic feature of the half-filled Hubbard model for large interaction strength76. It is important to note that the Slater peaks cannot be captured by the paramagnetic single-site reference problem in the DMFT-like approximation49.

Fig. 5: The Density of States.
Fig. 5: The Density of States.
Full size image

Imaginary part of the local Green’s function obtained for β = 10 and U = 5.6 for the half-field reference model (blue line) and the final lattice system with \({t}^{{\prime} }=-0.3\) and the chemical potential μ = − 1.8 (red line), which corresponds to 13% hole doping.

To construct the diagrammatic expansion, we integrate out the reference problem and transform the fermionic degrees of freedom to the dual space, c(†) → f (†). This leads to the following action:

$$\widetilde{{{\mathcal{S}}}}=-{{\rm{Tr}}}{\sum }_{1,2}{\varphi }_{1}^{* }{[\widetilde{{{\mathcal{G}}}}]}_{12}^{-1}{\varphi }_{2}+\widetilde{\Phi }[\,{f}^{* },f],$$
(3)

written in terms of Grassmann variables f (*) for dual fermions. The interaction part of the action \(\widetilde{\Phi }[\,{f}^{* },f]\) consists of the connected part of the exact four-point correlation function \(\langle {c}_{1,{\sigma }_{1}}{c}_{2,{\sigma }_{2}}^{* }{c}_{3,{\sigma }_{3}}{c}_{4,{\sigma }_{4}}^{* }\rangle\) of the reference problem45,49, where numbers represent the combined space-time {i, τ} indices. We perform calculations in the presence of an external superconducting d-wave field. Therefore, it is convenient to work in the Nambu representation by introducing spinors \({\varphi }^{* }=({f}_{\uparrow }^{* },{f}_{\downarrow })\) and \(\varphi ={({f}_{\uparrow },{f}_{\downarrow }^{* })}^{T}\). In (3), the bare dual Green’s function

$$\widetilde{{{\mathcal{G}}}}={\left[{\widetilde{t}}^{-1}-g\right]}^{-1}=\left(\begin{array}{cc}{\widetilde{{{\mathcal{G}}}}}^{\uparrow \uparrow } & {\widetilde{{{\mathcal{G}}}}}^{\uparrow \downarrow }\\ {\widetilde{{{\mathcal{G}}}}}^{\downarrow \uparrow } & -{\widetilde{{{\mathcal{G}}}}}^{\downarrow \downarrow }\end{array}\right)$$
(4)

is a 2 × 2 pseudospin matrix in the Nambu space, and the trace is taken over this space. In (4), g is the exact Green’s function of the reference system, and \(\widetilde{t}={t}^{(1)}-{t}^{(0)}\) represents the difference between the hopping matrices including the chemical potentials as the diagonal term of the original and reference problems in Eq. (2).

The correlations effects beyond the reference problem are taken into account by the first-order contribution to the self-energy in the dual space45,49:

$$\begin{array}{rcl}{\widetilde{\Sigma }}_{12}^{\uparrow \uparrow } & = & -{\sum }_{3,4}\left[\langle {c}_{1\uparrow }{c}_{2\uparrow }^{* }{c}_{3\uparrow }{c}_{4\uparrow }^{* }\rangle {\widetilde{{{\mathcal{G}}}}}_{43}^{\uparrow \uparrow }+\langle {c}_{1\uparrow }{c}_{2\uparrow }^{* }{c}_{3\downarrow }{c}_{4\downarrow }^{* }\rangle {\widetilde{{{\mathcal{G}}}}}_{43}^{\downarrow \downarrow }\right],\\ {\widetilde{\Sigma }}_{12}^{\uparrow \downarrow } & = & -{\sum }_{3,4}\langle {c}_{1\uparrow }{c}_{2\downarrow }^{* }{c}_{3\downarrow }{c}_{4\uparrow }^{* }\rangle {\widetilde{{{\mathcal{G}}}}}_{43}^{\uparrow \downarrow }, \hfill\end{array}$$
(5)

and similar for the remaining two spin components.

The perturbative parameter for the DF diagrammatic expansion can be estimated as follows. The dual Green’s function \(\widetilde{G}\)(4) is of the order of the next-nearest-neighbor hopping \({t}^{{\prime} }\) and the difference in the chemical potential μ, namely \(\widetilde{G} \sim \widetilde{t}=-4{t}^{{\prime} }\,\cos ({k}_{x})\cos ({k}_{y})-\mu\). Since the dual self-energy (5) has the same units as the exact reference Green’s function g, the perturbative parameter of the diagrammatic expansion is of the order of: \( < c{c}^{* }c{c}^{* } > \widetilde{G}/g \sim U{g}^{3}\,\widetilde{G}W\), where the bandwidth W appears due to integration over the energy. Since the reference Green’s function g ~ 1/W, the perturbative parameter is \(\sim U\widetilde{t}/{W}^{2}\). The latter estimate is correct, assuming that the Hubbard U is of the order of W/2, which is our case. In the formal limit of large U the expansion parameter is, instead, \(\widetilde{t}/U\).

In order to develop a practical computational scheme for a large lattice system, we stochastically evaluate (5). Within the determinant DQMC scheme, which employs auxiliary Ising fields {s}, the Wick theorem can be applied to compute four-point correlation functions:

$${\langle {c}_{1}{c}_{2}^{* }{c}_{3}{c}_{4}^{* }\rangle }_{s}={\langle {c}_{1}{c}_{2}^{* }\rangle }_{s}\,{\langle {c}_{3}{c}_{4}^{* }\rangle }_{s}-{\langle {c}_{1}{c}_{4}^{* }\rangle }_{s}\,{\langle {c}_{3}{c}_{2}^{* }\rangle }_{s}$$
(6)

In order to discard the disconnected part of the correlation function in (5), we subtract the exact Green’s function of the reference system, \({g}_{12}=-\langle {c}_{1}{c}_{2}^{* }\rangle\), obtained from the previous DQMC run, as follows:

$${\widetilde{g}}_{12}^{s}={g}_{12}^{s}-{g}_{12}.$$
(7)

We also utilize Fourier space for the efficient evaluation of (5). Within the DQMC framework, the lattice auxiliary Green’s function is not translationally invariant, meaning \({g}_{12}^{s}=-{\langle {c}_{1}{c}_{2}^{* }\rangle }_{s}\). To calculate \({\widetilde{g}}_{k{k}^{{\prime} }}^{s}\), we employ a double fast Fourier transform to momentum k and Matsubara frequency ν space, with k ∈ {k, ν}. It is important to note that the dual Green’s function, \({\widetilde{{{\mathcal{G}}}}}_{k}^{\sigma \sigma {\prime} }\), is translationally invariant. Additionally, after performing the Monte Carlo summation over the auxiliary spins {s}, the self-energy also becomes translationally invariant. The final expression for the dual self-energies then reads:

$$\begin{array}{rcl}{\widetilde{\Sigma }}_{k}^{\uparrow \uparrow } & = & -{\sum }_{s,{k}^{{\prime} }}\left[\left({\widetilde{g}}_{kk}^{\uparrow s}{\widetilde{g}}_{{k}^{{\prime} }{k}^{{\prime} }}^{\uparrow s}-{\widetilde{g}}_{k{k}^{{\prime} }}^{\uparrow s}{\widetilde{g}}_{{k}^{{\prime} }k}^{\uparrow s}\right){\widetilde{{{\mathcal{G}}}}}_{{k}^{{\prime} }}^{\uparrow \uparrow }+{\widetilde{g}}_{kk}^{\uparrow s}{\widetilde{g}}_{-{k}^{{\prime} },-{k}^{{\prime} }}^{\downarrow s}{\widetilde{{{\mathcal{G}}}}}_{-{k}^{{\prime} }}^{\downarrow \downarrow }\right],\\ {\widetilde{\Sigma }}_{k}^{\uparrow \downarrow } & = & -{\sum }_{s,{k}^{{\prime} }}{\widetilde{g}}_{k{k}^{{\prime} }}^{\uparrow s}{\widetilde{g}}_{-k,-{k}^{{\prime} }}^{\downarrow s}{\widetilde{{{\mathcal{G}}}}}_{{k}^{{\prime} }}^{\uparrow \downarrow }. \hfill\end{array}$$
(8)

Note that only \({\widehat{\widetilde{{{\mathcal{G}}}}}}_{k}\) contains both normal and anomalous contributions. In contrast, \({\widetilde{g}}_{k{k}^{{\prime} }}^{\uparrow s}\) and \({\widetilde{g}}_{k{k}^{{\prime} }}^{\downarrow s}\), obtained for the paramagnetic reference system in the auxiliary spins, are diagonal in the spin space. There is a normalization factor associated with the number of QMC steps (noting that there is no sign problem for the reference system), along with two additional factors of βN2 arising from the double Fourier transform and the summation over \({k}^{{\prime} }\). For relative high temperature (β = 3) we have directly checked accuracy of this scheme with numerically exact DQMC calculations for the same parameters μ and \({t}^{{\prime} }\) due to a mild sign problem (see Supplemental Material).

In superconducting calculations, we include a small external d-wave field:

$${\Delta }_{{{\bf{k}}}}=2\,{h}_{dw}\,(\cos {k}_{x}-\cos {k}_{y}),$$
(9)

which explicitly enters the perturbation hopping matrix in the Nambu space

$${\widetilde{t}}_{{{\bf{k}}}}({h}_{dw})=\left(\begin{array}{cc}{\widetilde{t}}_{{{\bf{k}}}} & {\Delta }_{{{\bf{k}}}}\\ {\Delta }_{{{\bf{k}}}} & -{\widetilde{t}}_{-{{\bf{k}}}}\end{array}\right).$$
(10)

The final expression for the lattice Green’s function of real fermions has the following matrix form49:

$${\widehat{G}}_{k}={\left[{\left({g}_{k}+{\widetilde{\Sigma }}_{k}\right)}^{-1}-{\widetilde{t}}_{{{\bf{k}}}}\right]}^{-1}=\left(\begin{array}{cc}{G}_{k} & {F}_{k}\\ {F}_{k}^{* } & -{G}_{-k}\end{array}\right),$$
(11)

where we introduced a shortened notations for the normal G = G↑↑ and anomalous F = G↑↓ Green’s functions in the Nambu space.

An example of the calculated normal local Green’s function for the realistic \({t}^{{\prime} }\) and the μ shift is presented in Fig. 5 in comparison with the reference case. We can clearly see the formation of a narrow quasiparticle peak from the “low Slater band”, while the Hubbard bands stay approximately on the same position due to the local nature of Mott-correlations. The intensity of the metallic upper Hubbard bands increases due to “merging” with the upper Slater band. We also examined the convergence properties of the dual perturbation series. A second-order DF-QMC expansion for a different set of system parameters (4 × 4 cluster for U = 2 and β = 5) shows that \({\widehat{G}}_{k}\) acquires only minor corrections beyond the first order.

D-TRILEX scheme

The second computational scheme is based on the dual triply irreducible local expansion (D-TRILEX) method56,57,58. For this approach we use the paramagnetic single-site impurity problem of DMFT as a reference system with the same chemical potential μ as in the original lattice problem. In this case, it is convenient to work in the momentum and Matsubara frequency space representation. The dual action in D-TRILEX is similar to (3), but the perturbation hopping matrix that enters the bare dual Green’s function (4) is given by the difference \({\widetilde{t}}_{k}={t}_{{{\bf{k}}}}^{(2)}-{\Delta }_{\nu }\) between the lattice dispersion \({t}_{{{\bf{k}}}}^{(2)}\), that includes both the NN and NNN hoppings, and the hybridization function Δν of DMFT. The interaction part of the action \(\widetilde{\Phi }\) exploits the partially-bosonized D-TRILEX form56, in which the interaction between fermions is mediated by charge density (ς = d) and magnetic (ς = m ∈ x, y, z) fluctuations, described by the bosonic fields bς:

$$\widetilde{\Phi }=-\frac{1}{2}{\sum }_{q,\varsigma }{b}_{-q}^{\varsigma }{\left[{\widetilde{{{\mathcal{W}}}}}_{q}^{\varsigma }\right]}^{-1}{b}_{q}^{\varsigma }+{\sum }_{\begin{array}{c}k,q,\varsigma ,\\ \sigma \sigma {\prime} \end{array}}{\Lambda }_{\nu \omega }^{\varsigma } \,{f}_{k\sigma }^{* }{\sigma }_{\sigma \sigma {\prime} }^{\varsigma }{f}_{k+q,\sigma {\prime} }{b}_{q}^{\varsigma }.$$
(12)

In this expression, \({\widetilde{{{\mathcal{W}}}}}^{\varsigma }\) is the interaction renormalized by the polarization operator \({\Pi }_{\omega }^{{{\rm{imp}}}\,\varsigma }\) of the reference impurity problem in the corresponding channel ς. \({\Lambda }_{\nu \omega }^{\varsigma }\) is the exact three-point vertex function of the single-site reference problem that describes the renormalized fermion-boson coupling. \({\sigma }^{{{\rm{d}}}}={\mathbb{I}}\), and σm are the corresponding Pauli matrices in the spin space. We note, that a recent attempt to investigate the role of a non-local (momentum-dependent) three-point vertex function in the electron pairing through spin fluctuations was proposed in Ref. 77.

The D-TRILEX approach self-consistently incorporates the effects of spatial collective electronic fluctuations through ladder-type diagrams for the self-energy and polarization operator in the dual space. In the presence of the superconducting field, the self-energy develops an anomalous contribution, while the normal part of the self-energy remains the same as in the paramagnetic case56,58:

$$\begin{array}{rcl}{\widetilde{\Sigma }}_{k}^{\uparrow \uparrow } & = & -{\sum }_{q,\varsigma }{\Lambda }_{\nu \omega }^{\varsigma }{\widetilde{G}}_{k+q}^{\uparrow \uparrow }{\widetilde{W}}_{q}^{\varsigma }{\Lambda }_{\nu +\omega ,-\omega }^{\varsigma }+2{\sum }_{{k}^{{\prime} }}{\Lambda }_{\nu ,0}^{{{\rm{c}}}}{\widetilde{{{\mathcal{W}}}}}_{0}^{{{\rm{c}}}}{\Lambda }_{\nu {\prime} ,0}^{{{\rm{c}}}}\,{\widetilde{G}}_{{k}^{{\prime} }}^{\uparrow \uparrow },\\ {\widetilde{\Sigma }}_{k}^{\uparrow \downarrow } & = & -{\sum }_{q,\varsigma }{\xi }^{\varsigma }{\Lambda }_{\nu \omega }^{\varsigma }{\widetilde{G}}_{k+q}^{\uparrow \downarrow }{\widetilde{W}}_{q}^{\varsigma }{\Lambda }_{-\nu ,-\omega }^{\varsigma }, \hfill\end{array}$$
(13)

where ξm/d = ± 1. Despite the Green’s function has an anomalous part, the polarization operator remains diagonal in the channel space, but receives an additional term originating from the anomalous Green’s function:

$${\widetilde{\Pi }}_{q}^{\varsigma }=2{\sum }_{k}\left({\Lambda }_{\nu +\omega ,-\omega }^{\varsigma }{\widetilde{G}}_{k}^{\uparrow \uparrow }{\widetilde{G}}_{k+q}^{\uparrow \uparrow }{\Lambda }_{\nu \omega }^{\varsigma }+{\xi }^{\varsigma }{\Lambda }_{-\nu ,-\omega }^{\varsigma }{\widetilde{G}}_{k}^{\downarrow \uparrow }{\widetilde{G}}_{k+q}^{\uparrow \downarrow }{\Lambda }_{\nu \omega }^{\varsigma }\right).$$
(14)

The dressed dual Green’s function and the renormalized interaction are obtained via the corresponding Dyson equations:

$${\left[{\widehat{\widetilde{G}}}_{k}\right]}^{-1}={\left[{\widehat{\widetilde{{{\mathcal{G}}}}}}_{k}\right]}^{-1}-{\widehat{\widetilde{\Sigma }}}_{k},$$
(15)
$${\left[{\widetilde{W}}_{q}^{\varsigma }\right]}^{-1}={\left[{\widetilde{{{\mathcal{W}}}}}_{q}^{\varsigma }\right]}^{-1}-{\widetilde{\Pi }}_{q}^{\varsigma },$$
(16)

and the lattice Green’s function is obtained using (11). The normal \({\Sigma }_{k}={\Sigma }_{k}^{\uparrow \uparrow }\) and anomalous \({S}_{k}={\Sigma }_{k}^{\uparrow \downarrow }\) parts of the lattice self-energy can be obtained through the Dyson equation for the lattice Green’s function.

To incorporate the effect of LMM formation in D-TRILEX, one has to assume spontaneous symmetry breaking, characterized by a non-zero condensate of the static (ω = 0) AFM (q = Q = {π, π}) component of the spin-z field in (12):

$$\langle {b}_{Q,0}^{z}\rangle =\pm {h}_{{{\rm{AFM}}}},$$
(17)

which serves as the Higgs field42,43,59 that introduces the AFM ordered LMMs in the system. In general, the value of the Higgs field can be chosen self-consistently from the minimum of the corresponding free energy, similar to the approach of Ref. 59. However, in this work, we determined the value of hAFM by comparing the results of the D-TRILEX and DF lattice DQMC schemes.

Averaging the dual Green’s function, dressed by the Higgs field, over the positive and negative values of hAFM effectively accounts for the singlet fluctuations and reproduces the paramagnetic case. The averaging procedure results in an additional contribution to both the normal and anomalous parts of the self-energy:

$${\widehat{\widetilde{\Sigma }}}_{k}^{{{\rm{AFM}}}}={\Lambda }_{\nu ,0}^{z}{\widehat{\widetilde{G}}}_{{{\bf{k}}}+Q,\nu }{h}_{{{\rm{AFM}}}}^{2}{\Lambda }_{\nu ,0}^{z}.$$
(18)

In this expression \({\widehat{\widetilde{\Sigma }}}^{{{\rm{AFM}}}}\) and \(\widehat{\widetilde{G}}\) are 2 × 2 matrices in the Nambu space. We note that a similar contribution, but only to the normal self-energy, has been considered in Refs. 42,43. Recalling the D-TRILEX form for the self-energy (13), the derived LMM contribution to the dual self-energy can be accounted for by adding the following contribution to the z component of the renormalized interaction:

$${\widetilde{W}}_{q}^{z}\to {\widetilde{W}}_{q}^{z}-{h}_{{{\rm{AFM}}}}^{2}{\delta }_{{{\bf{q}}},Q}{\delta }_{\omega ,0}.$$
(19)

This additional term corresponds to an effective classical AFM mode and is similar to the one in Ref. 78. Note that in our notations the renormalized interaction \({\widetilde{W}}_{q}\) is negative.

In general, the non-zero value of the field hAFM can be determined similarly to Ref. 59, namely from the minimum of the free energy of the system at a temperature, where the free energy develops a double-well form as a function of the bosonic field \({b}_{Q,0}^{z}\). In this work, we perform D-TRILEX at a rather high temperature (β = 10) and choose the value of the field hAFM = 0.485 to qualitatively reproduce the results obtained within the DF lattice DQMC scheme.