Abstract
A pairdensitywave (PDW) is a superconducting state with an oscillating order parameter. A microscopic mechanism that can give rise to it has been long sought but has not yet been established by any controlled calculation. Here we report a densitymatrix renormalizationgroup (DMRG) study of an effective tJV model, which is equivalent to the HolsteinHubbard model in a strongcoupling limit, on long two, four, and sixleg triangular cylinders. While a state with longrange PDW order is precluded in one dimension, we find strong quasilongrange PDW order with a divergent PDW susceptibility as well as the spontaneous breaking of timereversal and inversion symmetries. Despite the strong interactions, the underlying Fermi surfaces and electron pockets around the K and \({K}^{\prime}\) points in the Brillouin zone can be identified. We conclude that the state is valleypolarized and that the PDW arises from intrapocket pairing with an incommensurate center of mass momentum. In the twoleg case, the exponential decay of spin correlations and the measured central charge c ≈ 1 are consistent with an unusual realization of a LutherEmery liquid.
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Introduction
In a pairdensitywave (PDW) state, the Cooper pairs have a nonzero center of mass momentum^{1}. Longstudied versions of such states are the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phases^{2,3} that can arise in the context of BCS theory in the presence of a small degree of spin magnetization. However, it has been conjectured that a PDW could also arise from a strongcoupling mechanism and could be responsible for the dynamical layer decoupling phenomena observed in certain cuprate materials^{4,5,6}. While PDW states have been shown to be energetically competitive in certain meanfield and variational calculations^{7,8,9,10}, and evidence has been sought in numerical studies of generalized tJ^{4,11,12,13,14} and Hubbard models^{15,16}, PDW longrange order has not been established in any controlled calculation in two or higher dimensions. In onedimensional systems—which can be reliably treated using densitymatrixrenormalizationgroup (DMRG)—the closest one can get to such order is PDW quasilongrange order with a divergent susceptibility. However, even this has only been seen to date for the case of commensurabilitytwo PDW order in HeisenbergKondo chains^{17,18}; other reported DMRG sightings^{14,19,20} have found PDW correlations that fall off fast enough to yield only finite superconducting susceptibility.
Previously^{21}, we analyzed a strongcoupling limit of the HolsteinHubbard model, where the onsite direct electron–electron repulsion, U_{e–e}, is stronger than the phonon induced attraction, U_{e–ph}. We showed that the low energy physics is captured by an effective model of a form similar to the tJV model derived from the strongcoupling limit of the ordinary Hubbard model, but in an unfamiliar range of parameters. In the effective model, the electron hopping is exponentially suppressed by a Frank–Condon factor, \({}_{e}{{}\frac{U_e{\rm ph}}{2\omega_D}}\), relative to its bare value, t, where ω_{D} is the optical phonon frequency. Meanwhile, the nearestneighbor magnetic exchange coupling J and density repulsion V, are relatively unsuppressed. Thus one can readily access regimes in which the effective interactions are comparable to or larger than the electron kinetic energy. Specifically, we analyzed the model on the twodimensional triangular lattice in the adiabatic limit, where the suppression is large. We found a PDW when the electrons are dilute and the bare hopping integral t is negative (i.e., the band maximum occurs at k = 0 while minima occur at the K and \(K^{\prime}\) points). Unfortunately, in the adiabatic limit, the PDW is a BoseEinstein condensate (BEC) of realspace pairs, in which the same Frank–Condon factor suppresses the condensation temperature, meaning that this order would be destroyed at exponentially small temperatures.
In the present work, we employ DMRG to study the effective model in a range of parameters suggested by the abovediscussed analysis, but now for the case of a finite Frank–Condon factor. We perform DMRG on triangular cylinders with N_{x} × N_{y} sites with open boundary conditions in the xdirection and periodic boundary conditions in the ydirection. The xdirection is aligned with a primitive vector, as shown in Fig. 1a. \({\hat{e}}_{a = 1}=(1,0)\), \({\hat{e}}_{a = 2,3}=(\frac{1}{2},\pm\! \frac{\sqrt{3}}{2})\) are the three primitive vectors of the triangular lattice. The compactification of the lattice to a cylinder restricts us to N_{y} even. The momentum values in the ydirection take values \({k}_{y}=\frac{m}{{N}_{y}}\frac{4\pi }{\sqrt{3}}\), \(m\in {\mathbb{Z}}\), indicated in Fig. 1b. We will primarily focus on the twoleg case but some key findings are also verified in the four and sixleg cases. For all the cylinders we can treat in this way, we find a divergent PDW susceptibility, as shown in Fig. 2. The PDW state spontaneously breaks timereversal and inversion symmetries, resulting in a local pattern of equilibrium currents as shown in the inset of Fig. 3a. The state is far from the BEC limit as can be seen in Fig. 3b from the existence of a sharp drop in the electron occupation at welldefined Fermi points corresponding to electron pockets around the K and \(K^{\prime}\) points in the Brillouin Zone (BZ). However, the size of the pockets (i.e., the value of 2k_{F}) is a factor of two larger than what it would be for noninteracting electrons, which we will argue reflects spontaneous valley polarization. As shown in Figs. 2a, 3c, the PDW ordering wavevector, Q, is incommensurate and densitydependent. We will show that the ordering vector takes the value expected for intrapocket pairing.
Results
Pairdensitywave correlations
We first examine the equaltime pairpair correlations. The singlet pairing order parameter is defined on the nearestneighboring bonds to a given site r_{i} as \({{{\Delta }}}_{a}({{{{\bf{r}}}}}_{i})\equiv {\hat{c}}_{{{{{\bf{r}}}}}_{i},\uparrow }{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\downarrow }+{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\uparrow }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\downarrow }\). The pairpair correlator is defined as \({P}_{ab}({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {{{\Delta }}}_{a}^{{\dagger} }({{{\bf{r}}}}+{{{{\bf{r}}}}}_{0}){{{\Delta }}}_{b}({{{{\bf{r}}}}}_{0})\rangle\). Our principle finding is that P_{ab}(r) oscillates in sign and falls slowly in magnitude at a large distance, i.e.,
This can be seen in Fig. 2a, which is a plot of \({P}_{ab}({{{\bf{r}}}})/{A}_{ab} x{ }^{{\nu }_{{{\mathrm{SC}}}}}\) at large distance for a twoleg cylinder, taking ν_{SC} = 1.20. Without breaking the reflection symmetry across the xdirection, \({\hat{e}}_{2}\) and \({\hat{e}}_{3}\) are equivalent, so we only show pairing correlations involving Δ_{1} and Δ_{2}. The fact that all the components of P_{ab} oscillate inphase with the same wavevector, Q, indicates the local swave character of the PDW. An illustration of the pair correlation in the fourleg case can be seen in the inset of Fig. 2a, which matches our previous prediction in ref. ^{21} at a short distance.
Correspondingly, the Fourier transform of the correlator, i.e., the structure factor \({S}_{ab}^{\,{{\mathrm{PDW}}}\,}({{{\bf{k}}}})\equiv \frac{1}{N}{\sum }_{{{{{\bf{r}}}}}_{i},{{{{\bf{r}}}}}_{j}}{{{{\rm{e}}}}}^{{{{\rm{i}}}}{{{\bf{k}}}}\cdot ({{{{\bf{r}}}}}_{i}{{{{\bf{r}}}}}_{j})}\langle {{{\Delta }}}_{a}^{{\dagger} }({{{{\bf{r}}}}}_{i}){{{\Delta }}}_{b}({{{{\bf{r}}}}}_{j})\rangle\), has pronounced peaks at two nonzero momenta and vanishes at zero momentum, as shown in Fig. 2b. Around each peak but outside a cutoff window of width \(\delta {k}_{x} \sim \frac{2\pi }{{N}_{x}}\) (to avoid finitesize effects), the structure factor is wellfitted by a functional form \(AB {k}_{x}Q{ }^{{\nu }_{{{\mathrm{SC}}}}1}\) as is expected, given the longdistance correlation behavior in Eq. (1). It is important to note that the wavevector of the PDW, ∣Q∣, appears to be a smooth monotonic function of the electron density and is not locked to a multiple of the momenta at the K points, i.e., the PDW is incommensurate with the lattice.
Since 1D systems generically exhibit emergent Lorenz invariance, we can infer from the equal time correlator that the static susceptibility to PDW order should vary as \(\chi (Q) \sim \max {[\frac{1}{{N}_{x}},T]}^{(2{\nu }_{{{\mathrm{SC}}}})}\) as the system size tends to infinity and temperature to zero^{22}. Thus, χ(Q) diverges at zero temperature for ν_{SC} < 2, as we find is the case for a wide range of electron densities and model parameters. That the divergence of susceptibility is not restricted to the twoleg case can be seen in Fig. 2c, which shows the structure factor and the extracted exponents at the same electron density n = 0.05 for the four and sixleg cases.
Spontaneously broken timereversal and inversion symmetries
Another prominent feature of the groundstate is that it spontaneously breaks both timereversal and inversion symmetries. This can be directly seen from the currentcurrent correlation \({{{\Lambda }}}_{ab}({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {{{{\bf{J}}}}}_{a}({{{\bf{r}}}}+{{{{\bf{r}}}}}_{0})\cdot {{{{\bf{J}}}}}_{b}({{{{\bf{r}}}}}_{0})\rangle\), where \({{{{\bf{J}}}}}_{a}({{{{\bf{r}}}}}_{i})\approx {{{\rm{i}}}}{t}_{1}{\hat{e}}_{a}\,{\sum }_{\sigma }({\hat{c}}_{{{{{\bf{r}}}}}_{i},\sigma }^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\sigma }{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\sigma }^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\sigma })\) is the current on the bond directed in the \({\hat{e}}_{a}\) direction. (The actual current operator that we compute also receives minor contributions from t_{2} and τ, see Supplementary Discussion for the full expression). As shown in Fig. 3a, the currentcurrent correlation oscillates around a nonzero value at long distance, and we also confirm in Supplementary Fig. 3 that the peak of the structure factor of the currentcurrent correlation scales linearly with system size. These facts signify persisting currents in the ground states. The pattern of the current flows is shown in the inset, indicating that the groundstate is an orbital antiferromagnet.
Valley polarization
We further identify the broken symmetry states by investigating the occupation number in momentum space, \(n({{{\bf{k}}}})\equiv \frac{1}{N}{\sum }_{{{{{\bf{r}}}}}_{i},{{{{\bf{r}}}}}_{j},\sigma }{e}^{i{{{\bf{k}}}}\cdot ({{{{\bf{r}}}}}_{i}{{{{\bf{r}}}}}_{j})}\langle {c}_{{{{{\bf{r}}}}}_{i},\sigma }^{{\dagger} }{c}_{{{{{\bf{r}}}}}_{j},\sigma }\rangle\), for the twoleg cylinder. As shown in Fig. 3b, although the strong interactions shift a small fraction of occupation weight to the vicinity of zero momentum (far above the noninteracting Fermi surface), most of the weight is confined to narrow intervals of k about the two minima of the noninteracting bands, which occur at K = (4π/3, 0) and \(K^{\prime} =K\). However, the width of these intervals  labeled “2k_{F}” in the figure—is twice as large as it would be for a noninteracting system. Moreover, we have ruled out the possibility of a spinpolarized ferromagnet by checking that the groundstate has spin 0 (shown in Fig. 4b), and the possibility of spinvalley locked polarization by confirming that the spin current correlations exponentially decay in space (shown in Supplementary Fig. 3). We also checked that the triplet pairing correlations are weak and extremely shortranged, further disproving those possibilities. All the observations are rather consistent with the supposition that the groundstate is valleypolarized. We thus conclude that the presence of electrons at both K and \(K^{\prime}\) is an artifact of the fact that, for a finitesize system, the groundstate is a superposition of states with the two possible senses of valley polarization. This is also consistent with the observation that the maximum value of n_{k} is somewhat less than 1, while for an unpolarized noninteracting system it should equal 2 for all states below the Fermi energy. The occupancy of only one valley naturally explains the observed broken symmetries and the loop currents order, since the persisting current can be estimated as \({J}_{a}\approx \pm \!2{t}_{1}n\sin ({{{\bf{K}}}}\cdot {\hat{e}}_{a})\), where ± corresponds to K or \(K^{\prime}\) valley is occupied. Note that the current pattern is translationally invariant, and correspondingly the pattern of flux breaks point group symmetry but not translation symmetry —it is not induced by the PDW order.
The valley polarization can be understood with a simple meanfield theory. We propose a trial Hamiltonian
to capture the essence of physics. To simplify the problem, we neglect the constraint on no double occupancy. Further neglecting the weak τ and t_{2} terms, we solve the meanfield equation \({\tilde{t}}_{i,a,\sigma }\approx {t}_{1}\frac{J}{2}\langle {\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\bar{\sigma }}^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\bar{\sigma }}\rangle V\langle {\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\sigma }^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\sigma }\rangle\), to the leading order in n. We find that the valleypolarized solution, with complex hopping elements \({\tilde{t}}_{i,a,\sigma }={t}_{1}^{\prime}{{{{\rm{e}}}}}^{{{{\rm{i}}}}\theta }\) and \(\theta \approx \pm\! \frac{\sqrt{3}}{4}\frac{J/2+V}{{t}_{1}}n\), is always energetically favored. The band structure is renormalized to
The introduction of the complex phase θ thus energetically distinguishes the two valleys by \(\frac{9}{2}(J/2+V)n\), the amount of which is sufficiently large to fully valleypolarize the system while keeping the positions of the band minima. This mechanism is similar to that of Stoner magnetism, but here the density of states is divergent at the band bottom so the polarization always occurs in the dilute limit. (Conversely, finite critical interaction strength is necessary for dimensions d ≥ 2.)
Therefore, the pairing in each groundstate with valley polarization must happen between the two Fermi points located at \({k}_{x}=\widetilde{K}\pm {k}_{{{\mbox{F}}}}\) and the pair momentum should be twice the center momentum \(2\widetilde{K}\). This intravalley singlet pairing mechanism is distinct from the intravalley triplet or the intervalley pairing mechanism proposed for a spinvalley locked system^{16,23}, and is enforced by the broken symmetries. This weakcoupling, meanfield pairing mechanism is complementary to the strongcoupling, BECtype mechanism proposed in ref. ^{21}. Furthermore, due to the asymmetry of the noninteracting band structure around its band minima, \(\tilde{K}\) can be calculated to the leading order in n:
In Fig. 3c, we see that the observed Q indeed matches \(2\widetilde{K}\) calculated in this way modulo a reciprocal lattice vector, G = (4π, 0), up to an error of order \(\frac{\pi }{{N}_{x}}\). This accounts for the incommensurate nature of the PDW.
LutherEmery liquid
All the longdistance behaviors we have observed are consistent with a LutherEmery liquid^{24}, at least in the twoleg case. For instance, as shown in Fig. 4a, for electron density n = 0.15, the oscillatory piece of the charge correlator \(C({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {n}_{{{{\bf{r}}}}+{{{{\bf{r}}}}}_{0}}{n}_{{{{{\bf{r}}}}}_{0}}\rangle\) has wavevector 2k_{F} = 2nπ and exhibits a powerlaw decay with exponent ν_{CDW} = 0.90, such that the expected relation ν_{CDW} ⋅ ν_{SC} = 1.08 ≈ 1 is approximately satisfied. The spin correlator \(S({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {{{{\bf{S}}}}}_{{{{\bf{r}}}}+{{{{\bf{r}}}}}_{0}}\cdot {{{{\bf{S}}}}}_{{{{{\bf{r}}}}}_{0}}\rangle\) is shortranged corresponding to a correlation length ξ ≈ 4.36, as shown in Fig. 4b. To extract the central charge, we computed the von Neumann entanglement entropy \({S}_{E}(x)\equiv {{{\rm{tr}}}}({\rho }_{x}{{\mathrm{ln}}}\,{\rho }_{x})\) where ρ_{x} is the reduced density matrix of the subsystem on one side of a cut at x. For critical systems in 1 + 1 dimensions described by conformal field theory, it has been established^{25,26} that for an open boundary system with length N_{x},
where c is the central charge, A and B are modeldependent parameters, and q is an adjustable fitting parameter that should approach the Fermi momentum k_{F} in the thermodynamic limit. We see in Fig. 4c that this formula fits well with the observed data. As shown in the inset, the central charge c ≈ 1 and q → k_{F} = nπ when N_{x} → ∞, which confirms that there is only one gapless mode resulting from filling only one of the two valleys.
Discussions
With an increasing number of legs, the PDW correlations seem to weaken somewhat, as suggested by the growth of the exponent ν_{SC}. We speculate that this is an artifact in the fewleg system: at low electron densities, the Fermi pockets are small, so the bands with nonzero k_{y} and higher energies remain unoccupied for a range of n. The effect of adding legs only leads to an increase of the filling of the k_{y} = 0 bands. In other words, the true scaling process to the twodimensional limit has not started until the allowed values of k_{y} become sufficiently closely spaced that additional bands with nonzero k_{y} cross the Fermi surface. To have more than one band crossing the Fermi surface, the required number of legs scales as \(\sqrt{\frac{4\pi /\sqrt{3}}{n}}\), which is ~12 for n = 0.05. Going to larger n could in principle help improve the situation, but this will generally place the system outside of the range of n the PDW occurs—at least for ladders with more than two legs. We have not found a range of parameters amenable to DMRG in which the PDW exists and there are multiple Fermi points. Indeed, the results obtained here in the fewleg cases are essentially still onedimensional; the main insight we obtain from them that could help to understand the 2D case is that valley polarization physics accounts well for our DMRG results in the 1D case, and this physics should apply as well in two dimensions.
More generally, our findings suggest a new route to PDW order in two or higher dimensions—one that does not rest on complicated strongcoupling physics. Consider the situation in which, either due to explicit or spontaneous timereversal symmetry breaking, a low density of weakly interacting quasiparticles from a Fermi pocket about a band extremum at a single point in the BZ (such as the K point in the present context) about which the dispersion is not symmetric. Then from this starting point, it is natural to consider a pairing instability involving electrons on opposite sides of this pocket. To the extent that the dispersion relation can be treated as quadratic (i.e., in the effective mass approximation), these states would be perfectly nested, so deviations from nesting (e.g., trigonal warping) are small in proportion to the power of the electron density. Consequently, the pairing instability occurs for correspondingly weak couplings, and the resulting state is a PDW with ordering vector or vectors that are likewise continuously varying functions of n, as in Eq. (4). This is, in a sense, an orbital version of the original FFLO mechanism.
Methods
The effective Hamiltonian (with implicit projection to states with no doubly occupied sites) is:
where t_{1} is the renormalized nearestneighbor hopping, t_{2} is a weak nextnearestneighbor hopping term via an intermediate site m, 〈i, m, j〉 represents a triplet of sites such that m is a nearestneighbor of two distinct sites i and j, and (τ + 2t_{2}) is a weak singlet hopping term where \({\hat{s}}_{ij}=({\hat{c}}_{i,\uparrow }{\hat{c}}_{j,\downarrow }+{\hat{c}}_{j,\uparrow }{\hat{c}}_{i,\downarrow })/\sqrt{2}\) is the annihilation operator of a singlet Cooper pair on bond 〈ij〉. Explicit expressions are given in ref. ^{21} for the values of these effective couplings as functions of the parameters in the original HolsteinHubbard model. For the data shown in this study, we fix t = −1, ω_{D} = 3, U_{ee} = 22, and U_{eph} = 18, which leads to J ≈ 0.208, V/J ≈ 0.666, t_{1}/J ≈ −0.240, t_{2}/J ≈ 0.0329 and τ/J ≈ 0.0445 in the effective model. However, we have checked that in a broad range of parameters or setting t_{2} = τ = 0, the results we find do not change qualitatively. We include some results with different settings in the Supplementary Discussion. The range of V/J over which PDW order occurs was found to be [0.43, 1] in the adiabatic limit in^{21}. From DMRG we find that this range shifts somewhat with increasing Frank–Condon overlap factor. Nonetheless, the range in the adiabatic limit provided guidance in searching PDW, and indeed the data shown in this paper lies in this range.
Note that twoleg cylinders can be flattened to be a purely onedimensional chain by simply neglecting the y coordinate of each site, so we plot the data for all sites in the twoleg case. All the DMRG data collected are obtained from the lowest energy state out of five trials with independently randomized initial states and all the results shown (unless otherwise stated) are extrapolated to zero truncation error, utilizing data collected with six truncation errors ranging from 3 × 10^{−6} to 1 × 10^{−7}. In the twoleg case, we have checked our results do not change significantly down to truncation error 7 × 10^{−11}, corresponding to keeping bond dimensions up to m = 4320. The correlation functions shown are averaged over all legs of the ladder, taking N_{0} = 5N_{y} different reference sites r_{0} centered around the middle of the system. All data involving sites within N_{x}/4 to the open boundary are discarded, i.e. we only retain the data on the interval x ∈ [N_{x}/4, 3N_{x}/4], to reduce boundary effects. Thus, each structure factor we compute utilizes the data on N = N_{x}N_{y}/2 sites.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its supplementary information files.
Code availability
The codes implementing the calculations of this study are available from the corresponding author upon request.
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Acknowledgements
We are grateful to HongChen Jiang, YiFan Jiang, and Patrick Lee for helpful discussions. The DMRG calculations were performed using the ITensor Library^{27}. Part of the computational work was performed on the Sherlock cluster at Stanford. SAK was supported, in part, by the National Science Foundation (NSF) under Grant No. DMR2000987. KSH was supported, in part, by Stanford VPUE through an undergraduate major grant. HY was supported, in part, by NSFC Grant No. 11825404 at Tsinghua and by the Gordon and Betty Moore Foundations EPiQS through Grant No. GBMF4302 at Stanford.
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KSH and ZH contributed equally to this work. KSH performed the numerical investigation with the help from ZH. All authors analysed the data and contributed to the writing of the paper.
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Huang, K.S., Han, Z., Kivelson, S.A. et al. Pairdensitywave in the strong coupling limit of the HolsteinHubbard model. npj Quantum Mater. 7, 17 (2022). https://doi.org/10.1038/s4153502200426w
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DOI: https://doi.org/10.1038/s4153502200426w
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