Coexistence of charge-density-wave and pair-density-wave orders in underdoped cuprates
We analyze incommensurate charge-density-wave (CDW) and pair-density-wave (PDW) orders with transferred momenta / in underdoped cuprates within the spin-fermion model. Both orders appear due to exchange of spin fluctuations before magnetic order develops. We argue that the ordered state with the lowest energy has non-zero CDW and PDW components with the same momentum. Such a state breaks lattice rotational symmetry, time-reversal symmetry, and mirror symmetries. We argue that the feedback from CDW/PDW order on fermionic dispersion is consistent with ARPES data. We discuss the interplay between the CDW/PDW order and superconductivity and make specific predictions for experiments.
Introduction. The search for competitors to d superconductivity (d-SC) in underdoped cuprates has gained strength over the last few years due to mounting experimental evidence that some form of electronic charge order spontaneously emerges below a certain doping and competes with d-SC (Refs. mark_last (); ybco (); ybco_1 (); X-ray (); X-ray_1 (); davis_1 (); tranquada (); ber09 (); shen_a (); shen_2010 (); kerr (); bourges (); armitage (); ando (); hinkov (); taillefer_last ()) The two most frequently discussed candidates for electronic order are incommensurate charge density-wave (CDW) order (Refs. grilli (); ddw (); ms (); efetov (); greco (); laplaca (); charge (); tsvelik (); norman (); debanjan (); pepin (); charge_1 (); atkinson (); rahul (); debanjan_1 (); pepin_new ()) and incommensurate pair-density-wave order (PDW), which is a SC order with a finite Cooper pair momentum (Refs. kivelson (); agterberg (); agterberg_2 (); patrick (); lee_senthil (); corboz ()). Other potential candidates are loop current order varma () and CDW order with momentum near (Ref. sudip ()).
CDW order in underdoped cuprates has been proposed some time ago grilli () and has been analyzed in detail by several groups in the last few years within the spin-fluctuation formalism ms (); efetov (); charge (); tsvelik (); laplaca (); debanjan (); pepin (); charge_1 () and within model ddw (); greco (). The initial discussion was focused on near-equivalence between d-SC and d-wave charge bond order (BO) with momenta along zone diagonal ms (); efetov (); pepin (), but charge order of this type has not been observed in the experiments. It was later found laplaca (); debanjan (); charge (); charge_1 () that the same magnetic model also displays a CDW order with momenta or , which is consistent with the range of CDW wave vectors extracted from experiments mark_last (); ybco (); ybco_1 (); X-ray (); X-ray_1 (); davis_1 (); shen_a (); shen_2010 (); proust (). Such CDW order is also consistent with experiments that detect the breaking of discrete rotational and time-reversal symmetries in a range where competing order develops kerr (); bourges (); armitage (); ando (); hinkov (); taillefer_last (). In particular, when spin-fermion coupling is strong enough, the CDW order develops in the form of a stripe and breaks lattice rotational symmetry. A stripe CDW order with in turn gives rise to modulations in both charge density and charge current and breaks time-reversal and mirror symmetries charge (); charge_1 (); tsvelik (); rahul ().
The agreement with the data is encouraging, but two fundamental issues with CDW order remain. First, within the mean-field approximation, is smaller than the superconducting (and also the onset temperature for order. It has been conjectured that may be enhanced by adding e.g., phonons grilli (), or nearest-neighbor Coulomb interaction allias () or assuming the CDW emerges from already pre-existing pseudogap atkinson (); debanjan (). is also enhanced by fluctuations beyond mean-field charge (); tsvelik (), but whether such enhancements are strong enough to make larger than remains to be seen. Second, stripe CDW order cannot explain qualitative features of the ARPES data away from zone boundaries patrick ().
It has been argued patrick () that ARPES experiments for all momentum cuts can be explained by assuming that the competing order is PDW rather than CDW. PDW order was initially analyzed for doped Mott insulators kivelson (); lee_senthil (); corboz (), but it also emerges in the spin-fermion model charge_1 () with the same momentum as CDW order and its onset temperature is close to (the two become equivalent if one neglects the curvature of fermionic dispersion at hot spots pepin (); charge_1 ()). Given that PDW order explains ARPES experiments, it seems logical to consider it as a candidate for competing order. Just like CDW, the PDW order develops in the form of a stripe and breaks lattice rotational symmetry agterberg (); charge_1 (), if, again, the coupling is strong enough. However, it does not naturally break time-reversal and mirror symmetries agterberg_2 () (although it does so for a particular Fermi surface geometry agterberg ()), and the mean-field is also smaller than for d-SC.
In this communication we build on the results of the generic Ginzburg-Landau analysis charge_1 () and propose how to resolve the partial disagreement with experiments for pure CDW or PDW orders. We first re-iterate that pure CDW/PDW orders emerge in the forms of stripes only if the spin-fermion interaction is strong enough. In practice, has to be at least comparable to the upper energy cutoff of the spin-fermion model (see details below). For smaller couplings the system develops a checkerboard order for which symmetry is preserved comm (). The spin-fermion model is a low-energy model and it is rigorously defined only when the coupling is smaller than . In this respect, stripe CDW or PDW orders emerge, only at the edge of the applicability of the model. Here we consider spin-fermion model at smaller couplings, well within its applicability range, and allow both CDW and PDW orders to develop. We show that the system develops a mixed CDW/PDW order, in which a CDW component develops between hot fermions separated along, say, Y direction and a PDW component develops between fermions separated along X direction (see Fig. 1). Because the momentum carried by an order parameter is the transferred momentum for CDW and the total momentum for PDW, the CDW order along Y and the PDW order along X actually carry the same momentum . We argue that such a state further lowers its Free energy by developing (via an emerging triple coupling) secondary homogeneous superconducting orders charge_1 (). This effect favors the mixed CDW/PDW state over the pure checkerboard CDW or PDW states, which would otherwise all be degenerate. The mixed CDW/PDW state breaks symmetry because both orders carry either momentum or , but not both, and it also breaks time-reversal and mirror symmetries as the pure stripe CDW order with or does.
The presence of PDW component is relevant for the interpretation of the ARPES data. Without it, the fermionic spectrum in the CDW phase would contain the lower energy branch, which never crosses Fermi level, and the upper energy branch, which would approach the Fermi level from above as the momentum cuts enter the arc region. As discussed in patrick (), this is inconsistent with the data shen_a () which show that the dispersion approaches the Fermi level from below. We show that the presence of PDW component changes the structure of fermionic dispersion in such a way that now the lower branch crosses the Fermi level in the arc region (see Fig. 2), in full agreement with ARPES experiments.
We also consider the interplay between CDW/PDW order and d-SC and present the phase diagram in Fig. 3. The reduction of the superconducting in the coexistence region with CDW/PDW is the obvious consequence of competition for the Fermi surface. A small (of order ) drop of upon entering the coexistence region is the result of a weak first-order CDW/PDW transition. There exists, however, a more subtle feature of the phase diagram. Namely, a secondary SC order is generated by CDW/PDW order, which preserves the same sign of the gap along each quadrant of the Fermi surface. Below for d-SC, this secondary superconducting order couples with order, and the net result is the removal or shifting of the gap nodes. Simultaneously, the CDW order acquires an extra component with -form factor, i.e., the magnitude of its s-wave portion increases. We propose to verify these through experiments.
The model We follow previous works ms (); efetov (); charge (); charge_1 () and consider emerging charge order within the spin-fermion model acs (). This model describes interactions between itinerant electrons and their near-critical antiferromagnetic collective spin excitations in two spatial dimensions. Eight “hot” spots, defined as points on the Fermi surface separated by antiferromagnetic ordering momentum (points 1-8 in Fig. 1), are the most relevant for destruction of a normal Fermi liquid state. The known instabilities of the spin-fermion model include d-SC (e.g. , see Fig. 1) ms (); wang (); wang_el (), bond charge order (BO) with momenta (e.g. ) ms (); efetov (); pepin (), CDW order with momenta and (e.g. ) charge (); debanjan (); debanjan_1 () and PDW order with momenta and (e.g. ) pepin (); charge_1 (). The model has an approximate particle-hole symmetry ms (); efetov (); pepin (); charge_1 (); pepin_new (), which becomes exact once one linearizes the fermionic dispersion in the vicinity of the hot spots. This gives rise to near-degeneracy between d-SC and BO and between CDW and PDW.
The Ginzburg-Landau analysis We introduce four order parameters: for SC, for BO, for PDW, and for CDW respectively. SC and BO order parameters connects hot spots along diagonal bonds, which we label as and , while PDW and CDW connect hot spots along vertical and horizontal bonds, which we label as - in Fig. 1. We define the CDW order parameter on bond as and use analogous notations for other order parameters. The effective action has three terms:
The term is of our primary interest. Keeping the SU(2) symmetry exact, we follow Ref. charge_1 () and combine PDW and CDW orders on a given bond (say, bond A) into a matrix order parameter
where , , and is a matrix “phase”. The order parameters and phases are similarly defined (see Supplementary Material (SM) for details). Minimizing the Free energy, we obtain , , and . Under these conditions, the CDW/PDW action becomes
where and (Ref. charge ()). The prefactors , , and are determined by different convolutions of four fermionic propagators (the square diagrams charge (); debanjan_1 (); charge_1 ()). At we have , , and . We see that is the largest term, hence the action (3) is minimized when . Because , the action is unbounded, which implies that the transition is first-order and sixth-order terms (coming from six-leg diagrams) have to be included to stabilize the order. Including these terms we obtain a first order into CDW/PDW state at . We emphasize that this temperature is higher than the one for a pure CDW (or PDW) transition.
The constraint leaves the ground state hugely degenerate – the order parameter manifold is (Ref. charge_1, ). This manifold includes pure CDW and pure PDW checkerboard states and mixed CDW/PDW states. To select the actual ground state configuration we note that, if CDW and PDW orders have components which carry the same momentum , the Free energy is further lowered by creating a secondary order whose magnitude is a product of CDW and PDW order parameters. This secondary order is a homogeneous SC with equal sign of the gap along each quadrant of the FS charge_1 () One can straightforwardly check that the reduction of the Free energy is maximal when in a nominally checkerboard state CDW occurs along vertical bonds and PDW occurs along horizontal bonds or vise versa, i.e., each order develops in the form of a stripe. This corresponds to either (as in the inset of Fig. 1) or , the choice breaks lattice rotation symmetry. Furthermore, the stripe CDW order parameters and and PDW order parameters and get separately coupled by fermions away from hot spots, and the coupling between and locks the relative phase of and such that (Ref. charge ()). The choice of the sign breaks time-reversal and mirror symmetries. The coupling between and does not lock their phases.
Feedback from CDW/PDW order on fermions We now show that the feedback from stripe CDW/stripe PDW order on the fermionic dispersion at , taken as a function of for various , yields results in quite reasonable agreement with ARPES data shen_a (); shen_2010 (). Previous studies have shown charge () that a pure CDW order can explain the ARPES spectrum for a cut along the BZ boundary, but not for cuts that are closer toward BZ center (see Ref. patrick (); sm ()). To obtain the dispersion along various cuts in the presence of both CDW and PDW, we have extended our analysis of the CDW/PDW order to a finite momentum range away from the hot spots. We find that at the BZ boundary, the CDW order has a larger amplitude due to better FS nesting but the PDW component increases as the cuts move towards the hot spots. We present the details in SM and show the results in Fig. 2. There are three key features in our scenario that are qualitatively consistent with experiment: (1) at the BZ boundary (), the locus of minimum excitation energy shifts from to a larger value , where is the CDW momentum, (2) as decreases, the excitation approaches the Fermi level from below, and (3) at when the Fermi arc emerges, the fermionic dispersion becomes flat for . These features are also reproduced by pure PDW order patrick () and from a spatially homogeneous self-energy arising from a -wave CDW order peaked at greco (). However, both these scenarios do not immediately explain the observation of broken time-reversal symmetry or CDW order with small incommensurate momentum. To obtain quantitative agreement with the experiments, we would need to know how CDW and PDW order parameters depend on frequency. This would require one to model the bare dispersion far away from and solve complex integral equations for frequency-dependent order parameters.
Interplay between CDW/PDW order and superconductivity We next consider other terms in the effective action in Eq. (1). The term has been analyzed in efetov (); pepin (); debanjan_1 (). When symmetry is exact, d-SC and BO orders are degenerate and the action has four Goldstone modes. Once symmetry is broken by FS curvature, only d-SC order develops below . We assume that this is the case and keep only d-SC component in , i.e. reduce it to with , , and . The coupling between CDW/PDW and d-SC orders is again obtained by evaluating the square diagrams. The calculation yields with . Note that the magnitude of the coupling is phase sensitive, hence the phase locking between and at is important (see SM for details).
The analysis of the full action is straightforward and we show the results in Fig. 3. The mean-field temperature is comparable to near the SDW boundary but is enhanced by fluctuations debanjan (); atkinson (); charge (). We assume that this enhancement lifts above at large . Because CDW/PDW transition is first-order, jumps upon entering into the coexistence region, but the jump is again small in . Similar behavior has been recently observed in Fe-pnictides pnictides (). At small , the CDW/PDW and d-SC orders coexist.
The phase diagram in Fig. 3 is similar to that for pure CDW order charge (), but there are some extra features. First, the combination of CDW/PDW orders induces a secondary SC order charge_1 () with a non-zero gap along zone diagonal (-wave or ). In the coexistence region the order couples with d-SC order and, as a result, gap nodes either get shifted ( state) or removed ( state). A similar coupling has been examined in the context of the Fe-pnictides hinojosa (). A finite gap along zone diagonals has been observed in ARPES at doping (Ref. shen_b ()) and also inferred from Raman spectroscopy sakai (). Second, by the same logic, the d-SC and PDW orders induce a secondary -wave CDW order with the same momentum as the primary one. We propose to search for SC gap opening or node shifting and to examine the -component of CDW order in the coexistence region.
Conclusions In this letter we proposed a state with unidirectional CDW and PDW orders which carry the same momentum. We argued that this state is a member of the ground state manifold of the low-energy spin-fermion model and its energy is further reduced by induction of a secondary SC order. We further argued that CDW/PDW state has a number of features consistent with experiments: it breaks both and time-reversal symmetry and the feedback from CDW/PDW order on fermions reproduces the ARPES data from the BZ boundary to the tip of the Fermi arc. The transition into CDW/PDW state is weakly first-order and occurs at a higher transition temperature than that for a pure unidirectional CDW or PDW orders. We considered the interplay between CDW/PDW order and d-SC, and found that a SC gap becomes non-zero along zone diagonals. We proposed to search for this gap opening in the region where charge order and d-SC coexist.
We acknowledge useful discussions with W. A. Atkinson, D. Chowdhury, E. Fradkin, S. Kivelson, P. A. Lee, C. Pépin, and S. Sachdev, The work was supported by the DOE grant DE-FG02-ER46900 (AC and YW) and by NSF grant No. DMR-1335215 (DFA).
Appendix A I. Details of the Ginzburg-Landau action
a.1 A. CDW/PDW sector
The Ginzburg-Landau (GL) action for the CDW/PDW order has been derived and studied in detail in Ref. sm_pdw, . We briefly review the analysis here and apply it for our purposes.
When the curvature of the Fermi surface (FS) at hot spots can be neglected, spin-fermion model has symmetry which makes CDW and PDW orders degenerate. The action of the spin-fermion model can be rewritten in an explicitly -symmetric form in terms of particle-hole doublets at each of the hot spots 1-8,
In this notation, the CDW order parameter ’s and the PDW order parameter ’s involving the same pair of hot spots (labeled as , , , and in Fig. 1 in the main text) can be combined into a 22 matrix that couples bilinearly to particle-hole doublet ’. The four 22 matrices are
For convenience, we also define phases via . Each order parameter changes sign under a momentum shift of (e.g., between the pair 1,2 and the pair 3,4 in Fig. 1 in the main text) because spin-mediated interaction is repulsive. The magnitudes of the CDW and PDW order parameters between 1,2 and 3,4 do not have to match as these two pairs of hot spots are not equivalent. For simplicity, below we neglect this non-equivalence and assume that order parameters just change sign under a momentum shift by (this is often termed the wave approximation for the form-factor of the charge order).
The full effective action in terms of CDW and PDW order parameters up to quartic order is
In mean-field analysis the coefficient is proportional to , where is the effective four-fermion interaction and is the polarization operator (see Ref. sm_charge, for details). The polarization bubble increases as temperature decreases, and changes sign at the CDW instability temperature . By dimensional argument , where is the upper cutoff of the spin-fermion model, and is a dimensionless function. Then .
The calculation of requires more care. If one neglects momentum and frequency dependence of , one obtains that is non-zero only if exceeds some critical value sm_charge (); 23 (). This is a consequence of the velocities of hot fermions at and being generally not antiparallel. However, once one includes the fact that spin-fermion interaction is mediated by a boson with near-divergent dynamical susceptibility , one obtains sm_charge () that the threshold vanishes when the magnetic correlation length diverges. In this limit, the CDW/PDW instability occurs for arbitrary values of the spin-fermion coupling and . The spin-fermion model is justified as a low-energy model when interactions do not take a fermion outside of the low-energy subset, which holds when .
The coefficients are obtained by evaluating the four square diagrams in Fig. 4. In explicit form,
where is the Green functions for a fermion near hot spot , and momentum is defined as a deviation from this hot spot.
where and are and components of the fermi velocity at hot spot 1, and is the momentum cutoff . Using the fact that , we have , and .
where , , , and . This is Eq. (3) of the main text.
As we said in the main text, and the action is unconstrained at the quartic level. We assume that sixth order terms, given by convolutions of six Green’s functions have positive a prefactor and constrain the action. An order of magnitude evaluation of gave .
a.2 B. Stabilization of the mixed CDW/PDW order
In an -symmetric model, all states that satisfy and are degenerate ground states. These include pure checkerboard CDW and PDW states and a mixed CDW/PDW state with stripe CDW and PDW orders between pairs of hot spots along and directions respectively (or vise versa). We show that, when FS curvature is non-zero, the mixed CDW/PDW order generates a secondary homogeneous SC order, and this favors the mixed CDW/PDW order over pure CDW or PDW states. We also show that the FS curvature lifts the degeneracy between CDW and PDW orders and makes both quadratic and quartic terms in the effective action anisotropic. We show that this also favors the mixed “stripe” CDW/PDW state for some range of parameters.
1. Coupling to secondary homogeneous SC order
As we said in the main text, in the mixed state, CDW and PDW orders which carry the same momentum generically induce a secondary homogeneous SC order via triple coupling terms, and these terms can lower the Free energy. We show that the mixed CDW/PDW optimizes such coupling. The induced secondary SC order was shown in Ref. sm_pdw, to be a mixture of -wave and -wave. We define -wave SC and -wave SC order parameters as and . The action for the secondary SC order is given by sm_pdw ()
where is the susceptibility of the secondary SC orders (for simplicity, we take this susceptibility to be the same for and , a qualitatively similar mixed CDW/PDW ground will still result if the two susceptibilities are not the same). When both and are nonzero, superconducting orders and are induced and the Free energy is lowered.
To minimize the Free energy (10), we maximize the magnitude of the two combinations of CDW/PDW order parameters and . We define
where , , and .
The first CDW/PDW combination term in Eq. (10) hence becomes
where in the third line we have defined , , , and .
For generic the magnitude of Eq. (12) is maximized when
Repeating the same arguments for the second combination term we obtain one more set of conditions
Combining, we obtain
Then and . Maximizing the magnitude of both these terms, we obtain the conditions on ’s and ’s as
Before we proceed, we remind that ’s, ’s, and ’s are not free parameters – they are constrained by the condition , and Eqs. (15) and (16) have to be consistent with this condition. We recall that in our notations
So far for the minimization with respect to ’s we have assumed that are completely generic. For the special case when and , the condition on ’s are less strict – from Eq. (12) we see that in this case one only needs to satisfy
Doing the same for