Introduction

A pair density wave (PDW) is an exotic form of superconducting order in which Cooper pairs carry finite center-of-mass momentum1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19. Many recent experiments have reported possible signatures of PDW order in the absence of a magnetic field in correlated electronic systems, such as kagome metals20,21,22,23,24,25,26, NbSe227,28,29, UTe230,31,32, EuRbFe4As433, SrTa2S534 and rhombohedral graphene35. Certain La-based underdoped high Tc cuprates, such as La2−xBaxCuO4 (LBCO) and La2−x−yXySrxCuO4 with X=Nd (LNSCO) or X=Eu (LESCO) are the most intensely studied PDW candidate materials, where the bulk superconducting (SC) Tc has a deep minimum at x ≈ 1/8, while the ordering temperature for a stripe charge-density wave (CDW), Tcdw, is maximal36. At Tc < T < Tcdw transport measurements suggest a dynamical decoupling of the Cu-O layers37,38,39,40,41,42,43,44,45,46,47, which is plausibly explained by the existence of in-plane stripe PDW order with twice the period of the CDW. Below Tc, this PDW order most plausibly coexists with a d-wave SC order48.

However, obtaining direct experimental evidence of PDW order has proven difficult. Transport properties can be difficult to interpret uniquely in complex materials. STM is another commonly used technique to provide evidence of PDW order49,50,51. However, STM measurements provide information about surface states, and evidence of order can be difficult to disentangle from signatures of quasiparticle interference52. More fundamentally, it is unclear to what extent STM can distinguish a PDW state from a CDW+SC state. Given that a PDW is a “new phase of matter,” more direct and unambiguous experimental signatures are needed. An important experimental development in this direction is a recent X-ray study of underdoped LBCO and La2−xSrxCu1−yFeyO453 that apparently provides bulk evidence for the coexistence of PDW and uniform SC order in a range of T.

In this paper, we study low energy amplitude and phase collective modes in a system of unidirectional d-wave PDW order54,55,56 with and without coexisting d-wave SC order. In the pure PDW state we find the amplitude (Higgs) modes are overdamped. In the PDW+SC state, we find that (i) the system favors spontaneous time-reversal symmetry breaking (TRSB) and (ii) one of the amplitude (Higgs) modes is nearly delta-function-like. Such a sharp mode is predominantly from the synchronous motion of the two PDW amplitude modes with Q and − Q, and its damping is almost completely eliminated by the presence of uniform SC order. This mode should be visible in non-resonant Raman scattering measurements57,58,59,60,61,62,63,64,65,66,67, which then can be used as a versatile tool in the search for bulk evidence of PDW order. For comparison we also consider a coexisting CDW and SC state, which has been frequently discussed for La-based cuprates, see e.g.44,68,69,70,71,72,73. We argue that Raman experiments can distinguish between CDW +SC and PDW + SC states, at least when SC order is larger than the order with which it co-exists.

Results

Free energy at mean-field level

To enable explicit calculations, we consider a specific microscopic model of PDW order on a square lattice designed to represent one Cu-O layer. The fermion dispersion is taken to be \(\xi ({\boldsymbol{k}})=-2t\left[\cos {k}_{x}+\cos {k}_{y}\right]-4{t}^{{\prime} }\cos {k}_{x}\cos {k}_{y}-\mu\). We choose \({t}^{{\prime} }=-t/4\) and report all energies in units of t. The chemical potential μ is chosen to set x = 1/8, resulting in the Fermi surface in Fig. 1a. The PDW and SC order parameters Δ(R, τ; r) are taken to be of the form

$$\Delta ({\boldsymbol{R}},\tau ;{\boldsymbol{r}})=\sum _{\overline{{\boldsymbol{q}}}=0,\pm {\boldsymbol{Q}}}{\Delta }_{\overline{{\boldsymbol{q}}}}({\boldsymbol{R}},\tau ){e}^{i\overline{{\boldsymbol{q}}}\cdot {\boldsymbol{R}}}f({\boldsymbol{r}})$$
(1)

where τ is an imaginary time, \(\overline{{\boldsymbol{q}}}=0\) for the uniform SC order and \(\overline{{\boldsymbol{q}}}=\pm {\boldsymbol{Q}}\) for the two PDW orders74,75,76, and R and r denote the center-of-mass and relative positions of the two fermions in a Cooper pair. The Fourier transform of f(r) is the pairing form factor, which we take to have the \({d}_{{x}^{2}-{y}^{2}}\) form, \({f}_{{\boldsymbol{k}}}=\cos {k}_{x}-\cos {k}_{y}\). We assume a period-8 PDW order so \({\boldsymbol{Q}}=(\frac{\pi }{4},0)\).

Fig. 1: Fermi surface (FS) in the original Brillouin zone (BZ).
Fig. 1: Fermi surface (FS) in the original Brillouin zone (BZ).
Full size image

a FS without superconducting orders. b CEC for a d-wave SC in the original BZ for \(| {\overline{\Delta }}_{0}| =0.2t\). The red dot indicates the nodal point. The dashed and solid curves are for electron-like and hole-like dispersions, respectively. c FS in the folded BZ. d CEC for a pure PDW with momentum \({\boldsymbol{Q}}=(\frac{\pi }{4},0)\) in the folded BZ for \(| {\overline{\Delta }}_{\pm {\boldsymbol{Q}}}| =0.2t\). The red curves indicate the residual Fermi surfaces. e CEC for a PDW+SC case with φQ = φ−Q and φ0 = π/2 + φQ and with \(| {\overline{\Delta }}_{0}| =0.1t\) and \(| {\overline{\Delta }}_{\pm {\boldsymbol{Q}}}| =0.2t\). Different colors correspond to different energies, specified to the right of each panel.

At the mean field level, \({\Delta }_{\overline{{\boldsymbol{q}}}}({\boldsymbol{R}},\tau )={\overline{\Delta }}_{\overline{{\boldsymbol{q}}}}=| {\overline{\Delta }}_{\overline{{\boldsymbol{q}}}}| {e}^{i{\varphi }_{\overline{{\boldsymbol{q}}}}}\) does not depend on R and τ. The phase factors \({\varphi }_{\overline{{\boldsymbol{q}}}}\) have to be determined by minimizing the variational free energy. A conventional coupling of these mean field order parameters to band fermions (Δψ†ψ† + h.c) yields a set of Bogoliubov quasiparticle bands77. In Fig. 1b we show the constant energy contours (CEC) of the first Bogoliubov band above the Fermi level for a pure d-wave SC order. In Fig. 1c we show the FS in the folded BZ, and in Fig. 1d, e we show CEC for the PDW states. For a pure PDW order, there are multiple “Bogoliubov Fermi surfaces”39,42,78,79,80,81, as shown in Fig. 1d. These Fermi surfaces are further gapped if a uniform SC is also present, see Fig. 1e.

We obtain the effective Ginzburg-Landau (GL) action in terms of \({\Delta }_{\overline{{\boldsymbol{q}}}}({\boldsymbol{R}},\tau )\) by Hubbard-Stratonovich transformation of the underlying model with 4-fermion interactions (see ref. 77) we obtain

$$\begin{array}{ll}{{\mathcal{F}}}_{{\rm{MF}}}\,=\,{\alpha }_{1}(| {\overline{\Delta }}_{{\boldsymbol{Q}}}{| }^{2}+| {\overline{\Delta }}_{-{\boldsymbol{Q}}}{| }^{2})+{\alpha }_{2}| {\overline{\Delta }}_{0}{| }^{2}\\ \qquad\,\,+{\beta }_{1}(| {\overline{\Delta }}_{{\boldsymbol{Q}}}{| }^{4}+| {\overline{\Delta }}_{-{\boldsymbol{Q}}}{| }^{4})+{\beta }_{2}| {\overline{\Delta }}_{0}{| }^{4}\\ \qquad\,\,+{\beta }_{3}| {\overline{\Delta }}_{{\boldsymbol{Q}}}{| }^{2}| {\overline{\Delta }}_{-{\boldsymbol{Q}}}{| }^{2}+{\beta }_{4}| {\overline{\Delta }}_{0}{| }^{2}\left(| {\overline{\Delta }}_{{\boldsymbol{Q}}}{| }^{2}+| {\overline{\Delta }}_{-{\boldsymbol{Q}}}{| }^{2}\right)\\ \qquad\,\,+{\beta }_{5}\left({\overline{\Delta }}_{0}^{2}{\overline{\Delta }}_{-{\boldsymbol{Q}}}^{* }{\overline{\Delta }}_{{\boldsymbol{Q}}}^{* }+c.c.\right)+\ldots \end{array}$$
(2)

where all the coefficients are convolutions of fermionic propagators with the dispersion set by our microscopic model77. In particular, all βi turn out to be positive, at least at \(T\ll | {\bar{\Delta }}_{{\boldsymbol{Q}}}| ,| {\bar{\Delta }}_{0}|\). In this situation, \({{\mathcal{F}}}_{MF}\) is minimized when \(| {\overline{\Delta }}_{{\boldsymbol{Q}}}| =| {\overline{\Delta }}_{-{\boldsymbol{Q}}}|\). The relation between the phases of SC and PDW orders is determined by β5 > 0, so

$${\varphi }_{{\bf{0}}}-\frac{{\varphi }_{{\boldsymbol{Q}}}+{\varphi }_{-{\boldsymbol{Q}}}}{2}=\frac{\pi }{2}+n\pi$$
(3)

where n is an integer. Since φ0 → −φ0 and φ±Q → −φ∓Q under time reversal18, Eq. (3) is not time-reversal invariant. This implies that in a mixed PDW + SC state the system favors spontaneous TRSB (TRSB in the mixed state does not depend on the phase difference φQ − φ−Q, which is arbitrary to the order described by Eq. (2). This phase is fixed once we include the 8th order contribution to \({{\mathcal{F}}}_{MF}\): \(-\upsilon [{({\overline{\Delta }}_{{\boldsymbol{Q}}}{\overline{\Delta }}_{-{\boldsymbol{Q}}}^{* })}^{4}+c.c.]\), which is allowed for the commensurate, period 8 case we have treated. We have computed the prefactor v and found it is positive. Thus, φQ − φ−Q = mπ/2, where m is integer.), of which some experimental signatures have been reported in LBCO82,83.

Collective modes

We now go beyond mean-field and consider small fluctuations, which in momentum space we parameterize as54,74,84,85,86,87,88,89,90,91

$${\Delta }_{\overline{{\boldsymbol{q}}}}(q)\approx {\overline{\Delta }}_{\overline{{\boldsymbol{q}}}}\left[1+{{\mathcal{A}}}_{\overline{{\boldsymbol{q}}}}(q)+i{\theta }_{\overline{{\boldsymbol{q}}}}(q)\right]$$
(4)

where q = (ωm, Q) and \({{\mathcal{A}}}_{\overline{{\boldsymbol{q}}}}(q)\) and \({\theta }_{\overline{{\boldsymbol{q}}}}(q)\) are the amplitude and phase variations, respectively. They satisfy \({{\mathcal{A}}}_{\overline{{\boldsymbol{q}}}}(-q)={{\mathcal{A}}}_{\overline{{\boldsymbol{q}}}}^{* }(q)\), \({\theta }_{\overline{{\boldsymbol{q}}}}(-q)={\theta }_{\overline{{\boldsymbol{q}}}}^{* }(q)\) and \({{\mathcal{A}}}_{\overline{{\boldsymbol{q}}}}(0)={\theta }_{\overline{{\boldsymbol{q}}}}(0)=0\). It is convenient to introduce a vector basis \(\zeta (q)={[{{\mathcal{A}}}_{{\boldsymbol{Q}}}(q),{{\mathcal{A}}}_{-{\boldsymbol{Q}}}(q),{{\mathcal{A}}}_{{\bf{0}}}(q),{\theta }_{{\boldsymbol{Q}}}(q),{\theta }_{-{\boldsymbol{Q}}}(q),{\theta }_{{\bf{0}}}(q)]}^{T}\). Using this basis and truncating the fluctuating part of the GL action SGL at the Gaussian level, we obtain (see77 for details)

$${S}_{\,\text{GL}\,}^{(2)}=\sum _{q}{\zeta }^{T}(q){\hat{\Gamma }}^{-1}(q)\zeta (-q)$$
(5)

The matrix \({\hat{\Gamma }}^{-1}(q)\) can be thought of as the inverse matrix Green’s function for the fluctuating fields (see77 for details). The dispersions of the collective modes along the imaginary Matsubara frequency axis iωn can be found by solving for \(\det {\hat{\Gamma }}^{-1}(q)=0\). To obtain the dispersions along the real frequency axis ω and the spectral functions Bj(q, ω) (j labels the collective modes), we use Pade approximants92 to implement the analytic continuation iωn → ω + 0+. Here \({B}_{j}({\boldsymbol{q}},\omega )=\frac{-1}{\pi }\,{\text{Im}}\,{D}_{j}({\boldsymbol{q}},\omega )\) and Dj(q, ω) is the j-th eigenvalue of \(\hat{\Gamma }({\boldsymbol{q}},\omega )\). We will be interested in the spectral functions of the amplitude modes in the long-wavelength limit, and define Bj(ω) ≡ Bj(q = 0, ω). The calculations are done at T = 0.005t, which is in all cases that we studied is well below the values of \(| {\overline{\Delta }}_{{\boldsymbol{Q}}}|\), \(| {\overline{\Delta }}_{0}|\).

Pure PDW and pure SC

When only \({\overline{\Delta }}_{\pm {\boldsymbol{Q}}}\) are present, we find two phase and two amplitude collective modes

$${\theta }_{\pm }(q)=\frac{{\theta }_{{\boldsymbol{Q}}}(q)\pm {\theta }_{-{\boldsymbol{Q}}}(q)}{\sqrt{2}},\,\,{{\mathcal{A}}}_{\pm }(q)=\frac{{{\mathcal{A}}}_{{\boldsymbol{Q}}}(q)\pm {{\mathcal{A}}}_{-{\boldsymbol{Q}}}(q)}{\sqrt{2}}.$$
(6)

In Fig. 2a we plot the spectral functions B(ω) for the two amplitude modes. We see that both modes are strongly overdamped, and the damping is stronger for the \({{\mathcal{A}}}_{+}\) mode. The fact that B(ω) for the \({{\mathcal{A}}}_{+}\) mode peaks at a higher energy than that for the \({{\mathcal{A}}}_{-}\) mode can be understood analytically within the GL action54,77. For comparison we also present B(ω) for \({{\mathcal{A}}}_{{\bf{0}}}\) when only a d-wave SC is present (yellow curve). Unlike in an s-wave SC, where \(B(\omega )\propto 1/\sqrt{{\omega }^{2}-4| {\Delta }_{0}{| }^{2}}\), here for a d-wave SC there is no singularity because of nodal quadiparticles, and the broad maximum is at \(\omega /| {\overline{\Delta }}_{0}| \sim 6\).

Fig. 2: Spectral weights of the amplitude modes in various cases.
Fig. 2: Spectral weights of the amplitude modes in various cases.
Full size image

a Spectral functions B(ω) of \({{\mathcal{A}}}_{\pm }\) for a pure PDW order and \({{\mathcal{A}}}_{{\bf{0}}}\) for a pure SC order, both with d-wave form factor. b B(ω) for the three amplitude modes in Eq. (7). The inset shows \({C}_{{\mathcal{A}}}\) as a function of Matsubara frequency ωn at q = 0. In this case \({C}_{{\mathcal{A}}}\gg 1\). c is similar to (b) but with parameters such that \({C}_{{\mathcal{A}}}\ll 1\). d B(ω) for the two amplitude modes for the CDW+SC order. The numerical calculations are done at T = 0.005t.

PDW with SC

When both Δ0 and Δ±Q are present, we find three phase and three amplitude eigen modes

$$\begin{array}{ll}{\theta }_{+,\pm }(q)\,\,\propto \,\frac{{\theta }_{{\boldsymbol{Q}}}(q)+{\theta }_{-{\boldsymbol{Q}}}(q)}{\sqrt{2}}\pm {\left[{C}_{\theta }(q)\right]}^{\pm 1}{\theta }_{0}(q),\\ {\theta }_{-}(q)\,\,\,\,\,\,\propto \,{\theta }_{{\boldsymbol{Q}}}(q)-{\theta }_{-{\boldsymbol{Q}}}(q),\\ {{\mathcal{A}}}_{+,\pm }(q)\,\propto \,\frac{{{\mathcal{A}}}_{{\boldsymbol{Q}}}(q)+{{\mathcal{A}}}_{-{\boldsymbol{Q}}}(q)}{\sqrt{2}}\pm {\left[{C}_{{\mathcal{A}}}(q)\right]}^{\pm 1}{{\mathcal{A}}}_{0}(q),\\ {{\mathcal{A}}}_{-}(q)\,\,\,\,\,\,\propto \,{{\mathcal{A}}}_{{\boldsymbol{Q}}}(q)-{{\mathcal{A}}}_{-{\boldsymbol{Q}}}(q),\\ \end{array}$$
(7)

where Cθ(q) and \({C}_{{\mathcal{A}}}(q)\) are two dimensionless numbers. It can be shown that \({C}_{\theta }(0)=1/\sqrt{2}\), while \({C}_{{\mathcal{A}}}(0)\) depends sensitively on the relative gap magnitudes. We see from Eq. (7) that \({{\mathcal{A}}}_{-}\) is decoupled from \({{\mathcal{A}}}_{{\bf{0}}}\), although its propagator is affected by the presence of the SC order. For small q, we find \({C}_{{\mathcal{A}}}\,\gg\, 1\) for \(| {\overline{\Delta }}_{0}| \,\gg\, | {\overline{\Delta }}_{{\boldsymbol{Q}}}|\), and \({C}_{{\mathcal{A}}}\,\ll\, 1\) for \(| {\overline{\Delta }}_{0}| \,\ll\, | {\overline{\Delta }}_{{\boldsymbol{Q}}}|\). Moreover, when \({C}_{{\mathcal{A}}}\,\gg\, 1\), we have \({{\mathcal{A}}}_{+,+}\approx {{\mathcal{A}}}_{0}\) and \({{\mathcal{A}}}_{+,-}\approx {{\mathcal{A}}}_{+}\) [cf. Eq. (6)]; while in the other limit when \({C}_{{\mathcal{A}}}\,\ll\, 1\), we have \({{\mathcal{A}}}_{+,+}\approx {{\mathcal{A}}}_{+}\) and \({{\mathcal{A}}}_{+,-}\approx {{\mathcal{A}}}_{{\bf{0}}}\) instead. We show our numerical results of B(ω) for the three amplitude modes in Fig. 2b and c for \(| {\overline{\Delta }}_{0}| \,>\, | {\overline{\Delta }}_{{\boldsymbol{Q}}}|\), \({C}_{{\mathcal{A}}}\,\gg\, 1\) and \(| {\overline{\Delta }}_{0}| \,<\, | {\overline{\Delta }}_{{\boldsymbol{Q}}}|\), \({C}_{{\mathcal{A}}}\,\ll\, 1\), respectively. We note for both cases there exist an almost undamped peak for the lowest energy mode, which is predominantly \({{\mathcal{A}}}_{+}\).

A sharp Higgs peak is absent in the pure PDW and pure SC cases and thus appears to be a unique feature of mixed PDW+SC order. From an analytic perspective, the case of \(| {\overline{\Delta }}_{0}| \,\gg\, | {\overline{\Delta }}_{{\boldsymbol{Q}}}|\), \({C}_{{\mathcal{A}}}\,\gg\, 1\) is relatively easy to understand when \(| {\overline{\Delta }}_{{\boldsymbol{Q}}}|\) is treated perturbatively. To locate the modes, one has to (i) re-evaluate the frequencies of \({{\mathcal{A}}}_{+}\) and \({{\mathcal{A}}}_{-}\) from Eq. (6) in the presence of SC within GL action, (ii) include mode-mode coupling between \({{\mathcal{A}}}_{+}\) and \({{\mathcal{A}}}_{0}\) so that the eigen modes become \({{\mathcal{A}}}_{+,\pm }\) and (iii) re-evaluate the damping rates in the presence of SC. We show the calculations in77 and here list the results: On (i), the resulting \({{\mathcal{A}}}_{+}\) mode frequency remains comparable to \(2| {\overline{\Delta }}_{{\boldsymbol{Q}}}|\), as in a pure PDW state, while the frequency of the \({{\mathcal{A}}}_{-}\) mode increases in the presence of stronger SC and becomes comparable to \(| {\overline{\Delta }}_{0}|\). This is consistent with Fig. 2(b), which shows that the peak in the \({{\mathcal{A}}}_{-}\) mode is at a frequency set by \(| {\overline{\Delta }}_{0}|\) rather than by \(| {\overline{\Delta }}_{{\boldsymbol{Q}}}|\). We note in passing that this effect is caused by the same β5 term in (2) that is responsible for TRSB. On (ii), mode-mode coupling (level repulsion) shifts the frequency of the \({{\mathcal{A}}}_{+-}\) mode to a smaller frequency, comparable to \(| {\overline{\Delta }}_{{\boldsymbol{Q}}}|\). On (iii), the \({{\mathcal{A}}}_{-}\) mode is peaked above \(2| {\overline{\Delta }}_{0}|\) and its damping is not reduced compared to a pure PDW, but the damping rate of the \({{\mathcal{A}}}_{+,-}\) mode, peaked well below \(2| {\overline{\Delta }}_{0}|\), is strongly reduced by SC and also by the fact that even in a pure PDW state the damping is very small at \(\omega \sim | {\overline{\Delta }}_{{\boldsymbol{Q}}}|\). As a consequence, the \({{\mathcal{A}}}_{+-}\) mode becomes almost completely propagating and the corresponding B(ω) displays a near-δ-function peak (There is a certain similarity between our case and Morr-Pines scenario for the resonance peak in the cuprates93. We also note that the Higgs mode can, in principle, also decay into two quasiparticles of the phase mode θ−54,94. This is a higher-order process, not included in our analysis).

Numerical results77 for a generic ratio of \(| {\overline{\Delta }}_{0}| /| {\overline{\Delta }}_{Q}|\) and in particular, for the opposite limit \(| {\overline{\Delta }}_{0}| \,\ll\, | {\overline{\Delta }}_{Q}|\) as shown in Fig. 2c again show a sharp spectral peak for the lowest energy amplitude mode (note for \({C}_{{\mathcal{A}}}\,\ll\, 1\) as in Fig. 2c this undamped mode is \({{\mathcal{A}}}_{+,+}\)). To understand this analytically one needs to go beyond mode-mode coupling analysis (see ref. 77) since even an infinitesimal SC order parameter gaps out the entire Fermi surface (except for the nodal points) thus changing the susceptibilities in a non-perturbative way.

CDW with SC

We now discuss the case of uniform SC coexisting with CDW order. To permit a direct comparison to the PDW +SC state, we take the CDW ordering vector to be P = 2Q. We also assume the CDW has an electronic origin60,61,95,96, and neglect the presence of optical phonon modes (for comparison, see refs. 97,98,99,100,101,102). Thus, we parametrize the CDW fluctuations as

$$\rho (q)=2| {\overline{\rho }}_{{\boldsymbol{P}}}| (1+{{\mathcal{A}}}_{\rho }(q))\cos ({\boldsymbol{P}}\cdot {\boldsymbol{r}}+{\theta }_{\rho }(q)),$$
(8)

where \({{\mathcal{A}}}_{\rho }\) and θρ are the CDW amplitude and phase modes (amplitudon and phason). As before, we introduce the attractive interactions in the CDW and SC channels and obtain an effective action for CDW and SC orders. At the mean-field level, we find that the phase of a SC order parameter can be arbitrary, i.e., TRS is not broken. Importantly, this means that the phase with coexisting SC and PDW order is thermodynamically distinct from the phase with coexisting CDW and SC order! The part of the action describing fluctuations around mean-field is formally the same as Eq. (5) in a new basis \({\zeta }^{{\prime} }={[{{\mathcal{A}}}_{\rho }(q),{{\mathcal{A}}}_{{\bf{0}}}(q),{\theta }_{\rho }(q),{\theta }_{0}(q)]}^{T}\). There are two phase and two amplitude eigenmodes

$${\theta }_{\rho }(q),\,\,{\theta }_{0}(q),\,\,{{\mathcal{A}}}_{\pm }^{{\prime} }\propto {{\mathcal{A}}}_{{\bf{0}}}\pm {[{C}_{\rho }(q)]}^{\pm 1}{{\mathcal{A}}}_{\rho }.$$
(9)

In Fig. 2d we show the spectral functions for the two amplitude modes \({{\mathcal{A}}}_{\pm }^{{\prime} }\). First we note Cρ ≪ 1 for the chosen parameters. In fact, Cρ remains small even with larger \(| {\overline{\rho }}_{{\boldsymbol{P}}}|\) or \(| {\overline{\Delta }}_{0}|\), meaning \({{\mathcal{A}}}_{+}^{{\prime} }\approx {{\mathcal{A}}}_{{\bf{0}}}\) and \({{\mathcal{A}}}_{-}^{{\prime} }\approx {{\mathcal{A}}}_{\rho }\). As in the PDW + SC case, the mode, for which B(ω) displays a visible peak, largely describes fluctuations of a non-SC order (here, CDW). We see, however, that the peak is substantially broader than in the PDW+SC case.

Raman spectrum

We next check whether the sharp mode for the PDW+SC case and a more broadened \({{\mathcal{A}}}_{-}^{{\prime} }\) mode for the CDW+SC case can be detected in Raman scattering. For this, we compute dressed Raman susceptibilities defined as \({\chi }_{R}({\omega }_{n})=\int\,d\tau {e}^{i{\omega }_{n}\tau }\mathop{\lim }\nolimits_{{\boldsymbol{q}}\to 0}\left\langle {{\mathcal{T}}}_{\tau }\tilde{\rho }({\boldsymbol{q}},\tau )\tilde{\rho }(-{\boldsymbol{q}},0)\right\rangle\) where \({{\mathcal{T}}}_{\tau }\) is the time ordering and \(\tilde{\rho }({\boldsymbol{q}},\tau )={\sum }_{{\boldsymbol{k}},\sigma }\gamma ({\boldsymbol{k}}){\psi }_{\sigma }^{\dagger }({\boldsymbol{k}}+{\boldsymbol{q}}/2,\tau ){\psi }_{\sigma }({\boldsymbol{k}}-{\boldsymbol{q}}/2,\tau )\) is the Raman density with the Raman vertex γk. Applying linear response theory77,103,104,105,106,107,108, we obtain

$${\chi }_{R}({\omega }_{n})=K({\omega }_{n})-{\Lambda }^{T}(-{\omega }_{n})\hat{\Gamma }({\omega }_{n})\Lambda ({\omega }_{n}).$$
(10)

Here \(K={\chi }_{\tilde{\rho },\tilde{\rho }}\) is the bare Raman susceptibility, \(\Lambda ={\chi }_{\zeta ,\tilde{\rho }}\) is the coupling between the Raman density and a collective mode and \(\hat{\Gamma }\) is the susceptibility of the collective mode.

In the presence of either PDW or CDW order, the original four-fold rotational symmetry of a square lattice is broken down to two-fold, in which case the symmetry group for rotations in the XY plane becomes D2. Out of four one-dimensional representations of D2, the most relevant ones are \(A:\cos {k}_{x},\cos {k}_{y},\cos {k}_{x}\pm \cos {k}_{y}\) and \(B=\sin {k}_{x}\sin {k}_{y}\)105. We consider Raman vertices γ(k) with either A or B symmetry. For pure stripe PDW order in either x or y direction, the B1 Raman channel is inactive, and in the A channel only modes that are even under Q → −Q are visible. For the PDW+SC case, we find by directly computing χR that the \({{\mathcal{A}}}_{++}\) and \({{\mathcal{A}}}_{+-}\) modes are Raman active, while the \({{\mathcal{A}}}_{-}\) mode is Raman inactive. For the CDW+SC case, \({{\mathcal{A}}}_{\pm }^{{\prime} }\) are both Raman active.

In Fig. 3 we show the calculated Raman intensity \({I}_{R}(\omega )=\frac{-1}{\pi }\,\text{Im}\,{\chi }_{R}(i{\omega }_{n}\to \omega +i{0}^{+})\), in the A channel with \(\gamma (k)=\cos {k}_{x}\) and \(\cos {k}_{x}\pm \cos {k}_{y}\), for PDW, PDW +SC, and CDW +SC orders. We see that the Raman intensity is qualitatively the same for all γ(k). For a pure PDW [panel (a)], the Raman response is featureless at small frequencies because the \({{\mathcal{A}}}_{-}\) mode, which could potentially give rise to a peak in IR(ω)54, is Raman inactive. For PDW+SC order [panel (b)], IR(ω) reproduces the sharp peak in the spectral function of the \({{\mathcal{A}}}_{+-}\) mode. For the CDW + SC state, IR(ω) calculated with \(\gamma (k)=\cos {k}_{x}\) reproduces a small hump corresponding to \({{\mathcal{A}}}_{-}^{{\prime} }\) mode, but this feature is not observed when calculated using \(\gamma (k)=\cos {k}_{x}\pm \cos {k}_{y}\). Based on these results, we argue that Raman scattering can distinguish different ordered states.

Fig. 3: Raman intensities in the A-channel with \(\gamma ({\boldsymbol{k}})=\cos {k}_{x}\) and \(\cos {k}_{x}\pm \cos {k}_{y}\).
Fig. 3: Raman intensities in the A-channel with $$\gamma ({\boldsymbol{k}})=\cos {k}_{x}$$ γ ( k ) = cos k x and $$\cos {k}_{x}\pm \cos {k}_{y}$$ cos k x ± cos k y .
Full size image

The parameters are the same as those in Fig. 2(a), (b) and (d) respectively. Collective modes, which chiefly contribute to the peaks in IR(ω) are marked.

Effects of weak disorder

In the presence of weak disorder, the existence of 2D long-range CDW or PDW order is precluded (for incommensurate order this is true even in 3D), and we expect that PDW and CDW order only exists with finite (possibly large) correlation length. The nature of the remaining (vestigial) orders when a pure PDW state is disrupted by weak disorder is not entirely clear109,110. However, a PDW+SC state remains distinct from a CDW+SC state since the TRSB characteristic of the PDW+SC phase should survive as vestigial order for a finite range of disorder strengths.

Another feature of disorder is that it induces a new form of coupling between the uniform SC and PDW orders when they coexist. Such a coupling is realized by a possible 1Q CDW order

$${\rho }_{\pm {\boldsymbol{Q}}}({\bf{r}}) \sim \left[{\Delta }_{0}^{* }({\bf{r}}){\Delta }_{\pm {\boldsymbol{Q}}}({\bf{r}})+{\Delta }_{\mp {\boldsymbol{Q}}}^{* }({\bf{r}}){\Delta }_{0}({\bf{r}})\right].$$
(11)

In the absence of disorder, TRSB implies that this 1Q order vanishes. However, in the presence of a disorder V(r), there is an additive contribution to the effective action of the form \({{\mathcal{F}}}_{dis}=V({\bf{r}})\left[{\rho }_{{\boldsymbol{Q}}}({\bf{r}}){e}^{i{\boldsymbol{Q}}\cdot {\bf{r}}}+c.c.\right]\). Including this contribution in the PDW+SC case and assuming V(r) is short-range correlated we find77: 1) At least to lowest order in the disorder strength, the disorder coupling always favors spontaneous TRSB, enforcing the tendency already derived in the clean limit (The physics behind this is the same as that which underlies the disorder driven breaking of TRS in junctions in which the leading order Josephson coupling is frustrated111,112.); 2) It induces local 1Q CDW order which is weak (small amplitude) but can have a relatively long correlation length that diverges when \(\overline{{V}^{2}}\to 0\). This sort of disorder-stabilized density-wave correlations is reminiscent of Zn doping stabilized spin-stripe order in LSCO113 and YBCO114.

It is also important to mention that in materials which exhibit signatures of the coexistence of uniform SC and PDW or uniform SC and CDW orders, it is always questionable whether they coexist uniformly, or if instead they occur in distinct mesoscopic regions of the material, and only truly coexist at the interfaces between regions48. Such coexistence could exhibit features quite different from anything we have analyzed.

Discussion

We have found that there is a clear distinction between a state with coexisting uniform SC and PDW order and a state with coexisting uniform SC and CDW order. First, the former state is expected to spontaneously break TRS. Second, we studied amplitude collective modes for PDW +SC and CDW +SC states and found that for a PDW+SC state, the spectral function of one collective mode displays a near-δ-function peak. We found that this mode is Raman active and argued that a sharp peak should be visible in the Raman intensity. There is no such sharp peak in a CDW +SC state, i.e., the peak is apparently a unique feature of the PDW+SC order. A peak in the symmetric A Raman channel at ω ~ 50cm−1 has been reported in underdoped LSCO at x ≈ 0.157. It is tempting to associate this peak with low-energy peak in the PDW +SC state, Fig. 3(b). However, since establishing this channel requires careful subtraction across multiple raw datasets, more intensive and rigorous measurements are needed to conclusively identify whether this is the case.