Introduction
Pair-density-wave (PDW) superconductivity, of the kind originally discussed by Fulde, Ferrell, Larkin and Ovchinnikov1, 2 (FFLO), is believed to exist in the heavy-fermion superconductor CeCoIn5 (refs 3, 4) and in the organic superconductor 
-(bis(ethylenedithio)- tetrathiafulvalene)2Cu(NCS)2 (refs 5, 6). It is also believed to be relevant in cold atoms 7, 8 and in the formation of colour superconductivity in high-density quark matter9. In addition, PDW superconductivity with quite a different origin has been found in microscopic theories of correlated electronic materials10, 11, 12. Such order is believed by some to be competing with conventional d-wave superconductivity in underdoped high-temperature cuprate superconductors. Specifically, PDW order may appear as an alternative ground state near a hole doping of 1/8, where some sort of charge order is often observed13, 14. Given its relevance, it is important and useful to address the properties of PDW order from a phenomenological point of view. That is the primary goal of this work. In particular, we address two questions that apply to both FFLO superconductors and the underdoped cuprates. (1) What symmetries are broken by PDW order? (2) What are the properties of the vortex-like topological defects in the PDW ordered phases? The answer to the first question reveals that charge-density-wave (CDW) or spin-density-wave (SDW) order often must accompany the PDW order and provide a means to identify the PDW phase. The answer to the second question turns out to be non-trivial. It is clear that there will be superconducting vortices and these have been studied in the past (see ref. 6 for an overview). However, it is also well known that the periodic order has dislocations as natural topological defects. This has not been given much consideration in the context of PDW order. Here, we show that the topological defects of PDW superconductors contain not just vortices and dislocations, but also combinations of fractional vortices and fractional dislocations. We argue that these fractional defects play an important role when considering fluctuations in two dimensions (for example, leading to non-superconducting CDW or SDW phases) and that they play an important role when considering the physics of the usual superconducting vortices (for example, leading to the appearance of CDW or SDW order inside the vortex core). Finally, we address an issue that is mainly of relevance to the underdoped cuprates. In particular, we examine the role of competition between PDW order and translationally invariant d-wave superconductivity. We show that this competition preferentially selects PDW phases with CDW order. Furthermore, in addition to the CDW order stemming from the PDW order parameter, the coexistence of PDW and d-wave superconductivity leads to either SDW or extra CDW order.
Although our main results apply more generally to PDW superconductors, for concreteness we consider an example motivated by theoretical and experimental proposals for the underdoped cuprates that exhibit some type of charge order. In particular, we consider a PDW superconductor in a three-dimensional tetragonal system with a lattice spacing a for the two-dimensional square lattices. The PDW order is taken to be either commensurate, with a periodicity of 8a (refs 10, 11, 13, 14), or incommensurate (this also allows the theory to describe a variety of FFLO phases found to be stable in two dimensions6). Furthermore, we will take this order to be aligned along the 
(taken to be along a two-fold symmetry axis) or equivalent directions. The PDW order parameter is written as {
Qx,Qy,-Qx,-Qy}, describing PDW order with corresponding wave vectors {Qx,Qy,-Qx,-Qy}. The Ginzburg–Landau free-energy density is constructed by imposing translational, gauge, time-reversal, parity and tetragonal rotation symmetries. This yields
where (-1)i is -1 for Qi=
Qx and 1 for Qi=Qy, D=-i
-2eA and B=
A. The difference between the commensurate and incommensurate cases arises at eighth order. This implies that sufficiently near the mean field critical temperature, Tc, this difference can be ignored. However, there are situations where this difference can be important and this will be discussed later (in particular, when considering vortex-type excitations in a two-dimensional PDW superconductor).
An important property of the above free energy is that it contains a U(1)U(1)U(1) symmetry. It is this feature that gives rise to vortices and dislocations as well as the appearance of the combined fractional vortex and fractional dislocations mentioned above. Below, we will provide a physical picture for the topological defects of equation (1) (and also for the related half-flux vortices found in ref. 15). Before understanding these defects, it is important to first understand the possible PDW ground states and also useful to understand the induced CDW and SDW order that arises. To complete the latter goal, we include the free energy for CDW and SDW order and their coupling to the PDW order. These are also determined by the same symmetry requirements as above. The contribution to the free energy from the CDW order, 
Q, is
For our choice of PDW order, only SDW order with moments oriented along 
can be induced. Denoting the SDW order as SQz, the relevant free energy is
Throughout this work, we consider the case in which the CDW and SDW order is induced by the PDW order (
Q
,Qs>0) and thus is representative of the symmetries broken by the PDW phase transition. In this case, they are given by
Equations (2) and (3) reveal that the phase difference between two different components Q can be interpreted as the phase of either the CDW or SDW order. In general, it is possible that there is also an intrinsic CDW or SDW order (for which Q
0 or Qs0 is possible). This is also of interest, but is not considered here.
Remarkably, it is possible to find all of the possible PDW ground states of the general free energy defined by equation (1) (this is not true, for example, for a general phenomenological theory of superfluid 3He (ref. 16)). These ground states and the induced CDW and SDW orders are listed in Table 1. Note that we have listed the lowest Fourier components of the induced SDW and CDW order; higher Fourier harmonics will in general appear, but these should be smaller in magnitude. In principle, the induced CDW and SDW order can be measured and therefore used to identify what type of PDW order appears. One result that has not previously been highlighted is the existence of PDW phases that break time-reversal symmetry. In particular, phases 2 and 5 exhibit this behaviour and the broken time-reversal symmetry is readily apparent through the associated SDW order. At the mean-field level, this theory predicts that superconductivity appears at the same transition temperature as the SDW and CDW order. However, fluctuations can separate these transitions and a specific mechanism for this will be discussed later. It is interesting that related SDW and CDW order has been observed in a variety of cuprates17, 18, 19 and a systematic experimental study of such order in a single material would prove useful to identify any PDW order parameter.
Q=(
0/4 defects experience a confinement potential but can exist at short length scales and may consequently become physically relevant.![Table 1 - Properties of PDW ground states. All possible PDW ground states and accompanying CDW and SDW order. The third column shows the parameter regions for which these phases are stable. In the fourth and fifth columns, other modes can be found by using the relationships |[rho]|Q=(|[rho]||[minus]|Q)* and SQz=(S|[minus]|Qz)*. The fifth column shows the minimum flux contained by a topological defect. In phases 4 and 5, the |[Phi]|0/4 defects experience a confinement potential but can exist at short length scales and may consequently become physically relevant.](/common/images/table_thumb.gif)
We now address the nature of the competition between translationally invariant d-wave superconductivity and PDW superconductivity relevant only in the context of the cuprates. The lowest order coupling between these two order parameters is given by the following free-energy density:
The relevant feature of this coupling is that the term with coefficient 
c2 can always be made negative by an appropriate choice of phases of the different order parameters. Consequently, any PDW phase for which this coupling is non-zero will lower its energy through this coupling. Only phases 3 and 4 above lead to a non-zero 2c coupling; these are the two phases with PDW-induced CDW order. When PDW and d-wave superconductivity coexist, then the interplay between these two order parameters will lead to CDW or SDW order that appears in addition to the CDW order that already exists in phases 3 or 4. In particular, if the coefficient c2>0 then the extra order will be SDW order at the wave vectors of the PDW order; if c2<0, then there will be no induced SDW order and extra CDW order will appear at the wave vectors of the PDW order.
With the understanding of the possible PDW mean-field ground states, we now turn to understanding the topological defects. Single valuedness of the wavefunctions implies that these can be found by allowing the phases 
1,2 or 3 in Table 1 to have an integral multiple of phase winding 2
on encircling the core of the defect. To understand the energies and the magnetic properties of such defects, it is useful to consider the London theory. Here, we consider phase 3 explicitly and state the results for the other phases. Allow (1,2) to have phase windings of (n,m) times 2 respectively when encircling a defect. Taking 
, gives the London theory
where s,x=(1+2)|
|2, s,y=(1-2)||2 and s,z=3||2. Closely related London theories appear in a variety of other contexts, including in a theory motivated by superconducting UPt3 (ref. 20), in two-gap superconductors21 and a description of fractional vortices (with non-abelian vortex core states) in chiral superconductors22. Equation (4) yields a supercurrent with components Ji
s,i[
i(1+2)-2eAi]/2; far from the core of the defect, the minimum energy configuration has zero current and a contour integration then implies that the flux enclosed by a (n,m) defect is (n+m)
0/2. Consequently, the simplest defect encloses half a flux quantum. A similar analysis shows that phase 1 has only the usual superconducting vortices (the simplest containing flux 0), whereas phases 2, 3, 4 and 5 all have defects that contain flux 0/2. Equation (4) also shows that usual superconducting vortices have finite energy because the phase winding can be completely screened by the vector potential. The other defects have an energy that diverges as the logarithm of the system size. To help develop a physical picture of the fractional defects, it is useful to examine the induced CDW (and SDW) order near a 0/2 defect. Figure 1 reveals that near such a defect in phase 3 there is an edge dislocation in the CDW order when spatial uniformity of the defect along the 
direction is imposed. If the defect is taken to be spatially uniform along the direction parallel to Q, then a screw dislocation appears in the CDW order. Note that the CDW order in this phase has half the periodicity of the PDW order; consequently, a dislocation in the CDW order can be interpreted as half a dislocation in the PDW order. The 0/2 defects of the other phases have a similar origin. This leads to the prediction that a PDW superconducting vortex containing a fraction of a flux quantum will be pinned to a dislocation in the accompanying CDW or SDW order. It is worthwhile pointing out that if 4=0 in equation (1), then phases 4 and 5 will have 0/4 defects. A non-zero 4 leads to a confinement potential between these 0/4 defects. However, it is still possible that the 0/4 defects play a role at short length scales, for example in determining the vortex core structure of integer flux superconducting vortices.
Figure 1: Fractional vortex dislocation.
Dislocation in CDW order accompanying a flux 0/2 defect in phase 3. As the periodicity of the PDW order is twice that of the CDW order, a dislocation in the CDW order corresponds to half a dislocation in the PDW order.
We now turn to two physical consequences of the fractional flux defects. The first is in fluctuation-driven vortex and dislocation physics in two dimensions. In this case, the fractional flux defects lead to non-superconducting CDW/SDW phases. The second consequence is that conventional superconducting vortices can decay into a bound state of fractional flux defects. This provides a mechanism for the appearance of CDW or SDW order inside a vortex core.
The fact that single flux quanta vortices have short-range interactions and fractional flux defects have long-range interactions implies that novel vortex-related physics can occur in two dimensions. This physics differs for the commensurate and incommensurate cases and discussion of the commensurate case is left to the next paragraph. The theory for the incommensurate case follows from the London theory of equation (4) and resembles that for two-band superconductors considered in ref. 23. To understand the behaviour of the incommensurate case, consider initially taking the limit that the penetration depth 
. In this limit, the vector potential can be ignored, and a Berezinskii–Kosterlitz–Thouless (BKT) transition, corresponding to the unbinding of 0/2 defects, occurs at a transition temperature Tc*=(/2)s (refs 24, 25). In reality, because the penetration depth is finite, this will correspond to a crossover temperature at which the resistivity starts to fall. This is the case because superconducting vortices have a finite energy and therefore exist at any finite temperature. Note that this crossover temperature is half the value of the equivalent BKT transition temperature for a conventional superconductor with the same mean field Tc and superfluid density. Consequently, the corresponding fluctuation regime will be larger. Now consider the case in which there exist thermally excited unbound superconducting vortex pairs. Although such a state will not be superconducting, this does not imply that there is no longer any type of order present (as is the case for conventional superconductors). To see this, it is convenient to define 
=(1+2)/2 and =1-2. The field corresponds to conventional superconducting vortices and is disordered. The field is uncoupled from the vector potential and therefore has vortex-like defects that are not screened. Vortices in this field correspond to dislocations in the CDW order. These dislocations cost a logarithmically divergent energy and therefore are not present at sufficiently low temperatures. This implies that there can be quasi-long-range order in the CDW order parameter as defined through equation (2). The corresponding BKT transition at which the CDW order is removed occurs at TcdwTc*; above Tcdw, strictly speaking, there is no order of any type.
The commensurate case is more interesting because, in principle, it allows for the possibility to have Tcdw>T* (where T* corresponds to a crossover temperature at which the resistivity starts to fall). To carry out a proper treatment, it is important to include the following term in the free-energy density (which only exists in the commensurate case):
In the London theory for phase 3, this leads to an interaction term (
||8/8)cos(41-42). The renormalization group equations describing the CDW transition with such an interaction are the same as those derived in ref. 26, in which two-dimensional XY models subject to clock model-like symmetry-breaking fields are considered. These renormalization group equations imply that Tcdw for the CDW order is enhanced relative to that of the incommensurate case. In the limit that equation (5) can be treated as a perturbation, we find that the enhancement of Tcdw is given by 
, where h4=2a2||8 and Tcdw(0) corresponds to the BKT transition temperature with h4=0.
Finally, we turn to superconducting vortices containing a single flux quantum. These are the lowest energy vortices and they are created by magnetic fields. The key point is that a single 0 vortex can be either a conventional vortex or a bound state of fractional flux defects. This possibility is closely related to broken axisymmetric vortices discussed in the context of superfluid 3He (refs 27, 28), unconventional superconductors20, 29 and in FFLO superconductors15. The dissociation of a conventional vortex into a bound state of fractional flux defects provides a means to have a vortex core structure with CDW order. As a specific example, we have determined the structure of a vortex in a superconductor with non-vanishing d-wave and PDW superconductivity (the PDW order is that of phase 3 or 4). There exists a solution in which all components of the order parameter exhibit the same phase winding. Furthermore, the d-wave order parameter vanishes at the vortex core and the PDW order parameter is non-zero where the d-wave order vanishes. This vortex will have CDW order (at the wave vectors listed in Table 1) in the d-wave vortex core as follows from equation (2). Such a solution may be relevant for understanding the observed CDW order inside the vortex cores of some underdoped cuprates30.
In summary, we have examined the broken-symmetry phases of PDW superconductors and have shown that the coexisting CDW or SDW order provides a means to distinguish between these phases. We have also shown that PDW superconductors exhibit topological defects that include fractional superconducting vortices that are coupled to dislocations in the coexisting CDW or SDW order. These defects can play an important role in stabilizing non-superconducting SDW or CDW phases. They are also important in understanding the physics of superconducting single flux quantum vortices where they can lead to CDW or SDW order inside vortex cores. Finally, in the context of the cuprates, we have examined the competition between PDW and translational invariant d-wave superconductivity and have shown that this prefers PDW phases with intrinsic CDW order and that an extra SDW or CDW appears owing to the coexistence of d-wave and PDW superconductivity.
