Pseudo-spin Skyrmions in the Phase Diagram of Cuprate Superconductors
Topological states of matter are at the root of some of the most fascinating phenomena in condensed matter physics. Here we argue that skyrmions in the pseudo-spin space related to an emerging SU(2) symmetry enlighten many mysterious properties of the pseudogap phase in under-doped cuprates. We detail the role of the SU(2) symmetry in controlling the phase diagram of the cuprates, in particular how a cascade of phase transitions explains the arising of the pseudogap, superconducting and charge modulation phases seen at low temperature. We specify the structure of the charge modulations inside the vortex core below , as well as in a wide temperature region above , which is a signature of the skyrmion topological structure. We argue that the underlying SU(2) symmetry is the main structure controlling the emergent complexity of excitations at the pseudogap scale . The theory yields a gapping of a large part of the anti-nodal region of the Brillouin zone, along with phase transitions, of both nematic and loop currents characters.
The pseudo-gap (PG) phase in the under-doped region of cuprate superconductors remains one of the most mysterious known states of matter. First observed as a depression in the Knight shift of nuclear magnetic resonance (NMR) Alloul et al. (1989, 1991); Warren et al. (1989), it was soon established that, for a region of intermediate dopings around , part of the Fermi surface was gapped in a region close to the and points of the Brillouin zone, called anti-nodal region because of its remoteness from the point were the -wave superconducting gap changes sign on the segment of the Brillouin zone. In this anti-nodal region, the Fermi surface was found to be “wiped out”, and only some lines of massless quasiparticles known as Fermi arcs to be left out Campuzano et al. (1998); Vishik et al. (2012a, b); Yoshida et al. (2012); He et al. (2014); Vishik et al. (2014).
This puzzling situation became more complex with the observation of a reconstruction of the Fermi surface by quantum oscillation and other transport measurements in the same doping region Doiron-Leyraud et al. (2007); LeBoeuf et al. (2007, 2011); Laliberté et al. (2011); Sebastian et al. (2012); Doiron-Leyraud et al. (2013); Barišić et al. (2013); Grissonnanche and Laliberte (2015). This was attributed to the presence of incipient charge modulations with incommensurate wave vectors developing along the crystallographic axes: , where is the lattice spacing in a tetragonal structure, detected by X-ray scattering Ghiringhelli et al. (2012); Chang et al. (2012); Achkar et al. (2012); Blanco-Canosa et al. (2013); Blackburn et al. (2013a, b); Thampy et al. (2013); Blanco-Canosa et al. (2014); Tabis et al. (2014); Comin et al. (2014, 2015a); Comin and Damascelli (2015); Comin et al. (2015b). In real space, patches of charge modulation of a size of the order of twenty lattice sites have been observed at low temperatures ( K) using both scanning tunneling microscopy (STM) Hoffman et al. (2002); da Silva Neto et al. (2014a, b); Mesaros et al. (2016) and nuclear magnetic resonance (NMR) Wu et al. (2015) measurements. These take the form of oscillations of the charge density on the copper oxide planes of a frequency comparable to twice the lattice spacing. The amplitude of these oscillations decreases away from its centerpoint in real space and disappears around ten lattice lengths away from it.
Charge modulations were observed at the core of the superconducting vortices, below the superconducting transition temperature (). When voltage bias is increased, these modulations persist until the applied voltage reaches the energy scale corresponding to the formation of the pseudogap: Hamidian et al. (2015, 2015).
Below the pseudogap onset temperature , loop currents have been detected Fauqué et al. (2006); Mangin-Thro et al. (2014, 2015), and the areas exhibiting charge modulations coexist with zones with long-range nematic order Mesaros et al. (2011); Sato et al. (2017), reminiscent of the vicinity of a smectic-nematic transition. The latter are more and more numerous compared to charge-modulated areas when the temperature approaches Gomes et al. (2007a). Simultaneous measurements of the real and reciprocal space spectral functions however established that the opening of the pseudogap is correlated with the presence of charge modulations in real space Kohsaka et al. (2007). The whole real space picture has led to the image of an “ineluctable complexity” inherent to cuprate superconductors and driven by strong quantum fluctuations in the vicinity of the Mott transition in two dimensions Fradkin et al. (2015).
Here we argue that the presence of an underlying SU(2) symmetry in the under-doped region sheds light on the variety of observed phenomena and clarifies the mysteries of the real space picture. First, we describe the starting short-range antiferromagnetic model and its order-by-disorder treatment, which gives rise to a pseudogap phase governed by SU(2) fluctuations which stabilise -wave superconducting, nematic and axial orders. Then we geometrically interpret the proliferation of local defects by introducing an SU(2) order parameter which follows naturally from the previous derivation. Finally, we describe its topological structure and the cascade of phase transitions it generates.
Short-range antiferromagnetic model
We start by describing how an order-by-disorder treatment of a simple short-range antiferromagnetic model was shown to give rise to -wave superconducting, nematic and charge orders Montiel et al. (2017).
Our starting point is that short-range antiferromagnetic interactions, strongly coupled to conduction electrons, are the main ingredient of the physics of the cuprates above doping. This leads to the most simple Hamiltonian:
where is the hopping matrix from one site to another, creates an electron of momentum and spin , is the on-site spin operator and denotes the summation over nearest neighbours.
One can decouple the interaction term into the -wave superconducting channel described by and the -wave charge modulations channel at momentum , described by with , and the angle spanning the Brillouin zone. The charge modulation wave vectors, shown in Fig. 2A, are typically incommensurate, and taken either parallel to the crystal axes Montiel et al. (2017) or diagonal Metlitski and Sachdev (2010); Efetov et al. (2013). Indeed, it can take all the values connecting two hot-spots, which are the points where the Fermi surface crosses the line where the antiferromagnetic fluctuations diverge: (Fig. 2A).
This decoupling yields various possible order parameters which are all degenerate in magnitude at the Fermi surface hot-spots in the strong coupling limit, i.e. for much larger than the energy of the bottom of the electronic band in the anti-nodal region, as depicted in Fig. 2D-F Montiel et al. (2017). Note that one could also consider decoupling the interaction term in the spin-sector in the antiferromagnetic channel, in particular for the characterization of the low-doping properties of this model. It is however likely that the antiferromagnetic fluctuations would be gapped by the emergent orders which also compete for the hot-spots.
All the previously cited possible orders are -wave symmetric. This is dictated by the fact that antiferromagnetic interactions correspond to a wave vector, and therefore couple adjacent anti-nodal regions of the Brillouin zone in the gap equation corresponding to the diagram in Fig. 2C, of the typical form:
with , the electronic dispersion and and the fermionic and bosonic Matsubara frequencies and . The solutions of this equation are plotted in Fig. 2D-F for with three different choices for . In order to stabilise the gap equation there needs to be a change of sign between two such adjacent regions leading to , the simplest case of which is a -wave order Kloss et al. (2015).
It has been shown that the charge and superconducting order parameters and are related by an SU(2) symmetry in a region of the Brillouin zone, in the sense that one can define an SU(2) algebra relating the two Montiel et al. (2017). This symmetry is exact on a line of the Brillouin zone joining the hot-spots, and is broken away from it Montiel et al. (2017). This naturally causes the arising of the fluctuations associated to this symmetry, which we call SU(2) fluctuations Montiel et al. (2017).
The degeneracy of the various channels at the hot-spots introduced above has been shown to be lifted by considering the SU(2) fluctuations through the diagram in Fig. 2B Montiel et al. (2017), similarly to what happens in the order-by-disorder formalism, first described by Villain in the classical context Villain et al. (1980).
Remarkably, the choice of the starting charge modulation wave vector becomes irrelevant at this point, since it was found that the SU(2) fluctuations select three wave vectors characterizing respectively -wave nematic ordering at and axial modulations with or without symmetry breaking at and Montiel et al. (2017) (Fig. 2A). Both nematic and axial orders are therefore naturally selected by the SU(2) fluctuations.
In order to define the set of operators relating and , which form the SU(2) algebra, we use the order parameter with and . The operation is constrained by the closure condition of the SU(2) algebra to satisfy and Montiel et al. (2017). This causes to be k-dependent, that is, this causes the charge modulations to be multi-k Montiel et al. (2017). The lack of k-dependence would cause the arising of harmonics at multiples of Montiel et al. (2017). This can be chosen in several ways, for example by dividing the Brillouin zone in quadrants and assigning a single vector per quadrant, as mentioned implicitely above Efetov et al. (2013); Montiel et al. (2017). It can also correspond to charge modulations responsible for both the gapping of the anti-nodal region of the Brillouin zone and the formation of excitonic patches, as studied in Montiel et al. (2017).
These excitonic patches have been shown to proliferate in real space in some regions of the phase diagram Montiel et al. (2017). In the following section, we give a topological interpretation of the proliferation of local objects in real space, by introducing the SU(2) order parameter, which enables us to encompass many aspects of the phase diagram of the cuprates in an integrated manner.
SU(2) order parameter
The order parameter that naturally emerges from the previous discussion to describe the pseudogap is a composite of and , which can be cast into the form:
where , which is the constraint enforcing the SU(2) symmetry. Since and are complex fields, this constraint can be written as:
where the indices and denote the real and imaginary parts of the operators, respectively. In this picture, represents the energy scale below which the fluctuations between the two fields and are dominant; this scale is thus doping dependent. Notice that, by construction, this composite SU(2) order parameter is non-abelian.
At every doping , equation (4) describes a three dimensional hypersphere in a four-dimensional space. The transverse fluctuations of the order parameter on this hypersphere are naturally described by an O(4) non-linear -model Efetov et al. (2013)
where are the four-component vector subject to the constraint , with , , where , and the sign of the masses depends on the presence or absence of an applied magnetic field. The amplitude modes, or massive modes, can be safely neglected since the energy difference between the charge and superconducting states is much smaller than both their energies.
In the specific context of the sphere, no topological defect is generated, since a careful examination of the corresponding homotopy class gives Mermin (1979). In the following, we discuss the case where one degree of freedom is lost, allowing for topological defects to appear.
We now argue that, as the temperature is lowered, the phase of the charge modulations is frozen in some real space regions, as measured by phase-resolved STM Mesaros et al. (2011, 2016). This reduces this phase to a few integer values , with an integer and thus reduces the fluctuation space from to a set of halves of , indexed by Lee and Rice (1979). These regions are thus characterised by , and the effective non-linear -model becomes . The space of the fluctuations is depicted in Fig. 3A where a fluctuating hemi-sphere is shown; has been reduced to for the clarity of the representation, with phase and corresponding to the upper and lower hemi-spheres, respectively.
The second homotopy class of the sphere is , which yields the spontaneous generation of skyrmions Mermin (1979). In our case, in the regions of real space, fluctuates on a hemisphere and the boundary conditions corresponding to superconducting vortices give us half-skyrmions bearing a half-integer topological charge Lee and Wen (2001); Lee et al. (2006); Sheehy and Goldbart (1998); Goldbart and Sheehy (1998):
They are also called merons and correspond to a variation of the vector over one hemisphere, as illustrated in Fig. 1, 3A and 3C. They take two equivalent typical forms, of an edgehog and a vortex, and the proliferation pattern involves meron/anti-meron pairs such as the one presented in Fig. 1. Note that, contrary to the magnetic skyrmions observed in magnetic systems (see e.g. Muehlbauer et al. (2009); Romming et al. (2013)), here the topological structure acts on the pseudo-spin sector, with the three quantization axes corresponding respectively to (Fig. 3B). The choice of the quantization axis to be parallel to the charge modulation parameter is arbitrary but convenient, since the superconducting phase then corresponds to a simple easy plane situation (Fig. 3C).
Non-zero topological numbers are associated with the arising of edge currents. One can therefore imagine isolating one topological defect in order to examine these currents. In the most simple case of a single meron, such as displayed in Fig. 1, the SU(2) order parameter along the edge is purely in the superconducting plane, with a superconducting phase varying by when going around the full edge, exactly like in the case of a superconducting vortex. One can however engineer more interesting situations by taking opportunity, for example, of the proximity effect. Indeed, one could use four leads: two superconducting leads and two ferroelectric leads, as depicted in Fig. 4. This would constrain to be purely superconducting or purely charge-like at each of these points, and force the creation of a particular meron depicted in Fig. 4, corresponding to a rotation of the axes of the simple case considered in Fig. 1 and 3 such that would now be along . Note that this would mean that the phase of the charge order parameter changes along a line dividing the meron in two (the axis in Fig. 4), on which it has zero magnitude. Along this line, is maximal and the corresponding phase changes by when crossing the sample, as depicted in the inset of Fig. 4. Experimentally, this would give rise to a supercurrent along , as well as a peculiar charge pattern, measureable for example via STM. Applying a current along this line would also be a way to generate this gradient of superconducting phase, and consequently the whole meron Davis (2017). Moreover, one can define a specific order parameter along the edge as . The phase of this order parameter then rotates by when going round the sample, giving rise to a finite phase gradient, and hence to a peculiar type of current, which could also possibly be detected. Such a setup would provide a viable experimental test of the existence of the SU(2) order parameter through its direct manipulation.
The mechanism that allows one to see these topological defects is the freezing of the phase of the charge order parameter component of the SU(2) order parameter. This is caused by pinning to local defects or superconducting vortices and coupling of the charge order parameter to the lattice. In this formalism there is therefore coexistence, in the pseudogap phase, of two types of regions in real space: regions where the phase of is continuous, and which exhibit a nematic response, and regions where the phase of is quantised, where merons proliferate.
Measurements of superconducting vortices below found that they bear a very specific structure where charge modulations are observed at the core Wu et al. (2013). This corresponds to a pseudo-spin meron in whose centre the pseudo-spin vector is oriented along the -axis, producing charge modulations while the superconductivity order parameter vanishes, as detailed in Fig. 3C. The energy associated to the creation of this vortex is intrinsically of the order of the energy splitting between the superconducting and charge modulation orders, which is precisely the typical energy scale of the superconducting coherence . Hence pseudo-spin merons will proliferate around in the under-doped region of the phase diagram, acting as a Kosterlitz-Thouless (KT) transition towards the pseudogap state Wang et al. (2002); Alloul et al. (2010).
The size of the merons can be obtained from similar considerations. Indeed, if one only consider one mass for each field (setting, e.g., and ), the energy associated with this topological defect is , which gives us the size of the meron: , written in Fig. 3C. Note that if superconductivity dominates, .
We have introduced the general framework of the SU(2) theory, described how SU(2) fluctuations stabilise a pseudogap state, and how some regions in real space can freeze the charge modulation phase, causing the proliferation of pseudo-spin merons. We now turn to the consequences of this formalism on experimental observations. We start by discussing the simultaneous arising of nematic and loop current orders at , then we proceed to charge modulations in real space under applied electric field, and finally we examine the phase diagram under applied magnetic field.
Multiplicity of orders at : an ineluctable complexity?
It is a longstanding issue whether the pseudogap temperature , sketched in Fig. 5, corresponds to a phase transition or a cross-over. Solving this issue is made harder by the fact that the cause of the gapping of the Fermi surface also generates “collateral” orders which typically break discrete symmetries. Disintricating this cause and its collateral orders remains an outstanding problem.
Recently, the pseudogap line has been associated with the presence of orders in the form of intra unit cell orbital currents Fauqué et al. (2006); Mangin-Thro et al. (2014, 2015), and to the breaking of the rotational symmetry leading typically to a nematic signal Cyr-Choiniere et al. (2015). Ultra-sound experimental data following the pseudogap line have been strengthening the case for a phase transition Shekhter et al. (2013). Although orders cannot open a gap in the Fermi surface, they can induce time-reversal symmetry breaking, as in the case of loop currents, or symmetry breaking, as in the case of nematicity.
The SU(2) model is in very good posture to bring an ample clarification to the situation. Indeed it does predict the simultaneous gapping of the Fermi surface in the anti-nodal region and enhancement of the nematic susceptibility at , both emerging from the SU(2) fluctuations. Nematicity can then be stabilised as a collateral order below , for example by a small amount of symmetry breaking due to the oxygen chains in YBCO Orth et al. (2017). One can remark, interestingly, that the breaking of the -symmetry is not necessarily in competition with the charge modulations.
We now turn to the possible generation of loop current orders by the SU(2) fluctuations. For this, we rely on symmetry considerations, following reference Agterberg et al. (2015). The SU(2) symmetry involves strong coupling between and , which leads us to consider the scalar field , (). We consider the influence of both time-reversal () and parity () transformations on this order parameter (see Supplemental Material for details):
We now form a loop current order parameter out of the field : . Strikingly, this new order parameter transforms as (see Supplemental Material for details):
which is precisely the signature of the loop current observed experimentally. Following the argument in Agterberg et al. (2015), the field has the same symmetries as a pair density wave (PDW) order parameter, and by symmetry such a field can sustain loop currents described by . These are nil outside the pseudogap phase, since , and finite inside. The detailed study of the Ginzburg-Landau formalism for this field theory will be clarified elsewhere, and follow closely the study of the PDW detailed in reference Agterberg et al. (2015).
Charge modulations in real space
Charge modulations (CM) were observed up to the pseudogap energy scale by STM experiments under applied electric field Mesaros et al. (2011); Hamidian et al. (2015). Bulk X-ray probes and NMR, however, reported the presence of charge-modulated areas over a doping dome which doesn’t follow but decreases with as doping decreases Ghiringhelli et al. (2012); Blackburn et al. (2013a); Comin et al. (2014). Here lies a remarkable discrepancy : are the CM associated to the pseudogap energy scale , or to the superconducting coherence energy scale ?
To answer this question we first notice that the STM experiments are done at low temperature inside the superconducting phase ( K) whereas the X-ray probes can directly reach . The SU(2) scenario provides a simple explanation for this unusual situation. Indeed, in the SU(2) picture, vortices inside the ordered superconducting phase have a different structure than in standard superconductors where the normal state is a Landau metallic state. Fig. 3C and 3D show that the charge order coordinate of the SU(2) order parameter is finite at the center of the vortex core. This is similar to the case of SU(2) rotations between -wave superconductivity and a -flux phase Lee and Wen (2001); Einenkel et al. (2014). Here the SU(2) symmetry constrains charge modulations to be present inside the vortex core, as it has been actually seen by STM and NMR Wu et al. (2013). In terms of topological defects, it is as if a meron was sitting at the center of the vortex, with the quantization axis locked to the direction of the charge modulations (Fig. 3C). The emergence of a finite charge order parameter in the center of superconducting vortices is caused by the existence of this third degree of freedom of the SU(2) order parameter. This gives a natural explanation to the observation by bulk probes of charge modulations in the region of the phase diagram where vortices are present, below and within a dome above Wang et al. (2002).
At the operation temperature of STM experiments ( K), the quantization axis of the merons is still locked to the charge modulation direction, up to the energy scale , typical of the pseudogap. As the temperature is raised above the superconducting fluctuations dome, the distribution of quantisation axes across merons becomes fully random, as depicted in Fig. 5, and the CM are impossible to observe. Since the SU(2) theory features the coexistence of and real space regions, the first of which becoming less and less numerous when the temperature is raised, we can see that there are less and less CM as we get closer to , which elucidates experimental data Kohsaka et al. (2008); Gomes et al. (2007b). This disappearance of the CM patches when raising the temperature close to was previously explained by dislocations of the phase of the charge order, interpreted in the framework of a nematic/smectic transition where the nematic order competes with the CM patches Mesaros et al. (2011). Note that in our case, there is no competition between the two, and that the disappearance of the topological defects when temperature is raised, as regions make way for regions, simply free the phase of the charge order.
Phase diagram under magnetic field
The measured phase diagram as a function of applied magnetic field () and temperature () is particularly remarkable because it shows an abrupt phase transition at T from a superconducting phase to an incommensurate charge ordered phase Wu et al. (2011); LeBoeuf et al. (2013); Yu et al. (2014); Chang et al. (2016a); Gerber et al. (2015); Chang et al. (2016b). Both phases have a transition temperature of the same order of magnitude, and co-exist in a small region of the phase diagram. The abruptness of the transition at is reminiscent of a spin-flop transition, typical of non-linear -models Zhang (1997); Chakravarty et al. (1988), which could be related to a “pseudo-spin-flop” transition of our SU(2) order parameter, sketched in Fig. 5.
In the presence of an applied magnetic field, the non-linear -model can be written as:
where , is the electromagnetic vector potential, is the applied magnetic field and we have taken and . Here again, the vector with describes the pseudo-spin states with , , respectively. and are the phase stiffnesses which usually are temperature dependent. The masses can be taken at zero field such as to favour the superconducting state with for example , and .
When is large enough, vortices with charge modulation core are created until Wang et al. (2002). Meanwhile, the superconducting state becomes less favourable compared to the charge modulations . At a certain magnetic field , corresponding to , the spin-flop transition occurs (Fig. 5). Interestingly, for fields in the vicinity of the spin-flop transition, the SU(2) symmetry is almost perfectly realised and the model can be mapped onto the attractive Hubbard model at half-filling, known to possess an exact SU(2) particle-hole symmetry Yang (1989); Yang and Zhang (1990); Zhang (1990).
A prediction of our model is that inside the charge order phase for , vortices disappear and dislocations are the only type of topological defects left. The charge order parameter cancels along these dislocations, which means that the superconducting order parameter is then maximal. These “filaments” of superconductivity could also be caused by the application of a current Davis (2017), which would be reminiscent of transport measurements where glimpses of superconductivity have been seen at very high fields and low temperatures Grissonnanche et al. (2014); Yu et al. (2016); Hsu et al. (2017).
The complex phenomenology of the cuprates has led to the rise of more and more involved theoretical descriptions, some even untractable analytically. Here, we discussed a simple formalism which naturally gives birth to the wealth of observed phenomena, and enables us to embrace it all at once. We described the SU(2) parameter, which is a non-abelian composite of a charge and a superconducting order parameters, and its derivation from a short-range antiferromagnetic model. This order parameter is constrained by an SU(2) symmetry, which means that it sits on a three-dimensional hypersphere. As the temperature is lowered within the pseudogap phase, some real-space regions lose one charge degree of freedom, and the order parameter thus sits on a two-dimensional sphere. The SU(2) order parameter can then be seen as a pseudo-spin order parameter in these regions. This naturally leads to the formation of pseudo-spin skyrmions, which account for many puzzling features of STM, X-ray, and NMR data. Moreover, we discussed how the formation of these topological structures could be used to build experimental setups directly probing the existence of this SU(2) order parameter. Nematic and time-reversal symmetry-breaking features arise in the other regions where all the degrees of freedom still remain. Finally, we discussed the phase diagram under magnetic field, which could correspond to a pseudo-spin-flop transition.
In summary, general considerations on the SU(2) order parameter allow us to grasp the phase diagram of the cuprate superconductors in its complexity. We think this approach is an important step towards finally unravelling the mysteries of these wonderful materials and discuss realistic experimental setups for testing it.
We thank Y. Sidis for stimulating discussions. This work has received financial support from the ANR project UNESCOS ANR-14-CE05-0007, and the ERC, under grant agreement AdG-694651-CHAMPAGNE. The authors also like to thank the IIP (Natal, Brazil), where parts of this work were done, for hospitality.
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Supplemental material for:
Pseudo-spin Skyrmions in the Phase Diagram of Cuprate Superconductors
Here we give the details of the calculation of the loop current order parameter defined in the main text under the time-reversal () and parity () transformations. We start by considering the charge order parameter, which was derived from the - model in a previous work (Montiel et al. (2017) Eq. 7b):
where is the angle spanning the Brillouin zone and is the involution:
where is the ordering wave vector of the charge order parameter defined in the main text.
Applying time-reversal gives:
Let us now consider the -wave superconducting order parameter (Montiel et al. (2017) Eq. 7c):
Here we obtain:
Gathering these two results gives:
The case of parity is much simpler and yields:
We now define the loop current order parameter as:
which therefore transforms as: