A Universal Phenomenon in Unconventional Superconductivity?

S S symmetry Article Fluctuating Charge Order: A Universal Phenomenon in Unconventional Superconductivity? Erminald Bertel * and Alexander Menzel Institute of Physical Chemistry, University of Innsbruck, A-6020 Innsbruck, Austria; [email protected] * Correspondence: [email protected]; Tel.: +43-512-507-58000 Academic Editor: Sergey Borisenko Received: 6 May 2016; Accepted: 31 May 2016; Published: 10 June 2016 Abstract: Unconventional superconductors are characterized by various competing ordering phenomena in the normal state, such as antiferromagnetism, charge order, orbital order or nematicity. According to a widespread view, antiferromagnetic fluctuations are the dominant ordering phenomenon in cuprates and Fe based superconductors and are responsible for electron pairing. In contrast, charge order is believed to be subdominant and compete with superconductivity. Here, we argue that fluctuating charge order in the (0,π) direction is a feature shared by the cuprates and the Fe based superconductors alike. Recent data and theoretical models suggest that superconductivity is brought about by charge order excitations independently from spin fluctuations. Thus, quantum fluctuations of charge order may provide an alternative to spin fluctuations as a mechanism of electron pairing in unconventional superconductors. Keywords: iron-based superconductors; cuprates; charge density wave; charge order; spin order; fluctuations; phase transition; quantum critical point 1. Introduction Superconductivity arises from the interplay of several mechanisms. First, there is the exchange of bosonic excitations, which gives rise to an indirect electron-electron attraction competing with the Coulomb repulsion. In classical BCS superconductivity, the bosonic excitations are phonons. Basically, an electron with momentum k scatters into a state k ´ q creating a phonon with energy }wq. The phonon is absorbed by another electron, whereupon it changes its momentum from k1 to k1 ` q (see Figure1). The corresponding interaction is expressed as [1]: 1 1 1 xi|Hint | f y “ x f | He´ph |hy xh| He´ph |iy ` (1.1) 2 ˜ Ei ´ Eh Ef ´ Eh ¸ ¸h here, the initial state |iy “ k, k1; 0 is composed of the electron wave functions before scattering and phonon emission. The final state | f y “ k ´ q, k1 ` q; 0 contains the scattered electron wave functions ˇ D after scattering and phononˇ absorption. The intermediate, virtually excited states |hy “ k ´ q, k1; w ˇ D } q with energy E “ E ` E 1 ` w correspondˇ to the excitation of a phonon by one of the electrons. h k´q k } q ˇ D ˇ The perturbation operator He´ph is expressed as: íqr † ˚ iqr He´ph “ Aqe aq ` Aq e aq (1.2) † with aq and aq being the phonon creation and annihilation operators, respectively. Aq gauges the strength of the electron–phonon interaction. In order to qualitatively appreciate the meaning of Symmetry 2016, 8, 45; doi:10.3390/sym8060045 www.mdpi.com/journal/symmetry Symmetry 2016, 8, 45 2 of 14 Symmetry †2016, 8, 45 2 of 14 with aq and aq being the phonon creation and annihilation operators, respectively. Aq gauges † withthe strength aq and of atheq being electron the– phononphonon interaction.creation and In annihilation order to qualitatively operators, appreciate respectively. the meaningAq gauges of the strength of the electron–phonon interaction. In order to qualitatively appreciate the meaning of Equation (1.1), we assume for simplicity that the scattering amplitude is the same for f He ph h Equation (1.1), we assume for simplicity that the scattering amplitude is the same for f H h and i H h and is constant. This yields e ph e ph , and i He ph h and is constant. This yields Symmetry 2016, 8,, 45 2 of 14 12 1 1 i H f M int (1.3) 12 2 h EEEE 1 1 i h f h Equation (1.1), we assume for simplicityi Hint f thatthe M scattering amplitude is the same for x f | H |hy and(1.3) 2 EEEE e´ph h i h f h xThei| He ´summationph |hy, and is constant.is subject This to yields the constraint that the electrons can only be scattered into The summation is subject to the constrainth that the electrons can only be scattered into unoccupied states. Note that the virtual states1 are1 short-lived1 and hence need not obey energy xi|H | f y “ |M|2 ` (1.3) unoccupied states. Note that the virtualint states2 h areE ´ shortE -livedE ´ E and hence need not obey energy E h ˜ i h f h ¸ conservation. Since h includes the phonon energy¸ , q , resonance occurs for electron energies conservation.e Since E includes the phonon energy, , resonance occurs for electron energies The summationh is subject to the constraint that the electronsq can only be scattered into unoccupied EEh i q in the virtual state Eik2 . With hardly any empty states available below F e states. Note that the virtual states |hy are short-lived and hence need not obey energy conservation. ´ EEh i q in the virtual state Eik2 . With hardly any empty statese available below F EiqSince , Eith isincludes clear that the phonon the matrix energy, elemen}wq, resonancet Equation occurs (1.3) for is electron negative, energies i.e., leadsEh “ toEi án} wattractiveq in the virtual state Ei « 2#kF . With hardly any empty states available below Ei ´ }wq, it is clear that the interactionEiq, it (seeis clear Figure that 2). the matrix element Equation (1.3) is negative, i.e., leads to an attractive matrix element Equation (1.3) is negative, i.e., leads to an attractive interaction (see Figure2). interaction (see Figùre 2). ˘ Figure 1. Pairing interaction between electrons at the Fermi level via exchange of phonons q . FigureThe Figurediagram 1. Pairing 1. refersPairing interaction to interactionone term between in between the sum electrons electrons Equation at at the the(1.1) Fermi Fermi. The level leveltotal momentum via exchangeexchange of of ofthe phonons phonons electron}w pairq. q is. Theassumed diagramThe diagram to refersbe referszero to one to(this one term maximizes term in in the the sum sumthe Equation Equationphase space (1.1).(1.1). TheThefor total thetotal momentumsummation momentum of Equation theof the electron electron (1.1) pair). pair is Only is assumed to be zero (this maximizes the phase space for the summation Equation (1.1)). Only electrons assumedelectrons withinto be zeroa shell (this of widthmaximizes thearound phase EF canspace participate for the summationin the pairing Equation interaction. (1.1) ). Only within a shell of width }wq around EqF can participate in the pairing interaction. electrons within a shell of width q around EF can participate in the pairing interaction. FigureFigure 2. Electron 2. Electron energy energy diagram for for the the evaluation evaluation of matrix of matrix element eleme Equationnt Equation (1.3). Here, (1.3) the. widthHere, the Figurewidthof theof2. the FermiElectron Fermi distribution energy distribution isdiagram assumed is assumedfor to bethe negligible evaluation to be innegligible comparison of matrix in tocomparisoneleme the phononnt Equation to energy the (1.3)}phwqonon.. Hence, Here energy, the Ei “ Ef « 2#k . The configuration space available for the virtual states is shown as the shaded area. width . ofHence the ,Fermi EEF distribution2 . The is configurationassumed to be space negligible available in forcomparison the virtual to states the ph is ononshown energy as the q i f kF The. Hence attractive, EE interaction2 . The Equation configuration (1.3) is space a bare available interaction for the competing virtual states with is theshown Coulomb as the shadedq area. i f kF repulsion. Whether it can actually give rise to a sufficient Cooper pair density and a superconducting shaded area. phase transition depends on the response of the other electrons in the Fermi sea. The next step, therefore, is to calculate the Cooper pair susceptibility χpair pq “ 0, !q. As mentioned, the choice q “ 0 yields the largest interaction Equation (1.1), and, therefore, in the lowest energy state, the total Symmetry 2016, 8, 45 3 of 14 momentum of the pairs is zero, i.e., electron pairs with momentum k and ´k are formed. One obtains [2] χ0 p0, !q χ pair pair p0, wq “ 0 (1.4) 1 ` Uχpair p0, !q with 0 1 f pxkq ´ f p´x´kq cpair p0, wq “ (1.5) W w ´ xk ´ x´k ` id ¸k W being the volume of the system and f pxkq the Fermi factor at the energy xk “ #k ´ EF. Equation (1.4) 0 signals a phase transition once the denominator goes to zero. Since cpair p0, wq is positive, this may be the case for attractive interactions U pU ă 0q such as Equation (1.3). The exchange of phonons, however, is not the only possible source for attractive electron interactions. As mentioned above, other bosonic excitations could serve a similar purpose. In particular, there is a general consensus that paramagnon excitations are likely candidates in the case of cuprates and also of the iron-based unconventional superconductors. A third possibility arises in the presence of (fluctuating) charge density waves (CDW). The complex CDW order parameter is characterized by the amplitude and the phase of the CDW modulation. The normal modes of these two degrees of freedom are amplitudon and phason [3] (see Figure3). The dispersion of the normal modes resembles the dispersion of optical and acustic phonons, respectively. For an incommensurate CDW, a uniform phase shift of the charge modulation relative to the ionic lattice does not change the energy, thus the phason mode dispersion starts at wphason “ 0 for q “ 0. In contrast, the amplitudon has a finite energy even at q “ 0. Note that the phason involves a shift of the electronic charge with respect to the ionic background and therefore is associated with an optically excitable dipole moment, while the amplitudon changes the polarizability and is detectable in Raman spectroscopy.

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