Quantum oscillations in non-Fermi liquids:Implications for high-temperature superconductors

Quantum oscillations in non-Fermi liquids: Implications for high-temperature superconductors


We address quantum oscillation experiments in high superconductors and the evidence from these experiments for a pseudogap versus a Fermi liquid phase at high magnetic fields. As a concrete alternative to a Fermi liquid phase, the pseudogap state we consider derives from earlier work within a Gor’kov-based Landau level approach. Here the normal state pairing gap in the presence of high fields is spatially non-uniform, incorporating small gap values. These, in addition to -wave gap nodes, are responsible for the persistence of quantum oscillations. Important here are methodologies for distinguishing different scenarios. To this end we examine the temperature dependence of the oscillations. Detailed quantitative analysis of this temperature dependence demonstrates that a high field pseudogap state in the cuprates may well “masquerade” as a Fermi liquid.

The surprising discovery of quantum oscillations in the underdoped cuprate high-temperature superconductorsDoiron-Leyraud et al. (2007); Yelland et al. (2008); Bangura et al. (2008) can potentially elucidate the normal, non-superconducting state of these materials. However, a number of experiments seem to indicate the presence of phenomena that may be associated with the application of high magnetic fields, rather than with the intrinsic normal phase. The oscillatory frequency is very small, which suggests a small and possibly reconstructed Fermi surface.Millis and Norman (2007); Yao et al. (2011); Vignolle et al. (2011); Sebastian et al. (2012) More recently, static or quasi-static charge density wave order in the high-field regime has been observed, which also suggests substantial differences between the zero-field normal state and the high-field state.Wu et al. (2011); Ghiringhelli et al. (2012); Chang et al. (2012); Wu et al. (2013) Changes in magnetization, as well as transport coefficients, have also been found at high fields.Li et al. (2010); LeBoeuf et al. (2007); ?

It is notable that detailed studies of the temperature dependence of the oscillatory amplitudeSebastian et al. (2010) report excellent agreement with a Fermi-Dirac dependence, providing no sign of the pseudogap state that has been observed at magnetic field . Adding to the complexity, specific heat measurements suggest an overall temperature dependence of that is consistent with the presence of a -wave pairing gap. Riggs et al. (2011) These observations have led to theoretical proposals in which there are Fermi liquid-like features, perhaps co-existing with a pseudogap. Two notable scenarios both introduce a low frequency, third peak into the spectral function, while maintaining the two other peaks at finite which reflect a gap structure. Senthil and Lee (2009); Banerjee et al. (2013) However, the question of whether the quantum oscillation measurements require this peak (or more broadly, Fermi liquid-like behavior near the Fermi surface) is an open and very important one. The answer bears on the proper microscopic description of the cuprates.

In this paper we address this issue more phenomenologically. We quantitatively compare a high field pseudogap scenario and an alternative “co-existing pseudogap and Fermi liquid” approach with a strict Fermi liquid. We do so via the measured Sebastian et al. (2010) temperature dependences of the quantum oscillations which have been interpreted to strongly support a Fermi liquid phase at high . Our high field pseudogap scenario was derived earlier using a Gor’kov-based theory Scherpelz et al. (2013a, b) and incorporating Landau level physics. Although some model systems do display substantial deviations, discrepancies for the parameter sets appropriate to the cuprates are not large enough to be detected in existing experiments. Thus we conclude that a non-Fermi liquid phase supports quantum oscillations with dependence presently indistinguishable from that of a Fermi liquid.

Figure 1: (a) A density plot of for a pseudogap “blurred vortex” model.Scherpelz et al. (2013b) This indicates that the normal state gap structure (representing preformed pairs) is inhomogeneous as a result of high magnetic fields. (b) The density plot of for the -wave, system used throughout the paper. Here is based on a magnetic translation group. Both plots are normalized to .

Theory of the Pseudogap in High Magnetic Fields At the commonly used pseudogap self-energy , has been derived from a theory of pairing fluctuations,Janko et al. (1997); Maly et al. (1997) and has also been obtained phenomenologically by fitting angle resolved photoemission experiments.Norman et al. (1998, 2007) It is given by


where i is the fermionic Matsubara frequency, the single-particle dispersion, and and damping coefficients associated with the pairing gap () and single particles, respectively.

Using Gor’kov theory, we Scherpelz et al. (2013a) and others Dukan and Tesanovic (1994) have shown earlier that in the presence of large magnetic fields, with intra-Landau level pairing, the general BCS-like structure of the Green’s functions (and hence self energy as in Eq. (1)) is maintained. For this quasi-two dimensional pseudogap state, we incorporate Landau levels via , where is the Landau level and the degenerate quantum index. The latter is based on a magnetic translation group approach which is associated with the superconducting Dukan and Tesanovic (1994) and pseudogap phases.Scherpelz et al. (2013a)

Importantly, a gap squared contribution (as in Eq. (1)) persists Scherpelz et al. (2013a) into the normal phase. This reflects pairing (as distinct from phase) fluctuations which arise from short-lived, preformed pairs; they are to be associated with stronger-than-BCS attractive interactions (consistent with high transition temperatures) and they lead to a pseudogap. Furthermore, Gor’kov theory at high fields requires the introduction of inhomogeneity in the gap function .Scherpelz et al. (2013b) Similarly, in the normal high-field state the pairing gap must be dependent on the parameters defined above, and this will lead to real-space inhomogeneity. Physically, these inhomogeneities reflect excited pair states which were shown Scherpelz et al. (2013b) to correspond to small distortions or excitations of the optimal (condensate) vortex configuration. As such, they represent blurred lattice patterns. This “precursor vortex” stateScherpelz et al. (2013b) is illustrated in Fig. 1a.

For the purposes of this paper, the exact dependence of (where is the (real) magnitude of the gap) on need not be determined. Instead only the distribution of values over is fixed. Throughout the paper we use the normalization that , the specified gap magnitude. One can reasonably approximate this distribution by taking independent of , as the distribution should change only slightly for moderate to large Landau levels. While the high field normal state gap inhomogeneity (“pseudovortex”) is present for any pairing symmetry, it should be noted that nodal effects from -wave pairing in a Landau level basis also lead to real-space inhomogeneity. 1 For simplicity, we calculate the distribution for from previous work on -wave pairing at high magnetic fields within the magnetic translation group,Vavilov and Mineev (1998) and use this as a model distribution throughout the paper.

Fig. 1b shows the density plot of while the Fig. 2a inset shows the histogram used. We stress that deviations in vortex locations, pseudogap inhomogeneities, and inter-Landau level pairingTesanovic and Sacramento (1998) should only change this distribution slightly, and leave the conclusions unaffected. Thus, with this method we have extended the zero-field self energy, Eq. (1), to a form appropriate for the high-field pseudogap state.

It is useful to compare this high field normal state picture with others in the literature, which focus on short-range fluctuations of the phase, both in spaceStephen (1992) and time,Banerjee et al. (2013) caused by the disorder introduced by moving vortices within a vortex liquid. By contrast, our theory does not assume the presence of short-range phase coherence or vortices, but rather incorporates fluctuating pair states. We emphasize Landau level physics that should occur in any paired, high-field system and we focus on nodal gap states created by both the -wave pairing symmetry and by real-space inhomogeneity. Also important is the fact that real-space inhomogeneity is necessary to allow a phase transition (from the pseudogap to the superconducting state) in a field, owing to the one dimensionality associated with . Scherpelz et al. (2013b) It is currently unclear whether true vortices are present in the high-field quantum oscillation regime, as different experiments arrive at different conclusions. Li et al. (2010); Grissonnanche et al. (2013); Tan and Levin (2004) In some ways, however, the vortex liquid theories and the present pseudogap scenario are more similar than might be thought. In the pseudogap scenario there is significant weight at small gap values, while in the alternatives there is an additional gapless (or Fermi liquid) spectral function peak.

Quantum Oscillations in the Pseudogap State Using the self-energy, together with the form of , we address the amplitude of oscillations within this non-Fermi liquid pseudogap state via calculations of the density of states at the Fermi surface, . Here which will reflect the oscillations at zero-temperature. The spectral function for the self-energy (see Fig. 2b). Sample results for vs.  are shown in Fig. 2a.2 While the amplitude of the oscillations ranges from 3 to 20 times smaller (depending on ) than that in a Fermi liquid, they are still clearly visible. Thus, a non-Fermi liquid, pseudogap system can display robust quantum oscillations, as we have shown in similar work near zero field.He et al. (2013) This is consistent with the quantum oscillations seen previously within the superconducting phase of extreme type-II superconductors. Corcoran et al. (1994); ?; ?; ?; Stephen (1992); Dukan and Tesanovic (1995)

Figure 2: (a) A plot of the density of states at the Fermi surface vs.  (with ) for the 2D, high-field system with self-energy given by Eq. (1), with , , and . The black line shows the -wave used throughout this paper, while the dashed purple line shows a constant . The inset shows the probability distribution function of values (normalized so that ) for the high-field, -wave used here (black line) and for comparison a distribution of for zero-field -wave, (blue dots). (b) Example spectral functions for the system, with variable . In both the main plot and the inset, for the solid black, dashed red, and dash-dotted blue curve respectively. In the main plot, the self-energy in Eq. (1) is used with and , while in the inset Eq. (20) of Ref. Senthil and Lee, 2009 is used with and .

A critical component of these oscillations is the presence of nodes or near-nodal states in , as Figs. 1b and 2a inset show are present in this system. If on the other hand one presumed a constant , Fig. 2a (which plots the density of states) shows that the amplitude of the oscillations is reduced to zero. For the -wave case the near-nodal states, in contrast, preserve the single-particle Landau level dispersion and thus the quantum oscillations.

Temperature Dependence of Oscillations
To distinguish a non-Fermi liquid from a Fermi liquid on the basis of quantum oscillations, we next focus on the temperature dependence of the oscillation amplitude. Importantly, this temperature dependence has previously been measured in YBaCuO and shown to have excellent agreement with Fermi liquid theory.Sebastian et al. (2010) We compare the expected temperature dependence of the high field pseudogap state developed here with that of an admixed Fermi liquid/ pseudogap scenario for the normal state.Senthil and Lee (2009) Throughout we presume is roughly T-independent, since it depends on much larger temperature scales than those accessed by quantum oscillations.

Figure 3: (a) A plot of the rescaled temperature dependence of the oscillation amplitude, vs. . The green solid line is for Fermi-liquid theory; the black dashed line represents the present pseudogap model with physical parameters (Ref. Shibauchi et al., 2001), (consistent with Fermi arcsNorman et al. (2007); He et al. (2013)), and . The red dotted line represents an unphysical case for comparison, with small and . The inset shows the residual between the calculation and Fermi-liquid theory. (b) The non-oscillating density of states for the systems in part (a). The case is quite flat within the range . In contrast, the more unphysical case shows a significant curvature of the density of states. The normalization is used here.

The temperature dependence is computed by investigating the total energy of quasiparticles for a grand canonical system with fixed :Fetter and (1971)


where is the Fermi function. Because the oscillations are created by quasiparticles near the Fermi surface, and are both much less than , and we can approximate .

Following earlier work Abrikosov (1988); Dukan and Tesanovic (1995) we take the large Landau level limit and use the Poisson resummation formula to extract the fundamental frequency of oscillation. After integration by parts this yields Shoenberg (1984)


where . In a Fermi liquid system, in which only depends on , this becomes a convolution and gives a total amplitude equal to the Fourier transform of as in previous work.Shoenberg (1984); Sebastian et al. (2010) (Note that throughout we take to be temperature independent.) For a general non-Fermi liquid system, the above amplitude will not be a convolution, and the temperature-dependent amplitude can instead be calculated using an analytic form of .3

Results The calculated temperature dependence of this non-Fermi liquid with a large pseudogap is shown as the dashed line in Fig. 3a and gives excellent agreement with Fermi liquid theory (solid green line). In contrast, the dotted curve with a much smaller displays large deviations from the Fermi liquid. In the cuprates is much larger than . (For example, in Ref. Sebastian et al., 2010 meV, whereas can be tens of meV.Shibauchi et al. (2001)) This indicates that it can be possible for a pseudogapped system to “masquerade” as a Fermi liquid for quantum oscillation measurements in the cuprates.

The discrepancy in the energy scales of (Ref. Shibauchi et al., 2001) and is the primary cause of this similarity to Fermi liquid behavior. As shown by the dashed line in Fig. 3b, because these scales are so different, the underlying density of states has very little curvature. This, as well as the damping and inhomogeneity, creates a flat density of states near the Fermi surface which in turn leads to Fermi liquid-like behavior in the oscillations. Specifically, Fig. 3b plots the non-oscillating integrand , essentially a zero-field density of states. As quantum oscillations are primarily sensitive to the innermost Landau level to the Fermi surface, Eq. (3) is mostly sensitive to effects within of the Fermi surface. Although systems with do display variations on this scale, in more physical situations as studied here they are negligible.

FL Theory
Ref. Sebastian et al.,2010
(3) Ref. Senthil and Lee,2009
Table 1: A table of -limited moments for different cases. (1) and (2) use the parameters from the dashed black line and dotted red line of Fig. 2, respectively. (3) uses the spectral function in Ref. Senthil and Lee, 2009 with , and . Similarly to Ref. Sebastian et al., 2010 we fit the zero-temperature amplitude and the effective mass to the first two even moments which are obtained exactly. The next three even moments are non-fitted results, presented here. (Note that Ref. Sebastian et al., 2010 partially uses these higher moments in their fit.)

In order to more quantitatively determine the quality of fit, we calculate the moments of the Fourier transform of the amplitude, as in Ref. Sebastian et al., 2010, which are displayed in Table 1.4 In the table, Case (1) uses the large and considered throughout the paper, and shows excellent agreement with the theoretical Fermi liquid values. Case (2) provides an example of a non-Fermi liquid that does display large deviations from a Fermi liquid system, with a small , as can clearly be seen in Fig. 3a as well. This provides evidence that this technique can be useful for the discrimination of non-Fermi liquids in some systems. Finally, in Case (3) we consider a different theoretical proposal, using the self-energy in Eq. (20) of Ref. Senthil and Lee, 2009. This system produces a three-peaked spectral function as shown in the inset of Fig. 2b, which provides more weight at the Fermi surface in order to restore Fermi liquid behavior. Senthil and Lee (2009) Excellent agreement with Fermi liquid theory is obtained. This demonstrates that in the non-Fermi liquid model of Case (1) deviation from Fermi liquid behavior is present; however, it is not detectable with existing experiments.

Conclusions It should be stressed that the goal of the present paper was not to introduce a scenario for a high field non-Fermi liquid normal state (which was presented earlier in other contexts Scherpelz et al. (2013a, b)). This is admittedly a very controversial and still unresolved subject. Rather the aim of this paper is to design and present tests of different existing (and possibly future) theoretical scenarios by addressing the magnitude and temperature dependence of observed quantum oscillations.

By starting with a specific, non-Fermi liquid model of a high-field pseudogap, we have found that quantum oscillations can both be present in a non-Fermi liquid and display a temperature dependence remarkably similar to that of a Fermi liquid. Here we have focused on a model, built on Gor’kov theory, which incorporates Landau-level based pairing, real-space inhomogeneity, and -wave pairing symmetry. Our non-Fermi liquid scenario is to be contrasted with hybrid Fermi liquid-pseudogap approaches which view the “normal” high field phase as a vortex liquid, Senthil and Lee (2009); Banerjee et al. (2013) a concept about which there is not yet unanimity. Li et al. (2010); Grissonnanche et al. (2013) Within our model, two major components are responsible for the robust oscillations. First, the nodal or near-nodal effects leading to small values of are critical in retaining the visibility of quantum oscillations in these systems. Second, the large discrepancy in energy scales between the gapShibauchi et al. (2001) and the cyclotron frequency makes the system appear relatively Fermi liquid-like on the scale of the oscillations.

The fact that existing experimental work has not distinguished between these scenarios may allow for a simple resolution between oscillatory measurements and the specific heat measurements which suggest the continued presence of a -wave gap at high fields.Riggs et al. (2011) We note that the temperature dependent formalism outlined in this paper may serve as a vehicle for testing future theories of these oscillatory phenomena. Additionally, our work has shown that an extremely high level of precision will be required in future experiments to distinguish among different theoretical scenarios of the oscillations.

This work is supported by NSF-MRSEC Grant 0820054. P.S. acknowledges support from the Hertz Foundation.


  1. Note that in -wave gaps, nodal states are created solely by real-space inhomogeneity. However, in -wave contributions to nodal states from pairing symmetry and Landau level-based real-space inhomogeneity are both present and become essentially inseparable,Vavilov and Mineev (1998) which means we will not distinguish between their descriptions here.
  2. We take , which would be zero in the superconducting state, to be as large as based on estimates which use the Fermi arc size in zero-field.Norman et al. (2007); He et al. (2013) The choice of has little effect on the conclusions, but strikes a balance between computability (with no extremely sharp features, as opposed to ) and the visibility of distinct oscillations (as opposed to large ). Parameters used for the model from Ref. Senthil and Lee, 2009 preserve that paper’s choice of and , while itself is scaled to match the other models.
  3. For our calculation, we make one more transformation, by splitting up the integral in Eq. (3) (with the integrand): We know that as , the result of Eq. (3) must go to zero, and with , we find that . This latter formula is used to compute the constant part of the original integral, so that each individual calculation only requires computation of the term.
  4. Note that unlike Eq. (1) in Ref. Sebastian et al., 2010, no is used here.


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