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Cuprate Superconductors: Puzzle of the Pseudogap April 7, 2011

Posted by CMMP Section Chair in : Condensed Matter and Materials Physics (CMMP) , trackback Bookmark and Share

by Philip Phillips, University of Illinois Urbana-Champaign

It has now been 25 years since superconductivity was discovered in the copper-oxide ceramics (hereafter cuprates). One thing we have learned since then is that these materials defy explanation within the standard paradigms of solid state physics. In metals such as mercury, superconductivity emerges from a normal state in which the interactions between the electrons can be ignored. The only interaction which is relevant is that arising from the ions. When two ions move closer together, the electrons experience a net attraction which gives rise to charge 2e charge carriers.

In the cuprates, superconductivity emerges from the pseudogap state in which there is a depression of the single particle density of states in the absence of superconductivity. Straightforward application of the standard superconducting paradigm to a state of matter with no states at the chemical potential yields a vanishing superconducting transition temperature.

However, the transition temperature in the cuprates can be as high as 140K. Hence, something else must be going on in these materials. The experiments by He, et al [1] are designed to unlock the secrets of this mysterious pseudogap phase which sets in at a temperature T^\ast as shown in the figure below.

Phase diagram and anomalous transport in the cuprate high-temperature superconductors. Heuristic phase diagram as a function of holes doped into the copper-oxide plane. The pseudogap and strange metal are characterized by a depletion of the density of states and a T-linear resistivity, respectively. The pseudogap terminates at a zero-temperature critical point or quantum critical point (QCP). To the right is a Fermi liquid (conventional theory of metals) where weak-coupling accounts become valid. We currently have no theory of superconductivity in a state of matter that is a non-Fermi liquid. Even the most recent constructions of superconductivity using tools from string theory that are designed to get at strong correlations in quantum systems yield superconductivity [2] only from the Fermi liquid state. This is truly unfortunate and points to how rudimentary our understanding of superconductivity really is.

The phenomena surrounding the pseudogap in the cuprates used to be fairly simple. In zero magnetic field, lightly doped cuprates possess an incomplete Fermi surface, termed a Fermi arc, in the normal state. That is, the Fermi surface which is present in the overdoped, more conventional Fermi liquid regime is destroyed on underdoping leaving behind only a Fermi arc.

In actuality, the situation is much worse. That the Fermi arc does not represent a collection of well-defined quasiparticle excitations has been clarified by Kanigel, et al. [3] who showed that in Bi_2Sr_2CaCu_2O_{8-\delta}, the length of the Fermi arc shrinks to zero as T/T^* tends to zero. Consequently, the only remnant of the arc at T=0 is a quasiparticle in the vicinity of (\pi/2,\pi/2) and hence the consistency with nodal metal phenomenology.

Recently, however, new ingredients have been added to the pseudogap story in the underdoped regime which, on the surface, are difficult to reconcile with Fermi arcs. At high magnetic fields, quantum oscillations, indicative of a closed 2 Fermi surface, have been observed [4] in Y123 and Tl-2201 through measurements of the Hall resistivity, Shubnikovde Haas effect, and the magnetization in a de Haas-van Alphen experiment. Also attracting much attention is the recent experimental evidence for nematic order [5, 6] ( a state with broken translational symmetry but still possessing translational symmetry) at the onset of the pseudogap onset temperature, T^\ast.

The paper by He, et al. [1] reports a series of measurements (as others have previously [7]) which point to the pseudogap regime being driven by a phase transition. The most puzzling of these experiments is the Kerr effect which requires the breaking of time-reversal symmetry. The authors claim, however, that the magnitude of this effect is too small for it to be the dominant cause of the pseudogap. If this is so, then perhaps the order which is seen is really an epiphenomenon having no causal connection to the pseudogap. What then of the transport anisotropies which have been attributed to nematic order?

Interestingly, the models [8, 9] proposed to explain the Kerr effect do not result in transport anisotropies. It might turn out that the transport anisotropies observed in the Nernst signal are a red-herring, afterall since the orthorhombic lattice symmetry of the cuprates already has asymmetric x and y axes.

While trying to understand the origin of competing order in the pseudogap state is important, it is entirely likely that order has nothing to do with the efficient cause of the pseudogap, the suppression of the single-particle density of states at the chemical potential. Such a claim has been made recently by Yazdani and collaborators [10] who also observed electronic inhomogeneities at the onset of the pseudogap state. They state explicitly, “While demonstrating that the fluctuating stripes emerge with the onset of the pseudogap state and occur over a large part of the cuprate phase diagram, our experiments indicate that they are a consequence of pseudogap behavior rather than its cause.” [10]

I think it is in this context that the He, et al. [1] experiments must be placed. The disassociation of order from the origin of the pseudogap is not entirely surprising. After all, the phase diagram of the cuprates does tell us that the single theory of these systems must above the superconducting dome explain the pseudogap and at higher temperatures the strange metal. Hence, focusing on the pseudogap independent of the strange metal amounts to not facing up to the nature of the charge vacuum of the high-temperature phase.

It is in this regime that the strong correlations conspire to produce the anomalous properties of the normal state. As neither the pseudogap nor the strange metal appear necessarily as T=0 states of matter in the cuprate phase diagram, the standard guiding principle of model building in which only T=0 states are relevant fails in this problem.

Nonetheless, the relevant physics should emerge from correct implementation of the Wilsonian program. As Wilson has taught us, high and low-energy physics are linked through a series of recursion equations that arise once the high-energy degrees of freedom are integrated out. In weakly interacting systems (Hg for example), such an integration simply renormalizes the coupling constants in the low-energy sector. However, in strongly interacting systems, new degrees of freedom can be generated [11].

The theoretical resolution of the normal state of the cuprates rests in demonstrating how the degrees of freedom that are generated upon integrating out the high-energy scale mediate the strange metal and at lower temperatures the pseudogap regime. While significant progress has been made on this problem recently [11], the associated phenomena found by He, et al.[1] relating to the origin of time-reversal symmetry breaking have not been addressed. This stands as an open problem.

[1] R. He, et al., Science 331, 1579 (2011)
[2] T. Hartman and S. A. Hartnoll, arXiv:1003.1918
[3] A. Kanigel, et al. Nat. Phys. 2, 447 (2006)
[4] L. Taillefer, arxiv:0901.2313
[5] R. Daou, et al. arxiv:0909.4330
[6] D. Haug, et al., Phys. Rev. Lett. 103, 017001 (2009)
[7] B. Fauque, Phys. Rev. Lett. 96, 197001 (2006)
[8] Aji, V., et al., Phys. Rev. B 81, 064515 (2010)
[9] S. Chakravarty, et al., Phys. Rev. B 63, 094503 (2001)
[10] C. V. Parker, et al., Nature 468, 677 (2010)
[11] P. Phillips, Rev. Mod. Phys. 82, 1719 (2010)

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1. Igor Minich - April 27, 2011

1. Pseudogap is dielectric gap depending on wavevector of electron.
Pseudogap is at maximum in antinodal direction and is zero near nodal direction.
2. T* is the temperature of the phase transition of the first order.
3. Pseudogap and superconducting gap (SC gap) coexist in superconducting state.
4. SC gap is one electron gap, depending on wavevector of electron, and can exist only if pseudogap in this direction is zero (ie near nodal direction).
5. In underdoped state SC gap can’t exist in antinodal direction, where pseudogap is not zero.
6. In overdoped state pseudogap does not exist and SC gap is at maximum in antinodal direction.
7. In any case SC gap can be zero or nonzero in some cone near nodal direction.

2. Philipp - August 26, 2014

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3. CMMP Section Chair - August 26, 2014

I think this is fine.