arXiv:1001.1025v2 [cond-mat.supr-con] 8 Jan 2010 Magnetic quantum oscillations and multiple holon pockets in underdoped YBa$_{2}$Cu$_{3}$O$_{6+}$$_{y}$ Wei Chen $^{1}$, Oleg P. Sushkov$^{1}$, and Takami Tohyama2 $^{1}$School of Physics, University of New South Wales, Sydney 2052, Australia $^{2}$Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: November 20, 2018) We show that lightly doped YBa 2 Cu 3 O 6+$_{y}$ has multiple holon pockets with different areas, which lead to multiple frequencies of magnetic quantum oscillations. Using neutron scattering data on incommensurate spin ordering we determine these areas, which yields frequencies in good agreement with experiments. Divergence of the effective mass observed in magnetic quantum oscillations indicates a quantum phase transition at the oxygen content y ≈ 0 . 48. We argue that the transition is the onset of quasistatic incommensurate magnetic order predicted by theory and observed in neutron scattering. PACS numbers: 74.25.Jb 71.18.+y 74.72.Gh 75.30.Fv Topology of the Fermi surface is one of the central problems in the physics of cuprate superconductors. An undoped cuprate is antiferromagnetic Mott insulator. At a sufficiently small doping the system can be described by the extended two-dimensional t - J model [1, 2] that pre- dicts small holon pockets located at the “nodal points”. We use the term “holon” to stress that the quasiparti- cle does not carry usual spin, with more details to be explained later. On the other hand there is no doubt that at a sufficiently large doping the system behaves like a normal metal with a large Fermi surface and normal quasiparticles which carry simultaneously electric charge and spin S =1/2. Description of the transition between these two regimes is an open theoretical problem. Angle resolved photoemission spectroscopy (ARPES) studies indicate a large Fermi surface in overdoped regime and formation of Fermi arcs in the underdoped regime [3]. On the other hand, recent magnetic quantum oscillation (MQO) data [4–12] taken in YBa$_{2}$Cu$_{3}$O$_{6+}$$_{y}$(YBCO) in- dicate small Fermi pockets at y < 0 . 66 that corresponds to doping x < 0 . 125. Throughout the paper we denote doping by x and use results of Ref. 13 to relate the oxygen content y with the doping level x . Very recent ARPES studies [14, 15] also give some indications of small Fermi pockets. A theoretical interpretation of ARPES in terms of holons is a fairly involved issue. In a photoemission measurement partially separated spin and charge have to recombine to form a physical electron. So far the re- combination amplitude has only been calculated for the parent Mott insulator [16]. In contrast, MQO is sensi- tive mainly to the electric charge, hence interpretation of MQO in terms of holons is straightforward. With tilted magnetic field MQO can also probe spin of the quasipar- ticle. A very recent measurement of this kind [10] indi- cates suppression of spin and hence supports the holon picture. The sign of the Hall coefficient measured in Ref. 5 cor- responds to electron pockets instead of hole ones. How- ever, the field applied in the experiments is smaller than The typical measured value of F in YBCO is about 500 T, which gives a ratio of A$_{k}$ to the total area of the Brillouin zone A$_{BZ}$ = (2 π/a ) 2 ( a = 3 . 81 ˚ A is the lattice spacing) to be A$_{k}$/A$_{BZ}$ ≈ 0 . 0176. Our analysis of MQO is based on the spin spiral the- ory of a lightly doped Mott insulator. While the idea of the spin spiral has been suggested quite some time much bigger [23] in YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{35}$ ( x ≈ 0 . 065). MQO in a magnetic field B is described as cos(2 π F B + φ ). The period F and the area enclosed by a trajectory in the momentum space A$_{k}$ are related as [24] We will relate MQO with recent neutron scattering ob- servations [19–23]. These data demonstrate incommen- surate spin ordering pined to the tetragonal a ∗ direc- tion. Depending on doping the ordering can be static or dynamic. The QCP separating regions of dynamic and static ordering is located at x$_{QCP}$ ≈ 0 . 09. The spin-wave pseudogap Δ$_{sw}$ is opened at x > x$_{QCP}$ . The data [19] on YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{5}$ ( x ≈ 0 . 1) show Δ$_{sw}$ ∼ 10 meV, while in YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{6}$ ( x 0 . 12) Δ$_{sw}$ ∼ 20 meV [20]. On the other hand the quasistatic scattering has been ob- served at x < x$_{QCP}$ where Δ$_{sw}$ = 0. In YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{45}$ ( x ≈ 0 . 085) the signal is very weak [21, 22], and it is much bigger [23] in YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{35}$ ( x 0 . 065). In this paper we address the following issues related to MQO: (1)Typically more than one MQO frequency is ob- served. What is the origin for the multiple frequencies? (2)The main MQO frequency corresponds to the area of the pocket about 1.8% of the Brillouin zone. How to rec- oncile this very small area with the Luttinger sum rule? (3)Divergence of the effective mass has been observed [11] at doping about 9%. What is the physical origin for this quantum critical point (QCP)? the critical field H$_{c}$$_{2}$ , hence the contribution from the vortex liquid has to be taken into account, which may alter sign of the Hall coefficient [17, 18]. In the present work we consider only hole pockets. F = c ℏ 2 πe A$_{k}$ . (1)