Fermi surface reconstruction due to hidden rotating antiferromagnetism in n and p-type high-$T_C$ cuprates

The Fermi surface calculated within the rotating antiferromagentism theory undergoes a topological change when doping changes from p-type to n-type, in qualitative agreement with experimental data for n-type cuprate Nd$_{2-x}$Ce$_x$CuO$_4$ and p-type La$_{2-x}$Sr$_x$CuO$_4$. Also, the reconstruction of the Fermi surface observed experimentally close to optimal doing in p-type cuprates, and slightly higher than optimal doping in the overdoped regime for this n-type high-$T_C$ cuprate is well accounted for in this theory, and is a consequence of quantum criticality caused by the disappearance of rotating antiferromagnetism. The present results are in qualitative agreement with the recently observed quantum oscillations in some high-$T_C$ cuprates regarding the change in the size of the Fermi surface as doping evolves and the location of its reconstruction. This paper presents new results about the application of the rotating antiferromagnetism theory to the study of electronic structure for n-type materials.

Conclusions and a discussion of existing experimental data are given in Section 5.  50 We first focus on the normal (non superconducting) state where we review the derivation of rotating 51 antiferromagnetism (RAF). In section 2.4 we will review the interplay between SC and RAF. Consider In Section 2.3 below, an interpretation of RAF from a classical point of view will be given. 70 The parameter Q i in (2) is thus used to carry on a mean-field decoupling of the t-t ′ Hubbard model 71 (1). Consideration of the ansatz where φ i − φ j = π, with i and j labeling any two adjacent lattice sites, 72 and letting the phase φ i ≡ φ be site independent but assuming any value in [0, 2π] led to the following 73 normal state Hamiltonian in RAFT [1,6,7] where N is the number of sites, and n = n i,σ is the expectation value of the number operator. Because 75 of antiferromagnetic correlations a bipartite lattice with sublattices A and B is considered, even though 76 no long-range static antiferromagnetic order is taken into account. Note that RAFT is only valid away 77 from half-filling where this long-range order occurs. The summation runs over the reduced (magnetic) 78 Brillouin zone (RBZ). The Nambu spinor is Ψ † k = (c A † k↑ c B † k↑ c A † k↓ c B † k↓ ), and the Hamiltonian matrix is yielding the energy spectra where µ ′ (k) = µ − Un + 4t ′ cos k x cos k y , E q (k) = ǫ 2 (k) + (UQ) 2 , and ǫ(k) = −2t(cos k x + cos k y ).
Note that the thermal averages of S x i and S y i are given by and S z i = 0 for i in both sublattices. Because the phase φ assumes any value between 0 and 2π, 87 rotational symmetry will not look broken for times greater than the period of rotation as we will explain 88 below, when we review the calculation of the time dependence of the phase. However if the typical time 89 scale of a probe is much smaller than this period symmetry may appear broken. 90 The magnitude Q and electron occupation thermal average n are calculated by minimizing the phase-91 independent mean-field free energy. The following mean-field equations were obtained in the normal 92 state [1,6,7] The phase can thus be written as φ = Uτ /h modulo 2π when S + j (0) is identified with | S + j (τ ) |,

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(−| S + j (τ ) |), for sublattice A, (B), and e iφ with e iU τ /h . Using this result, the magnetic configuration The above derivation of RAF was supported by the following argument, which shows that rotating x-axis are placed on any two adjacent sites of a lattice. To relate RAF to spin flip processes, it is noted 129 that S ± = S x ± i S y =h 2 e ±iωt in this example. In a given model, a coupling is necessary for RAF. If the time spent by the neutron in the vicinity of the spin is smaller then there is a chance RAF 153 will be detected. Note that smaller times means higher energies for neutrons. This is an issue that is still 154 under investigation and will be reported on in the future.

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In RAFT, d-wave SC was introduced phenomenologically using an attractive coupling between 157 electrons on adjacent sites. The term −V i,j n i,↑ n j,↓ is now added to Hamiltonian (1), and is When both SC and RAF orders are taken into account, the mean-field Hamiltonian is written in terms 161 of an eight-component spinor given by and assumes the expression [1,6] 163 where H is an 8 × 8 matrix: with H ′ and U Q , two 4 × 4 matrices, given by

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The energy spectra obtained by diagonalizing the matrix H are ±E 1 (k) and where E q (k) = ǫ 2 (k) + Q 2 U 2 .

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Minimizing the free energy function with respect to Q and D 0 , and calculating the density of electrons 181 n led to the following mean-field equations that describe the interplay between RAF and SC for HTSCs 182 with tetragonal symmetry: In the case of crystals with orthorhombic symmetry it is possible that D i,j x = −D i,j y because the 185 superconducting coupling constants V x along the x axis and V y along the y axis may differ. Then, 186 the superconducting gap takes on the form D(k) = ψ s (cos k x + cos k y ) + ψ d (cos k x − cos k y ) with   RAF has been interpreted as a single q = (π, π) wave [10]. We conjecture that when temperature is 234 lowered across T N , the static magnetization sets in due to the three-dimensional coupling between the 235 copper-oxygen layers. The establishment of three-dimensional long-range order naturally allows other 236 spin waves with q = (π, π) to settle in along with the q = (π, π) spin wave present in RAF, a mechanism 237 which causes the loss of RAF. In this conjecture, the PG is a consequence of purely two-dimensional 238 physics, but the Néel order is as is well known due to three-dimensional physics. In future investigations, 239 we plan to seek the mechanism for the phase change from Néel order to RAF, and vice versa. As mentioned in the previous section, the appearance of RAF below a critical value of doping as the 243 latter is reduced from overdoped to underdoped regime for p-type or n-type systems at zero temperature 244 has been interpreted as a QCP. The case of p-type has been discussed before [1,6,8]. This QCP induces a . Also a small hole-like band is seen along the 249 diagonal around (π/2, π/2). The presence of the gap at (π, 0) for this doping and the small hole-like 250 band in the vicinity of (π/2, π/2) are due to the nonzero value of RAF's order parameter Q. This gap 251 is responsible for the PG behavior in the underdoped regime. The hole-like band is also seen along the 252 RBZ boundary [(π, 0) → (0, π)] as shown in Fig. 6 for p = 0.1. For p = 0.24 in the overdoped regime, 253 the PG has closed and the hole-like pocket has reached the (π, 0) and (0, π) points as can be seen along 254 the RBZ boundary in Fig. 6. Along this boundary E + (k) = E − (k), when Q = 0 at T = 0.1t, is above 255 the chemical potential all the way between (π, 0) and (0, π).

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For the n-type case, Figure 5 displays the spectra for doping n e = 0.06 in the underdoped regime, and 257 for n e = 0.2 well in the overdoped regime for T = 0.1t. The PG behavior is now a consequence of a gap 258 at (π/2, π/2), and a small electron pocket forms near (π, 0). For n e = 0.2, the PG is zero because Q has (π,0) (π/2,π/2) (0,π) vanished, and the electron pocket at (π, 0) joined that at (0, π). This can be understood by examining the 260 spectrum along the RBZ boundary which gives a completely full band along this direction. For example,

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for n e = 0.21 in Fig. 6, E + (k) = E − (k) < 0, which means that these bands are full. The above analysis 262 can be made even more transparent by calculating the FS, a task undertaken below. (π/2, π/2). Around optimal doping p = 0.2, the hole pockets reach the points (±π, 0) and (0, ±π). In 281 the overdoped regime, where the PG is zero, the FS is made of large contours around (0, 0) and (π, π) 282 as can be seen in Fig. 4 for p = 0.24. For the latter, because the PG is zero the upper band E + and 283 lower band E − touch at (±π, 0), (0, ±π) and (±π/2, π/2) to form a tight-binding spectrum given by tight-binding energy is a consequence of the limit Q → 0 in ǫ 2 (k) + U 2 Q 2 for the overdoped regime. for YBa 2 Cu 3 O 6+x supported the existence of small closed pockets in the underdoped regime as well.

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The calculated FS undergoes also a significant reconstruction when doping changes from p-type to

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As long-range antiferromagnetic order is ruled out in a single layer at finite temperature due to thermal 345 spin fluctuations, one needs to find an explanation for this result outside of the linear spin-wave theory.