Berry phase in cuprate superconductors

The geometrical Berry phase is widely recognized as having profound implications for the properties of electronic systems. Over the last decade or so, the Berry phase has been essential to our understanding of new materials such as graphene and topological insulators. In general, a nontrivial Berry phase is a result of band crossing as in the case of a massless Dirac point. The Berry phase can be accessed in quantum oscillation measurements as it contributes to the phase mismatch of electrons in their cyclotron orbits. With their enigmatic pseudogap and superconducting phases, the cuprates are materials where the Berry phase is thus far unknown. Based on quantum oscillation data in the high-field normal state of underdoped cuprates, we determined the Berry phase contribution to the phase mismatch unambiguously in this family of materials. In the hole-doped materials ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y, {\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$, and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$, a trivial Berry phase of 0 $\mathrm{mod}(2\pi$) is systematically observed, while the electron-doped ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$ exhibits a significant nonzero Berry phase of $1.4\pi$. Our results set significant constraints on the microscopic description of the high-field normal state and, in particular, do not support a nodal structure or broken time-reversal symmetry in the hole-doped compounds.

Figure 1

(a) Schematic zero-field temperature-doping phase diagram of cuprate superconductors (adapted from Ref. [13]). At $x=p=0$, the parent compound is an antiferromagnetic (AF) Mott insulator. Hole or electron doping suppresses the Néel temperature $T_{\mathrm{N}}$ and leads to superconductivity (SC) in a dome-shaped region bounded by $T_c$. $T^{☆}$ denotes the pseudogap temperature. Broken time-reversal symmetry is observed below $T_{\mathrm{mag}}$ by neutron-scattering [14] and Kerr-effect measurements [15]. Below $T_{\mathrm{CDW}}$ (green line), charge density wave (CDW) modulations are observed (see main text). The thick blue lines on the doping axes denote the regions where the QO data shown in Fig. 2 were measured. (b) Layered crystal structure of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$. (c) Brillouin zone (square) of the ${\mathrm{CuO}}_2$ plane. The Fermi surfaces from the band structure (gray) and reconstructed by ($\pi ,\pi$) antiferromagnetic order, thought to be relevant for electron-doped cuprates, are shown, with the hole and electron pockets in blue and red, respectively.

Figure 2

Left column: Quantum oscillatory part $\mathrm{\Delta }\rho$ of the electrical resistivity vs $1/B$ in (a) YBCO ($p=0.11$; this work), (b) YBCO ($p=0.108$; Ref. [29]), (c) YBCO ($p\approx 0.11$; Ref. [18]), (d) Y248 ($p=0.14$; Ref. [30]), (e) Hg1201 ($p\approx 0.09$; Ref. [21]), and (f) NCCO ($x=0.15$; Ref. [31]). In (a)–(c) and (f), the lines are data. (d),(e) The data are circles and the lines are fits to the data (see Refs. [21, 30]). (c) The resonance frequency shift $\mathrm{\Delta }f$ of a tunnel diode oscillator, proportional to the in-plane conductivity. The field is along the $c$ axis and the current as indicated. Numbers are the Landau-level index $n$. Right column: $n$ vs $1/B$. The lines are linear fits to the data points, whose equations are indicated. For NCCO, we also show $n$ vs $1/B$ for $x=0.165$ (gray; see Fig. 6; Ref. [32]). Insets: Zooms of the intercepts and their error bars from the linear fit (see Appendix C and Fig. 5).

Figure 3

Illustration of Berry phase accumulation in momentum space. (a) Parabolic dispersion of two uncoupled, gapped bands with a trivial phase accumulation of 0 $\mathrm{mod}(2\pi$). Two coupled, linear bands with (b) gapless and (c) gapped dispersion and a phase accumulation of $\simeq \pi$. Note that it is the topological Berry phase of coupled bands that can produce a phase mismatch of $\simeq \pi$ [2, 34].

Figure 4

Oscillatory part of the electrical resistivity $\mathrm{\Delta }{\rho }_{xx}$ (red) and conductivity $\mathrm{\Delta }{\sigma }_{xx}$ (blue) for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.54}$ at $T=4.2$ K as a function of inverse magnetic field $1/B$. The vertical gray lines indicate the positions of the maxima and minima in both quantities, showing that they match well.

Figure 5

Left column: Landau-level index $n$ vs $1/B$, as shown in Fig. 2 of the main text. Right column: Residual difference between the fan diagram data points and the linear fits. The values of $\beta$ and $B_f$ and their errors from the linear regression are indicated in the top of each panel.

Figure 6

Left column: Quantum oscillatory part of the electrical resistivity as a function of inverse magnetic field for NCCO at $x=0.15$ (from Ref. [31]) and 0.165 (from Ref. [32]). The magnetic field and electrical current are along the $c$ axis. The numbers are the Landau-level index. Right column: Landau-level index $n$ vs $1/B$. The lines are free linear fits to the data, whose equations are indicated.