Quantum Orders and Symmetric Spin Liquids

TL;DR

The paper introduces quantum order as a zero-temperature generalization of order beyond Landau's symmetry breaking and formulates a PSG-based framework to classify symmetric spin liquids. By applying projective representations of symmetries, it systematically catalogues a wide landscape of Z2, U(1), and SU(2) spin liquids, including numerous gapless variants whose stability is protected by quantum order rather than symmetry. It demonstrates continuous, symmetry-preserving transitions between spin liquids and links these classifications to measurable neutron-scattering signatures through the two-spinon spectrum. The work further connects these quantum-ordered spin liquids to high-Tc superconductors, arguing that quantum order underpins both the stability of exotic spin-liquid phases and the phenomenology of underdoped cuprates. Overall, PSG and quantum order provide a unifying lens to understand and predict diverse spin-liquid states and their experimental fingerprints.

Abstract

A concept -- quantum order -- is introduced to describe a new kind of orders that generally appear in quantum states at zero temperature. Quantum orders that characterize universality classes of quantum states (described by {\em complex} ground state wave-functions) is much richer then classical orders that characterize universality classes of finite temperature classical states (described by {\em positive} probability distribution functions). The Landau's theory for orders and phase transitions does not apply to quantum orders since they cannot be described by broken symmetries and the associated order parameters. We find projective representations of symmetry groups (which will be called projective symmetry groups) can be used to characterize quantum orders. With the help of quantum orders and the projective symmetry groups, we construct hundreds of symmetric spin liquids, which have SU(2), U(1) or $Z_2$ gauge structures at low energies. Remarkably, some of the stable quantum phases support gapless excitations even without any spontaneous symmetry breaking. We propose that it is the quantum orders (instead of symmetries) that protect the gapless excitations and make algebraic spin liquids and Fermi spin liquids stable. Since high $T_c$ superconductors are likely to be described by a gapless spin liquid, the quantum orders and their projective symmetry group descriptions lay the foundation for spin liquid approach to high $T_c$ superconductors.

Figures (15)

• Figure 1: Contour plot of the spinon dispersion $E_+(\v k)$ as a function of $(k_x/2\pi, k_y/2\pi)$ for the $Z_2$-linear spin liquids. (a) is for the Z2A$0013$ state in Z2lC, (b) for the Z2A$zz13$ state in Z2lA, (c) for the Z2A$001n$ state in Z2lEa, (d) for the Z2A$zz1n$ state in Z2lG.
• Figure 2: Contour plot of the spinon dispersion min$(E_1(\v k), E_2(\v k))$ as a function of $(k_x/2\pi, k_y/2\pi)$ for the $Z_2$-linear states. (a) is for the Z2B$0013$ state in Z2lF, (b) for the Z2B$zz13$ state in Z2BlA, (c) for the Z2B$001n$ state in Z2BlE, (d) for the Z2B$zz1n$ state in Z2BlG.
• Figure 3: Contour plot of the spinon dispersion $E_+(\v k)$ as a function of $(k_x/2\pi, k_y/2\pi)$ for (a) the $U(1)$-linear state U1C$n00x$ in U1lB, and (b) the $U(1)$-quadratic state U1C$x10x$ in U1qA. In the $U(1)$-quadratic state, the spinon energy vanishes as $\Del \v k^2$ near two points $\v k =(\pi, 0), (0,\pi)$.
• Figure 4: Contour plot of the spinon dispersion $E_+(\v k)$ as a function of $(k_x/2\pi, k_y/2\pi)$ for the $U(1)$-gapless states. (a) is for the U1C$n01x$ state U1glA, (b) for the U1A$0001$ state U1glB, (c) for the U1A$0011$ state U1glC, and (d) for the U1A$x10x$ state U1glD.
• Figure 5: Contour plot of the spinon dispersion $E_+(\v k)$ as a function of $(k_x/2\pi, k_y/2\pi)$ for the $Z_2$ spin liquids. (a) is for $Z_2$-gapless state Z2A$x2(12)n$ in Z2glA, and (b) is for $Z_2$-quadratic state Z2B$x2(12)n$ in Z2qA. Despite the lack of rotation and parity symmetries in the single spinon dispersion in (a), the two-spinon spectrum does have those symmetries.