Quantum Spin Liquids

TL;DR

This review surveys quantum spin liquids as highly entangled, nontrivially correlated phases that lack conventional order yet host emergent gauge structures and non-local excitations. It develops a framework built on gauge theories and parton constructions to describe gapped (e.g., $Z_2$) and gapless (e.g., U(1)) QSLs, including their symmetry fractionalization via projective symmetry groups. It connects exact solvable models like Kitaev's honeycomb and the toric code to broader phenomena such as fractional quantum Hall states, and surveys computational methods, experimental probes, and material candidates. The work emphasizes how entanglement, topological order, and gauge structure underpin the physics of QSLs and highlights experimental strategies and challenges in identifying QSLs in real materials. The insights aim to guide future theoretical, numerical, and experimental efforts toward definitive realizations and characterizations of QSLs and related topological phases.

Abstract

Quantum spin liquids may be considered "quantum disordered" ground states of spin systems, in which zero point fluctuations are so strong that they prevent conventional magnetic long range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons that are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments to study quantum spin liquids, and to the diverse probes used therein.

Figures (8)

• Figure 1: Operators in the toric code on the square lattice. Blue and red links denote $\sigma_i^x$ and $\sigma_i^z$ operators, respectively. A star $S_s$ and plaquette $P_p$ are shown, as is a loop operator on the contour $\mathcal{L}$ enclosing the area $\mathcal{A}$.
• Figure 2: Anyons in the toric code. A pair of e anyons is created at the ends of a string of $\sigma_i^x$ operators as shown. A pair of m anyons is likewise created at the ends of a string of $\sigma_i^z$ operators, as indicated. The $\varepsilon$ anyon is simply an e and m in close proximity.
• Figure 3: Statistics and mutual statistics in topological phases. In (i), two identical quasiparticles of type a are exchanged. This exchange may be accompanied by an arbitrary phase in two dimensions, which defines the self-statistics of an anyon. In (ii) a quasiparticle of type b is moved around a quasiparticle of type a, and the wavefunction accumulates a phase which defines the mutual statistics of the two anyons in two dimensions. In three dimensions, as shown in (iii), the mutual statistics of particles is replaced by the phase accumulated on moving a pointlike quasiparticle b around a closed loop which links with a line-like defect b. The pointlike particles, however, must be bosons or fermions.
• Figure 4: Illustration of how entanglement supports anyons in the toric code. A pair of e anyons at the ends of a line $\mathcal{L}$ is created by the action of a string of $\sigma_i^x$ along $\mathcal{L}$. It is crucial that this string does not modify the ground state away from the ends, otherwise the state created would have an energy proportional to its length. Here we show the action of this string, shown in red, away from its ends, on the toric code state, which is a superposition of loops. Each component in the second line corresponds to the result of the action of the string operator on the component directly above it, in the first line. We can see that, while the action of the string modifies each component of the wavefunction all along the line, the result is simply another component of the original state, as shown by the arrows. Consequently, the highly entangled superposition state is not modified by the string (except at its ends, which are not shown here).
• Figure 5: Area law and tensor network states. In (a), we schematically show the division of space into a region A and its complement B. Nearly all ground states of local Hamiltonians obey the "area law", where the leading term in the entanglement entropy for a large region A is proportional to the $d-1$-dimensional area of the boundary in $d$ dimensions, i.e. the length $L$ of the boundary in two dimensions, shown. One can visualize the area law term as arising from the sum of local contributions due to entangled pairs (e.g. singlets) spanning the boundary, as shown in (a). In a gapped system the next correction in two dimensions is a constant, which is known as the topological entanglement entropy, see Eq. (\ref{['eq:10']}). In (b) we show the representation of a quantum state by a Tensor Network State (TNS), specifically a Projected Entangled Pair State (PEPS), as in Eq. (\ref{['eq:75']}). One elementary tensor is shown in red, and contains here four internal indices and one physical one. By construction, the entanglement between a region A (in green) and its complement is communicated only through the tensor legs crossing the boundary, so that the TNS construction automatically satisfies the area law.