Quantum spin liquid states

This is an introductory review of the physics of quantum spin liquid states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may develop many exotic behaviors once we leave the regime of semiclassical approaches. The purpose of this article is to introduce these developments. The article begins by explaining how semiclassical approaches fail once quantum mechanics become important and then describe the alternative approaches for addressing the problem. Mainly spin-$1/2$ systems are discussed, and most of the time is spent in this article on one particular set of plausible spin liquid states in which spins are represented by fermions. These states are spin-singlet states and may be viewed as an extension of Fermi liquid states to Mott insulators, and they are usually classified in the category of so-called $SU(2)$, $U(1)$, or $Z_2$ spin liquid states. A review is given of the basic theory regarding these states and the extensions of these states to include the effect of spin-orbit coupling and to higher spin ($S>1/2$) systems. Two other important approaches with strong influences on the understanding of spin liquid states are also introduced: (i) matrix product states and projected entangled pair states and (ii) the Kitaev honeycomb model. Experimental progress concerning spin liquid states in realistic materials, including anisotropic triangular-lattice systems [$\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$ and ${\mathrm{EtMe}}_3\mathrm{Sb}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$], kagome-lattice system [${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$], and hyperkagome lattice system (${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$), is reviewed and compared against the corresponding theories.

Figure 1

Berry’s phase with a magnetic monopole.

Figure 2

Geometric frustration. The spin $C$ is frustrated because either the up or down orientation will give rise to the same energy in the AFM Ising limit.

Figure 3

A spin-singlet dimer configuration covering a lattice. An RVB state is a superposition of such configurations.

Figure 4

A spinon excitation on top of an RVB ground state.

Figure 5

(a) Schematic zero-temperature phase diagram for the Mott transition. $U$ is the strength of the Hubbard interaction, and $t$ is the hopping integral. The electron quasiparticle weight and the quasiparticle charge current $\sim 1+F_1^s/d$ vanish at the critical point, whereas the effective mass remains finite. (b) Schematic phase diagram showing finite-temperature crossovers and possible instability toward gapped phases at lower temperatures. There exists a (finite-temperature) critical region around $U_c$ where the phenomenology is not applicable. From [345].

Figure 6

A $Z_2$ vortex created by flipping the signs of the $u_{ij}$ on the bonds cut by the dashed line (indicated by thick lines).

Figure 7

Variational phase diagram for the Hamiltonian presented in Eq. (73). From [215].

Figure 8

Two classical planar Néel states ($q=0$ and $\sqrt{3}\times \sqrt{3}$) on a kagome lattice. $A$, $B$, and $C$ specify three coplanar spin orientations with intersection angles of 120°.

Figure 9

A valence-bond solid construction of the AKLT state.

Figure 10

Graphical representation of a PEPS in terms of contracted tensors (tensor network). Each box denotes a tensor $T$ with components $T_{lrud}^{s_{ij}}$ at site $ij$, where $l$, $u$, $r$, and $d$ are tensor indices related to left, right, up, and down bonds, respectively, linking to their neighbors; the open lines represent the physical spin states $s_{ij}$, and the connected lines represent the contraction of the tensors.

Figure 11

The VBS construction of an $S=2$ AKLT state on a square lattice and the corresponding tensors.

Figure 12

Kitaev honeycomb model. $x$, $y$, and $z$ denote three types of links in the honeycomb lattice.

Figure 13

Loop operator $W_p={\sigma }_1^x{\sigma }_2^y{\sigma }_3^z{\sigma }_4^x{\sigma }_5^y{\sigma }_6^z$ on a lattice plaquette (hexagon).

Figure 14

Graphic representation of the four-Majorana-fermion decomposition of the Hamiltonian expressed in Eq. (106).

Figure 15

Phase diagram of the Kitaev honeycomb model. The triangle is the section of the positive octant ($J_x,J_y,J_z\ge 0$) that lies in the plane $J_x+J_y+J_z=1$. The $A$ phase contains three gapped subphases. The $B$ phase is gapless.

Figure 16

Temperature-pressure phase diagram of the spin liquid compound with a quasitriangular lattice $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$, which undergoes a Mott transition at moderate pressure. From [158].

Figure 17

(a) In-plane structure of the ET layer in $\kappa \text{−}(\mathrm{ET}{)}_{2}X$. It is modeled to (b) an anisotropic triangular lattice. (c) First principles calculations of transfer integrals in $\kappa \text{−}(\mathrm{ET}{)}_{2}X$ for $X=\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Cl}$, $\mathrm{Cu}(\mathrm{NCS}{)}_2$, and ${\mathrm{Cu}}_2(\mathrm{CN}{)}_3$; squares ([220]), circles ([130]), and triangles ([154]).

Figure 18

Magnetic susceptibility of polycrystalline $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$ and $\kappa \text{−}(\mathrm{ET}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Cl}$. The core diamagnetic susceptibility has already been subtracted. The solid and dotted lines represent the results of the series expansions of the triangular-lattice Heisenberg models using [$6/6$] and [$7/7$] Padé approximations with $J=250\mathrm{K}$. The susceptibility of $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$ below 30 K is expanded in the inset. From [277].

Figure 19

${}^1\mathrm{H}$ NMR spectra for single crystals of $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$ and $\kappa \text{−}(\mathrm{ET}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Cl}$. From [277].

Figure 20

${}^{13}\mathrm{C}$ nuclear spin-lattice relaxation rate for a single crystal of $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$. The open triangles and circles represent the relaxation rates of two separated lines coming from two nonequivalent carbon sites in an ET. At low temperatures below 5 K, the two lines merge and are not distinguished. The inset shows the exponent in the stretched-exponential fitting to the relaxation curves of the whole spectra, whose relaxation rates are plotted using closed diamonds. From [278].

Figure 21

Low-temperature specific heat $C_p$ of $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$ for several magnetic fields up to 8 T in $C_p/T$ vs $T^2$ plots. Those of antiferromagnetic insulators $\kappa \text{−}(\mathrm{ET}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Cl}$, deuterated $\kappa \text{−}(\mathrm{ET}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Br}$, and $\beta \text{−}(\mathrm{ET}{)}_2{\mathrm{ICl}}_2$ are also plotted for comparison. From [328].

Figure 22

Low-temperature thermal conductivity $\kappa$ of $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$ (samples $A$ and $B$) in $\kappa /T$ vs $T^2$ plots. Sample $A$ was investigated at 10 T applied perpendicular to the basal plane, as well as at 0 T. From [326].

Figure 23

First principles calculations of (a) bandwidth $W$ and (b) transfer integrals in $A[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$ for various cations $A$. The $\mathrm{Pd}(\mathrm{dmit}{)}_2$ layers are modeled to triangular lattices characterized by transfer integrals $t_B$, $t_S$, and $t_r$. $t_3$ is the interlayer transfer integral. AF, QSL, and CO stand for antiferromagnet, quantum spin liquid, and charge-ordered insulator. From [297].

Figure 24

Magnetic susceptibility of an antiferromagnet ${\mathrm{Me}}_4\mathrm{Sb}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$, a spin liquid ${\mathrm{EtMe}}_3\mathrm{Sb}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$, and a charge-ordered insulator ${\mathrm{Et}}_2{\mathrm{Me}}_2\mathrm{Sb}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$. The core diamagnetic susceptibility has already been subtracted. From [137].

Figure 25

${}^{13}\mathrm{C}$ nuclear spin-lattice relaxation rate $1/T_1$ of ${\mathrm{EtMe}}_3\mathrm{Sb}{[\mathrm{Pd}(\mathrm{dmit}{)}_2]}_2$. The inset shows the $1/T_{1}T$ vs $T$ plots. The circles indicate the values determined from the stretched-exponential fitting to the relaxation curves and the squares denote the values determined from the initial decay slopes of the relaxation curves. From [118].

Figure 26

Low-temperature specific heat $C_p$ of ${\mathrm{EtMe}}_3\mathrm{Sb}{[\mathrm{Pd}(\mathrm{dmit}{)}_2]}_2$ for several magnetic fields up to 10 T in $C_p/T$ vs $T^2$ plots. The data of other insulating systems, i.e., ${\mathrm{Et}}_2{\mathrm{Me}}_2\mathrm{As}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$, ${\mathrm{EtMe}}_3\mathrm{As}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$, and ${\mathrm{EtMe}}_3\mathrm{P}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$, are also plotted for comparison. A large upturn below 1 K is probably attributable to the rotational tunneling of Me groups. The low-temperature data are expanded in the inset. From [329].

Figure 27

Low-temperature thermal conductivity $\kappa$ of ${\mathrm{EtMe}}_3\mathrm{Sb}{[\mathrm{Pd}(\mathrm{dmit}{)}_2]}_2$ (dmit-131) in $\kappa /T$ vs $T^2$ and $\kappa /T$ vs $T$ (inset) plots. The data of other insulators, i.e., ${\mathrm{Et}}_2{\mathrm{Me}}_2\mathrm{Sb}{[\mathrm{Pd}(\mathrm{dmit}{)}_2]}_2$ (dmit-221, nonmagnetic) and $\kappa \text{−}(\mathrm{ET}{)}_2{\mathrm{Cu}}_2(\mathrm{CN}{)}_3$, are also plotted for comparison. From [327].

Figure 28

Temperature variation of the spin-frozen fraction determined by muon spin rotation experiments for ${\mathrm{Zn}}_x{\mathrm{Cu}}_{4-x}(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$. The inset shows the $x$ dependence of the spin-frozen fraction at a low temperature. From [200].

Figure 29

Temperature dependence of the inverse magnetic susceptibility ${\chi }^{-1}$ of ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$. The line denotes a Curie-Weiss fit. Inset: ac susceptibility (at 654 Hz) at low temperatures. From [103].

Figure 30

${}^{17}\mathrm{O}$ NMR shift of two lines ($M$ and $D$) decomposed from the observed spectra for a powder of ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$. The $M$ and $D$ lines are considered to come from the oxygen sites depicted in the inset. The solid (red) curve represents the trace of one-half of the value of the $M$ line. The sketch in the lower left corner illustrates the environment of a Zn substituted on the Cu kagome plane, and thick lines represent Cu-Cu dimers. From [234].

Figure 31

${}^{17}\mathrm{O}$, ${}^{63}\mathrm{Cu}$, and ${}^{35}\mathrm{Cl}$ nuclear spin-lattice relaxation rates $1/T_1$ for a powder of ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$. The inset shows ${}^{17}\mathrm{O}$ $1/T_1$ vs $1/T$ plots. From [234].

Figure 32

(a) Temperature dependence of ${}^{17}\mathrm{O}$ Knight shift and (b) the field dependence of the spin gap deduced from the Knight shift for a single-crystal ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$. From [73].

Figure 33

(a) Specific heat $C$ of ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$ in various applied fields. The inset shows $C$ over a wider temperature range in applied fields of 0 T (squares) and 14 T (stars). (b) $C$ in a zero field at low temperatures. The lines represent power-law fits. From [103].

Figure 34

(a) Instantaneous magnetic correlations at 4 and 10 K for a time scale corresponding to approximately 6.5 meV. The solid lines are a guide to the eye. (b) The $Q$ dependence in the dynamic correlations with the energy integration interval indicated in the legend. The dotted lines in (a) and (b) are the structure factors for dimerlike AF correlations. The dashed line, a single-ion contribution corresponding to the 6% antisite spins in this system, is added. (c) The energy and temperature dependence at $Q=1.3{\AA }^{-1}$. D7, IN4, and MARI in the legends stand for the types of spectrometers used. From [56].

Figure 35

Contour plot of dynamical structure factor, $S_{\mathrm{m}\mathrm{a}\mathrm{g}}$ ($\mathbf{Q}$, $\omega$), integrated over $1\le \hslash \omega \le 9\mathrm{meV}$ for a single crystal ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$ at 1.6 K. The intense scattering is extended in a green-colored region, without peaking at any specific points. From [98].

Figure 36

(a) Temperature dependence of the inverse magnetic susceptibility ${\chi }^{-1}$ of polycrystalline ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ under 1 T. The inset shows magnetic susceptibility $\chi$ in various fields up to 5 T; for clarity, the curves are shifted by $3\times {10}^{-4}$, $2\times {10}^{-4}$, and $1\times {10}^{-4}\mathrm{emu}/\mathrm{mol}$ Ir for 0.01, 0.1, and 1 T data, respectively. (b) Magnetic specific heat $C_m$ divided by temperature $T$ of polycrystalline ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$. To estimate $C_m$, data for ${\mathrm{Na}}_4{\mathrm{Sn}}_3{\mathrm{O}}_8$ is used as a reference of the lattice contribution. The inset shows $C_m/T$ vs $T$ in various fields up to 12 T. (c) Magnetic entropy. From [233].

Figure 37

Thermal conductivity $\kappa$ of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ in $\kappa /T$ vs $T^2$ plots for magnetic fields of 0 and 5 T. The inset shows the low-temperature part of the data. From [286].

Figure 38

The line width (FWHM) of Gaussian-broadened ${}^{17}\mathrm{O}$ and ${}^{23}\mathrm{Na}$ NMR spectra and the mean value of the distributed local fields detected based on $\mu \mathrm{SR}$ ([52]). For the NMR line width, its deviation from the value at 15 K is plotted. Inset: ${}^{23}\mathrm{Na}$ spectra at 78.937 MHz for 7 T (empty circles) and 45.046 MHz for 4 T (solid line) with the horizontal axis shifted by 3.005 T at 1.3 K. The square blue line shows the expected powder pattern of the spectrum with every Ir site carrying the same moment. From [280].