Field-induced superfluids and Bose liquids in projected entangled pair states

In two-dimensional incompressible quantum spin liquids, a large enough magnetic field generically induces “doping” of polarized $S=1$ triplons or $S=1/2$ spinons. We review a number of cases such as spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) or spin-1/2 resonating valence bond (RVB) liquids where the projected entangled pair states (PEPS) framework provides very simple and comprehensive pictures. On the bipartite honeycomb lattice, simple PEPS can describe Bose condensed triplon (AKLT) or spinon (RVB) superfluids with transverse staggered (Néel) magnetic order. On the kagome lattice, doping the RVB state with deconfined spinons or triplons (i.e., spinon bound pairs) yields uncondensed Bose liquids preserving $U\left(1\right)$ spin-rotation symmetry. We find that spinon (triplon) doping destroys (preserves) the topological ${\mathbb{Z}}_2$ symmetry of the underlying RVB state. We also find that spinon doping induces longer range interactions in the entanglement Hamiltonian, suggesting the emergence of (additive) log corrections to the entanglement entropy.

Figure 1

(a) The honeycomb $S=3/2$ AKLT state under finite external magnetic field. Each spin is “split” into three spins 1/2 (red dots). A “valence bond” configuration is constructed by pairing neighboring spins 1/2 into singlets (red) or polarized triplets (green). (b) A $D=3$ generalization of the AKLT PEPS is obtained by considering, in addition to the maximally entangled states $|00\rangle +|11\rangle$ on the bonds, new $|22\rangle$ bonds which represent triplets. Subsequently, the spins at each site are symmetrized, and a configuration with $p$ triplets is picked with weight ${\alpha }_p$. (c) The tensors on the A and B sublattices are grouped together to form an effective square lattice. On the bonds, $X$ matrices transform the $|00\rangle +|11\rangle$ states into $|01\rangle -|10\rangle +\beta |11\rangle$. (d) A cylinder geometry with boundaries “vectors” $B_L$ and $B_R$ is used and the cylinder length $N_h$ is taken to $\infty$. A vertical bipartition (dashed line) is used to compute the boundary Hamiltonian and the entanglement spectrum.

Figure 2

Energy per site (different symbols are used when either $\lambda$ or $\beta$ is fixed to 0) of the triplon-doped $S=3/2$ AKLT state versus reduced magnetization computed on an infinite cylinder with perimeter $N_v=6$. The PEPS gives the exact asymptotic behavior close to saturation (i.e., slope $h_{c,2}=3$ at $m=m_{\mathrm{sat}}$) and the low-field slope ($h_{c,1}\simeq 0.113$) is in good agreement with Ref. 14. Inset: Comparison between the energies of AKLT PEPS with soft-core and hard-core triplons. PEPS results are also compared with ED data (Ref. 20).

Figure 3

Honeycomb lattice: Square of the transverse magnetization ${(\langle S_i^x\rangle /S)}^2$ versus reduced magnetization. All computations involve PEPS defined on an effective square lattice, and are performed on an infinite cylinder with perimeter $N_v=6$. (a) Magnon-doped $S=3/2$ AKLT $D=2$ and $D=3$ states and $D=3$ spinon-doped $S=1/2$ NN RVB state. (b) Comparison between $S=3/2$ AKLT PEPS with doped soft-core [as in (a)] and hard-core triplons. (c) Fugacity of the kagome RVB $D=3$ PEPS vs $m/m_{\mathrm{sat}}$.

Figure 4

Spinon-doped NN $S=1/2$ RVB wave functions on the honeycomb (a) and kagome (b) lattices. An equal-weight superposition of all spinon (green dots)/VB (red ellipses) configurations is assumed. In the kagome RVB PEPS (c), three sites are grouped together (see Ref. 5 for details). Spinon doping in the honeycomb (d) and kagome (e) RVB PEPS is introduced by adding an extra nonzero tensor element. Its magnitude $\lambda$ plays the role of a fugacity for the spinons.

Figure 5

Weights $|c_{\nu }|$ and $|d_{\nu \mu }{\left(r\right)|}^2$ of the one-body (i.e., $r=0$) and two-body operators in the expansion of $H_b$ of the kagome spinon-doped RVB PEPS as a function of distance $r$, for increasing $\lambda$ values (corresponding to reduced magnetization $x\sim {10}^{-3},0.007,0.06,0.53$, respectively).

Figure 6

$\left[E\right(x)-E(0\left)\right]/x$ versus $x=m/m_{\mathrm{sat}}$ for small $x$ compared to ED data (Ref. 20) for several honeycomb doped AKLT (a) and kagome RVB (b) states on $N_v=6$ cylinders. Shallow minima (if any) are shown by vertical arrows. The estimated extrapolated spin gaps of Refs. 14 and 24 (multiplied by the spin $S$)—to be compared with the $m/m_{\mathrm{sat}}\to 0$ limits of the various curves—are shown by horizontal (red) arrows.

Figure 7

Entanglement spectrum of a bi-partitioned $N_v=6$ kagome RVB cylinder as a function of the momentum along the cut, for different values of the spinon fugacity $\lambda$ corresponding to $m/m_{\mathrm{sat}}\sim {10}^{-3},0.007,0.02$, respectively. Different symbols are used for different $S_z$ sectors of the edge. The same $S_z=0$, $K=0$ state is used as energy reference ${\xi }_0$ (bold $+$ symbol).