Abelian ${\mathrm{SU}\left(N\right)}_1$ chiral spin liquids on the square lattice

In the physics of the fractional quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin like systems, we propose a family of $\mathrm{SU}\left(N\right)$-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary $N$ of an SU(3) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of ${\mathrm{SU}\left(2\right)}_1$ and ${\mathrm{SU}\left(3\right)}_1$ chiral spin liquids (CSL) on similar models we search for topological $\mathrm{SU}({N)}_1$ CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to $N=10$ and an innovative $\mathrm{SU}\left(N\right)$-symmetric version of iDMRG to compute entanglement spectra on (infinitely long) cylinders in all topological sectors provide unambiguous signatures of the $\mathrm{SU}({N)}_1$ character of the chiral liquids. An SU(4)-symmetric chiral PEPS, constructed in a manner similar to its SU(2) and SU(3) analogs, is shown to give a good variational ansatz of the $N=4$ ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying $N$ provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.

Figure 1

We considered various system topologies: (a) periodic cluster topologically equivalent to a torus; (b) open cluster topologically equivalent to a disk; (c) cylinder with left and right boundaries. We used (a) and (b) in ED and the infinite-length version of (c) in DMRG and iPEPS. The chiral modes of the CSL are schematically shown on the system edges.

Figure 2

Low-energy spectra computed by ED in the SYT singlet basis on periodic clusters of $N_s=kN$ sites, $k\in \mathbb{N}$, and [(a)–(i)] for a fixed value of $\theta =\pi /4$ as a function of $\varphi$, for $N$ ranging from 2 to 10, or [(k)–(n)] for a fixed value of $\varphi =\pi /2$ as a function of $\theta \in [0,\pi /2]$, for $N=4,7,8,9$. Only 10 (40) lowest singlet levels are shown at small $N$ in (a)–(e) and (k) [larger $N$ in (f)–(i) and (l)–(n)). The $\varphi$ and $\theta$ axes being discretized, lines connecting the data points are used as guides to the eye (hence, levels crossings around ${\varphi }_{\mathrm{lc}}$ may look like anticrossings). $N$ degenerate or quasidegenerate singlets (see Figs. 3 and 4 and text) are separated from the higher energy states by a gap, in an extended $(\varphi ,\theta )$ region around $(\pi /2,\pi /4)$. The energy of the (fully polarized) ferromagnetic state ($E_{\mathrm{ferro}}=2\sqrt{2}{N}_s(2cos\varphi +sin\varphi )$), crossing the singlet GS at $\varphi ={\varphi }_F$, is shown as a dashed line in (a)–(i). The location of the CSL and ferromagnetic phases along the cuts (c)–(i) and (k)–(n) are schematized in (j). Note that for $N=2\text{and}3$ [(a) and (b)] the CSL is expected to extend all the way to $\varphi =0$.

Figure 3

Zoom of the singlet low-energy spectra at $\theta =\pi /4$ and $\varphi =\pi /2$, for $N$ ranging from 2 to 10, and the same cluster sizes as in Fig. 2. The GS energy is subtracted off for better comparison between the various spectra. The exact degeneracy $g$ of each level is indicated on the plot as $\times g$. The first nonsinglet excitation belonging to the adjoint IRREP above the $N$ quasidegenerate low-energy singlets is shown as a filled triangle (see text).

Figure 4

Low-energy spectra on periodic clusters at fixed $\varphi =\pi /2$ and for $\theta =\pi /4$ [(a)–(d)] or $\theta =\pi /6$ (e)–(h)]. Clusters with site numbers $N_s=kN$ (left) or $N_s=kN-1$ (right), $k\in \mathbb{N}$, are chosen to obtain 0 and 1 quasihole, respectively, in the putative CSL. The respective BZ with the allowed discrete momenta is shown on each plot as a gray square—only nonequivalent momenta are labeled—and the number of equivalent momenta appearing are listed as grayed squared numbers. For $N_s=kN$ (left), the GS manifold is composed of $N$ singlets (open circles). For $N_s=kN-1$ (right), it is composed of $N_s$ quasidegenerate levels, one level at each cluster momentum. Each level is comprised of $N$ degenerate states forming a $\overline{N}$ antifundamental IRREP (open triangles).

Figure 5

Low-energy spectra on open $C_4$-symmetric clusters depicted on the right-hand side of the figure, as a function of the angular momentum $l$ (with respect to the GS angular momentum $l_0$), at fixed $\varphi =\pi /2$ and for $\theta =\pi /4$ (a)–(d)] or $\theta =\pi /6$ [(e)–(h)]. Symbols labeling the various SU($N$) IRREPs entering the chiral mode are shown in the legends. The Young diagrams for the corresponding IRREPs can be identified using the tables in Appendix pp3. The GS IRREPs are fully antisymmetric, and labeled by Young diagrams consisting of a single column of $r_0=\mathrm{mod}(N_s,N)$ boxes, with degeneracy $\frac{N!}{(N-r_0)!r_0!}$. Identifying $l-l_0$ with the Virasoro level $L_0$, all low-energy ToS in (a)–(h) for $0\le l-l_0\le 3$ follow exactly the WZW CFT predictions of Tables 8, 9, 12, 13, 16, 15, 17, and 21, respectively. The only exception is the SU(6) $\overline{\mathbf{15}}$ (SU(8) $\mathbf{1}$) tower, for which two multiplets $\overline{\mathbf{15}}$ and $\overline{\mathbf{21}}$ ($\mathbf{1}$ and $\overline{\mathbf{63}}$) are missing in the $L_0=3$ Virasoro level.

Figure 6

Band structures of the parton Hamiltonian on the torus along high symmetry directions for $N=2,3$ and 4. We set $t_x=t_y$ for $N=2$ and 4, and $t_x=t_y/2$ for $N=3$.

Figure 7

The parton single-particle levels including the edge states on a wide cylinder for $N=2$ to 9. Filling the Fermi sea up to zero energy corresponds to a filling fraction $1/N$. This fully occupies the lowest parton band as well as the edge states up to the degenerate zero modes at the single-particle momentum $k_y=\pi /N$. These exact zero modes, denoted by $d_{L\sigma }$ and $d_{R\sigma }$, are localized at the left and right boundaries of the cylinder, respectively.

Figure 8

The entanglement spectra on width-6 cylinders for SU(2) CSLs. (a) Identity sector. (b) Semion sector ($\otimes \frac{1}{2}$). Identifying $K_y$ with the Virasoro level $L_0$, the content of the chiral branches agrees exactly with the CFT predictions of Tables 6 and 2 up to $K_y=4$ (mod[6]).

Figure 9

The entanglement spectra on width-6 cylinders for SU(3) CSLs. (a) Identity sector. (b)  (c)  Identifying $K_y$ with the Virasoro level $L_0$, the content of the chiral branches agrees exactly with the CFT predictions of tables 6 and 26 up to $K_y=3$ (mod[6]). Note that the towers of the $\mathbf{3}$ and $\overline{\mathbf{3}}$ sectors are identical, apart from an overall conjugation of all IRREPs.

Figure 10

The entanglement spectra on width-8 cylinders for SU(4) CSLs. (a) Identity sector. (b)  (c)  (d)  Note that the towers of the $\mathbf{4}$ and $\overline{\mathbf{4}}$ sectors are identical, apart from an overall conjugation of all IRREPs. Identifying $K_y$ with the Virasoro level $L_0$, the content of the chiral branches agrees exactly with the CFT predictions of tables 8, 27 and 28 up to $K_y=3$.

Figure 11

PEPS on the square lattice involving site $\mathcal{A}$ tensors and bond $\mathcal{B}$ tensors. The bond dimension on the black links is $D$, up to $D^{*}=4$ ($D=15$), and the vertical red segments correspond to the physical space $\mathcal{F}$ spanned by the $d=N$ ($d^{*}=1$) physical degrees of freedom. All indices (i.e., legs or lines) carry arrows which indicate whether legs enter or leave a tensor in terms of state space fusion. This can be translated into co- and contravariant tensor index notation, respectively [76, 77]. Note that reverting an arrow also flips all affected IRREPS into their dual representations.

Figure 12

Entanglement spectra on an infinitely long width-4 cylinder obtained from an SU(4) ($D=15$) PEPS wave function optimized for $\theta =\pi /4$, $\varphi =\pi /2$ and environment dimension $\chi =1350$. Spectra are plotted vs perimeter momentum $k_y$ and, to better evidence their chiral nature, the $k_y=-\pi /2$ spectrum is replicated at $k_y=3\pi /2$. Appropriate ${\mathbb{Z}}_4$ charge boundaries $Q=2$, 0 and $\pm 1$ are set up to select the $\mathbf{6}$ (a), $\mathbf{1}$ (b), and $\mathbf{4}/\overline{\mathbf{4}}$ [(c) and (d)] topological sectors, showing one, two and four branches, respectively. Note that the $\mathbf{4}$ and $\overline{\mathbf{4}}$ spectra are identical apart from an overall charge conjugation of all IRREPs (and small finite-$\chi$ numerical errors).

Figure 13

Leading correlation lengths obtained from the transfer matrix (in the absence of gauge flux) for the case of SU(4) plotted versus the number of multiplets ${\chi }^{*}$ kept in the environmental tensors, normalized by $D^{2*}=26$ which represents the number of multiplets in the product space of $D\times D$ states with $D$ fixed. The SU(4) IRREPs associated to these correlation lengths are indicated.

Figure 14

Eigenspectrum of a $2\times 2$ cluster described by the plaquette Hamiltonian $H_p$ in Eq. (A1) vs $\varphi$ using the parametrization in Eq. (2), with $\theta =\pi /4$ fixed as in the main text [e.g., see Fig. 2]. To focus on energy per site, the energies are divided by the number of sites $N_s=4$ as indicated. (a)–(d) refer to case of $N=2,3,4,5$ symmetric flavors, respectively. Colors indicate symmetry sectors as indicated with the legend based on Dynkin labels. The small numbers on top of each line in panel (a) indicate the degeneracy of multiplets which shows that the green line only is twofold degenerate. This also holds for all data in the other panels.

Figure 15

The (co)root lattice ${\mathrm{\Lambda }}_{\text{r}}\left(\mathfrak{g}\right)$ (black circles) and and the weight lattice ${\mathrm{\Lambda }}_{\text{w}}\left(\mathfrak{g}\right)$ (red circles) of $\mathfrak{g}=\mathfrak{su}\left(3\right)$. The root (weight) lattice is an integer span of two simple (co)roots ${\mathit{\alpha }}^{\left(1\right)}$ and ${\mathit{\alpha }}^{\left(2\right)}$ (two fundamental weights ${\mathit{\omega }}_{\left(1\right)}$ and ${\mathit{\omega }}_{\left(2\right)}$). In $\mathfrak{su}\left(3\right)$, ${\mathit{\omega }}_{\left(1\right)}$ and ${\mathit{\omega }}_{\left(2\right)}$ respectively correspond to the highest weights of $\mathbf{3}$ and $\overline{\mathbf{3}}$.

Figure 16

Weights of $\mathbf{6}$ and $\mathbf{10}$-dimensional representations of $\mathfrak{su}\left(3\right)$. The representations $\mathbf{6}$ and $\mathbf{10}$ have highest weights (shown as “hws”) with Dynkin labels $(d^1,d^2)=(2,0)$ and (3,0), respectively. Red (blue) arrows show the action of the roots (“lowering operators”) $-{\mathit{\alpha }}^{\left(1\right)}$ ($-{\mathit{\alpha }}^{\left(2\right)}$) to the weights (see Fig. 15 for the definitions of ${\mathit{\alpha }}^{(1,2)}$).

Figure 17

Energy difference between the ground state of the antisymmetric IRREP ${\text{aIR}}_N\left(r_0\right)$ and the completely symmetric (ferromagnetic) state for $\theta =\pi /4$, $N=4$ (red) and $N=8$ (blue), $\varphi =\pi /2$ (filled symbols) and $\varphi =\pi$ (open symbols). In all cases, the energy difference scales approximately linearly with $N_s$, revealing a macroscopic energy difference.

Figure 18

Low-energy spectra of the SU(4) model computed on 8-site (a) and 16-site (b) periodic clusters at $\theta =\pi /4$, plotted vs $\varphi$. A few of the lowest energies of the SU(4) singlet subspace are shown (in blue) on both panels as well as the lowest energy (in red) within the subspace of the higher Casimir [4400] (or 105) (a) and [8800] (or 825') (b) IRREPs, which can also be viewed as SU(2) singlets. Other lowest-energy excitations are also shown for completeness.

Figure 19

Graphical representation of MPO matrix elements with U(1) spin symmetry for spin-$1/2$ fermions. Numbers in brackets indicate the possible values of $S_z$ quantum numbers, $0,-1/2$ and $1/2$ representing the $|\rangle$, $|\downarrow \rangle$ and $|\uparrow \rangle$ at each physical site, respectively. Double occupancy, $|\uparrow \downarrow \rangle$, is excluded.

Figure 20

Graphical representation of fusing edge virtual indices of two MPOs, $d_{k\uparrow }^{†}$ and $d_{k\downarrow }^{†}$.

Figure 21

(a) Illustration of the parton Hamiltonian of the SU(2) CSL. The phase of nearest neighbor hopping is 0 $\left(\pi \right)$ along the solid (dashed) edges. The phase of next-nearest-neighbor hopping is $\pi /2$ $(-\pi /2)$ along the green (red) arrows. [(b) and (c)] The entanglement spectra on a $4\times 12$ cylinder for the parton wave function.