Simultaneous Dimerization and SU(4) Symmetry Breaking of 4-Color Fermions on the Square Lattice

Using infinite projected entangled-pair states, exact diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg model undergoes a spontaneous dimerization on the square lattice, in contrast with its SU(2) and SU(3) counterparts, which develop Néel and three-sublattice stripelike long-range order. Since the ground state of a dimer is not a singlet for SU(4) but a 6-dimensional irreducible representation, this leaves the door open for further symmetry breaking. We provide evidence that, unlike in SU(4) ladders, where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is additionally broken, leading to a gapless spectrum in spite of the broken translational symmetry.

Figure 1

Sketch of dimer and color order realized in the ground state of the SU(4) Heisenberg model on the square lattice.

Figure 2

Sketch of some color and bond-energy configurations found by LFWT (a)–(c) and iPEPS (d)–(e). In all plots, the thickness of the lines is proportional to the square of $⟨P_{ij}⟩$ on the corresponding bond. The color content reflects that of the Hartree starting point for LFWT, and the actual result for an appropriate, infinitesimal symmetry breaking field for iPEPS. (a) “Plaquette” state selected by quantum fluctuations within LFWT. Its energy per site $E_0/N=-3J/2$. (b) “Dimerized” state with LFWT energy per site $E_0/N=-1.293J$. (c) Stripe-ordered state, with much higher LFWT energy per site $E_0/N=(4/\pi -2)J\simeq -0.727J$. (d) iPEPS results with $D=2$ and a unit cell $4\times 4$. At this order, the bond-energy pattern of the ground state is similar to that of the plaquette state selected by LFWT. (e) iPEPS results for $D=12$ and a unit cell $4\times 2$. The bond-energy pattern of this state is similar to that of the dimerized state of LFWT. Two types of dimers $A$ and $B$ with different local color densities can be identified.

Figure 3

iPEPS results as a function of inverse bond dimension $1/D$. Dotted lines are only a guide to the eye. (a) Energy per site in units of $J$ for different unit cell sizes in the iPEPS, compared to the VMC energy from Ref. 13, and exact diagonalization results. (b) Color order parameter (COP), which is zero if on each site the color density $n_k$ of each color $k$ is the same, i.e., $n_k=0.25$. Since the data do not seem to extrapolate to zero for $D\to \infty$, we expect the SU(4) symmetry to be spontaneously broken. (c) Bond energies of the bonds $H$, $V$, $A$, and $B$ marked in Fig. 2 obtained from the $4\times 2$ unit cell for $D>4$ and from the $4\times 4$ unit cell for $D\le 4$. The dimerization (strong $A/B$ bonds) is seen to persist for $D\to \infty$.

Figure 4

(a) Color structure factor for the $N=16$ square sample in momentum space. The maxima of the structure factor (filled circles) are located at momentum ($\pi$, $\pi /2$) and equivalent points. (b) Connected bond-energy correlations $⟨P_{ij}{P}_{kl}⟩-⟨P_{ij}{⟩}^2$ for $N=16$. (c) $⟨{\mathcal{P}}_{(a,b)}(i,j){\mathcal{P}}_{(a,b)}(k,l)⟩$. In (b) and (c) the line width is proportional to the absolute value of the correlations and blue, solid (red, dashed) lines denote positive (negative) values. (d) Low energy spectrum of the $N=16$ sample plotted as a function of the quadratic SU(4) Casimir operator $C$. Expected levels in the tower of states of the SU(4) symmetry breaking state sketched in Fig. 1 are highlighted by dashed circles. In the grey shaded region there are additional levels (not shown) which have not been resolved as a function of the Casimir.