Topological crystals and soliton lattices in a Gross-Neveu model with Hilbert-space fragmentation

TL;DR

The paper tackles the finite-density behavior of the one-flavour Gross-Neveu-Wilson model using nonperturbative tensor-network methods. It combines large-N mean-field analysis with matrix product state simulations in both grand-canonical and canonical ensembles to reveal a wealth of inhomogeneous phases. On the symmetry line $ma=-1$, Hilbert-space fragmentation gives rise to topological crystals of immobile defects that bind dopants, and at larger interactions a parity-broken solitonic phase with anti-kinks; away from the line, chiral spirals with wavevector $k=2\pi\rho$ emerge nonperturbatively. Off symmetry, the results demonstrate a continuous connection from fragmented crystals to smooth chiral textures, highlighting exotic orders in lattice field theories and motivating quantum simulations of QCD-like phenomena in cold-atom systems.

Abstract

We explore the finite-density phase diagram of the single-flavour Gross-Neveu-Wilson (GNW) model using matrix product state (MPS) simulations. At zero temperature and along the symmetry line of the phase diagram, we find a sequence of inhomogeneous ground states that arise through a real-space version of the mechanism of Hilbert-space fragmentation. For weak interactions, doping the symmetry-protected topological (SPT) phase of the GNW model leads to localized charges or holes at periodic arrangements of immobile topological defects separating the fragmented subchains: a topological crystal. Increasing the interactions, we observe a transition into a parity-broken phase with a pseudoscalar condensate displaying a modulated periodic pattern. This soliton lattice is a sequence of topological charges corresponding to anti-kinks, which also bind the doped fermions at their respective centers. Out of this symmetry line, we show that quasi-spiral profiles appear with a characteristic wavevector set by the density $k = 2πρ$, providing non-perturbative evidence for chiral spirals beyond the large-N limit. These results demonstrate that various exotic inhomogeneous phases can arise in lattice field theories, and motivate the use of quantum simulators to confirm such QCD-inspired phenomena in future experiments.

Figures (17)

• Figure 1: Pictorial representation of the models.(a) GNW model in the rotated $\Psi_{n}$ (upper panel) and rung $\Phi_{n}$ (lower panel) bases. By splitting the spinor indices $\sigma$ and $s$, the different terms contained in the Hamiltonians \ref{['eq:GNW']} and \ref{['eq:rung_hamiltonian']} can be depicted in 2-leg ladders, where the upper and lower legs correspond to the $\uparrow$ ($+$) and $\downarrow$ ($-$) indices of $\Psi_n$ ($\Phi_n$). The solid lines stand for the hopping terms, which in the $\Psi_n$ basis read $\Psi^\dagger_{n, \sigma}\Psi_{n',\sigma'}+{\rm H.c.}$; the wavy ones correspond to the density-density interaction associated with the quartic term, $\sum_{\sigma} \Psi^\dagger_{n, \sigma} \Psi^\dagger_{n, \sigma'} \Psi_{n,\sigma'}\Psi_{n, \sigma}$; and the uniform single-orbital lobes are proportional to the number operatros $\Psi^\dag_{n,\sigma}\Psi_{n,\sigma}$, with a global action identical to adding a interaction-dependent chemical potential $\mu(g^2)=g^2/2a$, relevant only when assuming a grand-canonical ensemble. (b) Su–Schrieffer–Heeger (SSH) model: the chain is divided into unit cells of two sites, usually named $A$ and $B$, so that interaction consists of the tunnelings and density-density repulsions between the nearest neighbor alternating species.
• Figure 2: Schematic representation of the GNW phase diagram at zero temperature $T$ and density $:\!n_{\rm f}\!:$: depending on the values of the bare mass $ma$ and the interaction strength $g^2$ there are three distinct phases: a SPT phase (shaded in red) characterized by the topological Zak phase $\varphi_{\rm Z}=\pi$ and a zero pseudoscalar condensate $\pi_0$; a parity broken (Aoki) phase (shaded in yellow) where $\pi_0$ is spontaneously broken and $\varphi_{\rm Z}$ adopts non-quantized values; and a trivial band insulator (shaded in blue) where the previous quantities are null. The scalar condensate $\sigma_0$ takes non-vanishing values in all the phase diagram except in the symmetry line $ma=-1$ (vertical dotted line). These three phases are delimited by second-order phase transitions (green lines). The red dotted lines indicate the SPT-Trivial critical lines, and follow from the condition $m(g^2)=0$ (left) and $m(g^2)=-2$ (right), with $m(g^2)=m+\sigma_0$.
• Figure 3: Large-$N$ prediction for the GNW phase diagram near $\boldsymbol{ma=-1}$ . Phase diagram in the $(\mu,g^2)$ plane obtained by minimising the large-$N$ effective potential on a 1024-site lattice assuming spatial homogeneity. The parameters are $\beta=100,\,ma=-0.999$. From left to right, it is shown the density $\rho$, the product $v_F\kappa$, and the pseudoscalar condensate $\Pi$. According to large-$N$ calculations, there is a bounded compressible phase that takes place at $\mu a=1$ in the free theory and widens for non-zero values of the interactions, ending at intermediate values of $g^2$. Regarding the pseudoscalar condensate, it takes non-vanishing values only inside the Aoki phase at zero density, a region that occurs at $g^2\geq 2$ according to the large-$N$ approximation.
• Figure 4: Grand-canonical phase diagram of the GNW model at zero temperature. We have computed the ground state for several values of $(\mu a, g^2)$ along the symmetry line $ma=-1$ for chains composed of $N_s=128$ lattice sites. To characterize the phase diagram we have represented the average fermion density $\rho(\mu a, g^2)$ (left panel), the compressibility $\kappa(\mu a, g^2)$ (center panel), and, in order to detect inhomogeneities in the pseudoscalar condensate, we work out its discrete Fourier transform (DFT) and retain the wavevector $k_{\rm max}(\mu a, g^2)$ with maximum amplitude (right panel). To pick out the relevant spatial modulations in $\pi_n$ from the ones associated with the boundary effects around $n=1$ and $n=N_s$, we have used a cutoff criterion: bearing in mind that $\pi_n$ is homogeneous in the bulk at zero density, we calculated the maximum amplitude of its DFT --extracted its mean-- at $g^2=4$, since the boundary effects are expected to be largest at the critical point. In this manner, only those $k_{\rm max}$ with greater amplitudes are considered. Clarified this, three qualitatively different regions are observed in the panels: first of all, the ones with either zero or unit density, which do not have spatial modulations on $\pi_n$. For intermediate values of $\rho$ there are two types of spatially-modulated phases as we increase the chemical potential: first, we find a compressible phase at low densities for $g^2>0$, while for larger ones plateaus around the fillings $\rho = (:\!n_{\rm f0}\!: \!a)^{-1}$ take place, being these more stable when corresponding to commensurate fillings. In all cases, we observe that the wavevector and the density are related by the expression $k_{\rm max}\approx 2\pi\rho$, which is exact for commensurate fillings while it may be approximate for the incommensurate ones.
• Figure 5: Grand-canonical ensemble of the GNW model at zero temperature and $N_s=128$ lattice sites. Density $\rho$ (green solid line) and compressibility $\kappa$ (red solid line) at (a)$g^2=2.0$, (b)$g^2=4.0$, (c)$g^2=6.0$ and (d)$g^2=8.0$. First of all, we observe that for $g^2\lesssim 4.0$ the density acquires a non-zero value at $\mu a \ll 1$, because of the existence of a zero-energy topological edge state which is populated as we turn on the chemical potential. Additionally, for $g^2>0$, once a given threshold in $\mu a$ is reached, the system enters a compressible phase, whose extension is $g^2$-dependent. This compressible phase is followed by an incompressible one consisting in plateaus around the densities $\rho = (:\!n_{f0}\!: \!a)^{-1}$ (dashed horizontal lines), being those associated with commensurable fillings (coloured in green) more stable.