Simple Fermionic Model of Deconfined Phases and Phase Transitions

Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with $N$ flavors of fermions per site for $N=1$, 2, and 3. The models have an extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm. In particular, one of the prominent phases for $N>1$ corresponds to $2N$ gapless Dirac fermions coupled to an emergent ${\mathbb{Z}}_2$ gauge field in its deconfined phase. However, unlike a conventional ${\mathbb{Z}}_2$ gauge theory, we do not impose “Gauss’s Law” by hand; instead, it emerges because of spontaneous symmetry breaking. Correspondingly, unlike a conventional ${\mathbb{Z}}_2$ gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss’s law constraint at zero temperature. By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for $N=2$ and a valence-bond solid for $N=3$. Furthermore, for $N=3$, the valence-bond solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.

Popular Summary

A rich variety of phases beyond solid, liquid, and gas are found in nature, and these phases originate from simple rules that encode interactions among elementary constituents. Quantum mechanics provides a window to new phases in which constituents can be in several different patterns all at once, much like Schrödinger’s cat can be dead and alive simultaneously. Here, we introduce a simple model, free of the fermion sign problem, that harbors fascinating new phases and can be used to conduct unbiased quantum Monte Carlo simulations.

The basic rules of our model are simple: Fermions of up to three flavors tunnel between sites of a square lattice, and each tunneling event changes the sign of an Ising field (subject to a magnetic field) residing on the links of the lattice. One fundamental question is the following: Does our model host a phase in which strong quantum fluctuations do not allow the system to pick any one ordering pattern? We find that the answer to this question is indeed yes. There exists a regime in which there is no ordering, and the description is in terms of fluctuating strings whose ends are the fermions. Additionally, we find two symmetry-breaking phases that are separated by a continuous transition that lies beyond the Landau paradigm of phase transitions. The transition from the symmetry-unbroken phase to the symmetry-broken phase is again seemingly continuous.

We expect that our findings will motivate work to determine a larger theoretical framework encompassing our model.

Figure 1

Schematic zero-temperature phase diagram of the model in Eq. (1) (a), as well as cartoons (b)–(e) of a selected number of phases. (a) We observe a ${\mathbb{Z}}_2$ Dirac deconfined phase ($Z_2\mathrm{D}$), a Néel antiferromagnet phase (AFM) [or a superconductor (SC), depending on the pattern of particle-hole symmetry breaking], a charge density wave (CDW) phase, and a valence-bond solid (VBS). For $N=1$, we do not find evidence for a $Z_2\mathrm{D}$ phase beyond $h=0$, consistent with the arguments in the main text. The phase transitions from the $Z_2\mathrm{D}$ to AFM/SC ($N=2$) and VBS ($N=3$) are seemingly continuous. At $N=3$, we observe a deconfined quantum critical point (DCQP) between the VBS and AFM phases. (b)–(e) Cartoons of the corresponding phases. Circles correspond to fermions, and the color code corresponds to the flavor index. Circles with two colors represent a pair of fermions on a site with corresponding flavors. The low-energy properties of the $Z_2\mathrm{D}$ phase resemble $\mathrm{SU}(N)$ fermions propagating freely in space-time and connected by a $Z_2$ gauge string (b). The symmetry-broken phases correspond to the confined phases of the model. At $N=3$, the AFM phase (e) has the fundamental (conjugate) representation of SU(3) on sublattice A (B). The corresponding Young tableaux are included. The VBS phase (d) corresponds to a pattern of intersite SU(3) singlets.

Figure 2

Dynamical spin structure factor at $h=0$. At the considered temperature, we can safely neglect the thermal fluctuations of the Ising fields. Only Ising field configurations with a $\pi$ flux per plaquette contribute.

Figure 3

Single-particle spectral function $A(\mathit{k},\omega )$ for the $N=1$ model. As argued in the text, $A(\mathit{k},\omega )$ is $\mathit{k}$ independent.

Figure 4

(a) Staggered order parameter of Eq. (28) at finite symmetry-breaking field $h_s=0.01$. (b) Staggered order parameter of Eq. (31) again at finite symmetry-breaking field $h_s=0.01$. The temperature scale is normalized by $T_{1/2}$ defined in Eq. (30). (c) $T_{1/2}$ as a function of $h$. (d) Data collapse of the density-density correlation function defined in Eq. (32). This correlation function is related to the CDW order parameter via Eq. (34). A reasonable data collapse is obtained using the 2D Ising exponents.

Figure 5

(a) Electric field and (b) magnetic flux for the $N=1$ model as a function of the transverse field. After taking into account the strong finite temperature and size effects in the small $h$ limit, the data are consistent with a smooth behavior.

Figure 6

(a) Size extrapolation of the charge density wave order parameter ${\mathrm{\Delta }}_{\mathrm{CDW}}$ performed with a second-order polynomial in $1/L$. (b) Extrapolated value of ${\mathrm{\Delta }}_{\mathrm{CDW}}$ at different temperatures. Below $h=0.1$, our lattice sizes are too small to unambiguously extrapolate to the thermodynamic limit.

Figure 7

Dynamical charge structure factor for the $N=1$ model.

Figure 8

Energy as a function of temperature for the $N=2$ model at various values of $h$.

Figure 9

(a) Average value of $⟨{\widehat{Q}}_{\mathit{i}}⟩$ in the presence of a small symmetry-breaking field, $U=-0.0625$. (b) Average value of $⟨{\widehat{p}}_{\mathit{i}}⟩$ at the same value of the symmetry-breaking field. Unless mentioned otherwise, we have set $L=8$. (c) Estimate of $T_c$ below which we expect $⟨{\widehat{Q}}_{\mathit{i}}⟩$ to order ferromagnetically. $T_E$ stems from the energy data of Fig. 8, and the temperature scale $T_{1/2}$ is defined by $⟨{\widehat{Q}}_{\mathit{i}}⟩(T_{1/2})=1/2$.

Figure 10

(a) Electric field and (b) magnetic flux per plaquette as a function of $h$ for the $N=2$ model.

Figure 11

(a) Magnetization as a function of system size [see Eq. (43)]. In the ordered phase, we have used a second-order polynomial form in $1/L$ to extrapolate to the thermodynamic limit. At $h=0.1$ and $h=0.075$, we have set $\beta =120$ and fitted the data to the form $f(x)=a{L}^{-b}$. (b) Estimated value of the magnetization in the thermodynamic limit.

Figure 12

Dynamical spin structure factor for the $N=2$ model as a function of $h$. For the smallest value of $h$, we have considered $\beta =60$ as opposed to $\beta =40$ for other values of the transverse field. This choice of temperature places us below $T_c$ such that ${\widehat{Q}}_{\mathit{i}}\simeq 1$.

Figure 13

Uniform spin susceptibility at (a) $h=0.075$ (${\mathrm{Z}}_2\mathrm{D}$ phase) and (b) $h=1$ (AFM phase).

Figure 14

(a) $⟨\widehat{Q}{⟩}_{\mathit{K}}=\frac{1}{L^2}\sum _{\mathit{i}}{e}^{i\mathit{K}·\mathit{i}}⟨{\widehat{Q}}_{\mathit{i}}⟩$ and (b) $⟨\widehat{p}{⟩}_{\mathit{K}}=\frac{1}{L^2}\sum _{\mathit{i}}{e}^{i\mathit{K}·\mathit{i}}⟨{\widehat{p}}_{\mathit{i}}⟩$ in the presence of a small staggered chemical potential term, which breaks particle-hole symmetry and introduces an antiferromagnetic Zeeman term for the Ising Hamiltonian of Eq. (24). (c) Electric field and (d) magnetic flux per plaquette as a function of $h$.

Figure 15

(a) SU(3)-spin staggered moment as a function of system size. The data are consistent with the following choices of fit functions. For $h>3/8$, we have used a second-order polynomial in $1/L$. At $h=3/8$, we have fitted the data to the form $m(L)=a{L}^{-(1+\eta )/2}$ to obtain $\eta \simeq 0.64$, which turns out to be substantially larger than the exponent obtained for the J-Q model for the SU(3) case, $\eta \simeq 0.4$ [54]. For lower values of $h$, we again use a polynomial fit and impose a zero intercept with the $y$ axis. In panel (b), we plot the extrapolated value of the magnetization as well as of the VBS order parameter. At $h=3/8$, both VBS and AFM order parameters are consistent with a power-law decay.

Figure 16

Uniform spin susceptibility in the ${\mathrm{Z}}_2\mathrm{D}$, VBS, and AFM phases.

Figure 17

SU(3) VBS order parameter. At $h=1/12$, corresponding to a rough estimate of the transition from the ${\mathbb{Z}}_2$ Dirac deconfined to the dimerized phase, we fit the data to the form $a{L}^{-b}$. The transition from the VBS phase to the AFM one occurs approximately at $h=2/3$. For this value of $h$, the data are consistent with a large $\eta =2b-1$ exponent, $\eta \simeq 0.6$, which is larger than the one obtained for the J-Q model at SU(3) $\eta \simeq 0.5$ (see Ref. [54]).

Figure 18

Temperature dependence of the dimer-dimer and spin-spin correlations in the ${\mathrm{Z}}_2\mathrm{D}$, VBS and AFM phases of the $N=3$ model. The arrow in Fig.(c) corresponds to the energy scale at which $⟨\widehat{Q}{⟩}_{\mathit{K}}$ plotted in Fig. 14 14 takes a value of $1/2$ (see Eq. (30).

Figure 19

Dynamical spin structure factor for the SU(3) model. The simulations were carried out on a $16\times 16$ lattice at $\beta =30$ for $h\ge 1/6$ and at $\beta =60$ for $h=1/12$.