Two-dimensional O(3) model at nonzero density: From dual lattice simulations to repulsive bosons

We discuss the thermodynamics of the O(3) nonlinear sigma model in $1+1$ dimensions at nonzero chemical potential (equivalent to a magnetic field). In its conventional field theory representation the model suffers from a sign problem. By dualizing the model, we are able to fully access the nonzero density regime of an asymptotically free theory with dynamical mass gap at arbitrary chemical potential values. We find a quantum phase transition at zero temperature where as a function of the chemical potential the density assumes a nonzero value. Measuring the spin stiffness we present evidence for a corresponding dynamical critical exponent $z$ close to 2. The low energy O(3) model is conjectured to be described by a massive boson triplet with repulsive interactions. We confirm the universal square-root behavior expected for such a system at low density (and temperature) and compare our data to the results of Bethe Ansatz solutions of the relativistic and nonrelativistic one-dimensional Bose gas. We also comment on a potential Berezinskii-Kosterlitz-Thouless transition at nonzero density.

Figure 1

Energy density versus the coupling at ($\mu =0$). The analytical results for the strong and weak coupling expansions from Eq. (11) agree very well with the numerical results (having very small error bars) obtained from simulating the dual ensemble on a ($10\times 10$) lattice and using Eq. (16).

Figure 2

Top: Expectation value of the particle number density versus the chemical potential $\mu$, both in lattice units, at different couplings $J$ for lattice size $90\times 90$. Bottom: Logarithm of the corresponding critical chemical potentials as a function of the coupling. We compare our data to the strong coupling result $\ln (3/J)$ and the weak coupling expansion of the mass gap and Eq. (8), having obtained $C=102(2)$ from a fit of the $J\ge 1.4$ data.

Figure 3

Phase transition in the $J–\mu$ plane for $N_s=30$ and $N_t=1000$. At large $J$, i.e., towards the continuum limit, the critical chemical potentials with increasing index (from red to green to black to blue) induce individual transitions between integer charges as utilized in [1]. At small $J$ the first $N_s$ of them join to a single transition, ${\mu }_1=\dots ={\mu }_{Ns}$, increasing the particle density $n$ in lattice units from 0 to 1 (see text). The next critical chemical potential ${\mu }_{N_s+1}$ (blue) is separated from those, and above it a density $n=2$ will be induced. At large $J$, on the other hand, the change from ${\mu }_{Ns}$ to ${\mu }_{N_s+1}$ plays no particular role. The strong coupling predictions for the transitions $0\to 1$ and $1\to 2$ are included as solid curves.

Figure 4

Phase transition as in Fig. 3, but with a modified quantity on the $y$-axis (and some more data points for higher $J$’s); see text.

Figure 5

Expectation value $K$ for the third dual variable [for its definition and interpretation see Eq. (17) and below] also displaying a transition at $\mu =m$ ($J=1.3$, $N_s=200$, $N_t=1000$). The value at small $\mu$ is the isotropic one related to the energy density by $K=J(4-e)/6$.

Figure 6

Dependence of the particle number density and its susceptibility on the chemical potential at a fixed low temperature $T/m=0.023$ for three spatial sizes’ volumes (coupling $J=1.3$, lattices with $N_t=200$ and $N_s=80$, 120, 160).

Figure 7

Illustration of the four scaling trajectories in the $T–L^{-1}$ plane towards the zero temperature and infinite volume limit (black dot) which we use in this study [see discussion around (19)].

Figure 8

Emergence of a phase transition in the limit $T\to 0$ for free one-dimensional fermions in an infinite volume. We show that the smooth curves for $T/m=1/10$ (green) and $T/m=1/50$ (blue) approach the $T=0$ square-root behavior from Eq. (21). This is a model calculation for the scaling trajectory $\alpha =0$, especially its second limit $T\to 0$ at $L^{-1}=0$.

Figure 9

Particle density $n$ in units of the mass as a function of $\mu$ for different temperatures at a fixed volume of $L=22/m$ ($J=1.3$, $N_s=100$, $N_t=200$, 400, 1000, 2000). When lowering the temperature one observes [1] the emergence of plateaus at integer values of the particle number, which for the normalized density $n/m$ used here corresponds to integer multiples of 0.046.

Figure 10

Illustration of how the low temperature density with its steps at any finite volume approaches its infinite volume limit. Top: The Tonks-Girardeau gas similar to free fermions at zero temperature (see text) for $Lm=4.4$, 6.6, 19.8 (red, green, blue) being a model calculation for the scaling trajectory $\alpha =\infty$ (second limit). Bottom: Our results for the O(3) model at $T/m=0.0023$. For the different scales recall that free fermions approximate the O(3) model only at low densities.

Figure 11

Particle number density for the $\alpha =1$ (top) and $\alpha =2$ (bottom) scaling trajectories, both at $J=1.3$. For $\alpha =1$ we approach $T=0$ using $L=1/T$, whereas for $\alpha =2$ $L\sim T^{-1/2}$ increases less rapidly. In the $\alpha =2$ panel we also include the lowest $T$ data from $\alpha =1$ for comparison. For the lattice parameters see Table 2.

Figure 12

Spin stiffness for the $\alpha =1$ (top) and $\alpha =2$ (bottom) scaling trajectories. As in Fig. 11 we include the lowest $T$ data from $\alpha =1$ in the $\alpha =2$ panel. For the lattice parameters see Table 2.

Figure 13

Agreement of stiffness and density (in mass units) close to zero temperature in a large volume $L\sim 1/\sqrt{T}$ ($\alpha =2$, time elongated lattice): $T/m=0.0007$, $Lm=36$.

Figure 14

Numerical checks of statement (a) for exponents $z=1$ (top) and $z=2$ (bottom). For the lattice parameters see Table 2.

Figure 15

Numerical checks of statement (b) for $z=1$ (top) and $z=2$ (bottom) assuming $\nu =1/2$. In the bottom panel we also include the square-root behavior in these variables. For the lattice parameters see Table 2.

Figure 16

Numerical checks of statement (b) assuming $z=2$ for $\nu =1/3$ (top) and $\nu =3/4$ (bottom). For the lattice parameters see Table 2.

Figure 17

The density $n/m$ vs chemical potential $\mu /m$ at temperatures $T/m=0.228$, 0.091, 0.046, 0.005 from top to bottom. We compare the numerical evaluation of the relativistic Bethe Ansatz equation (29) (green curve) and the nonrelativistic Bethe Ansatz equation of (28) (blue curve), with the lattice data (red symbols, $J=1.3$, $N_s=100$ and $N_t=20$, 50, 100, 1000). The black line in the bottom plot is the universal square-root behavior given in (31).