--- abstract: 'Recent calculations using chiral effective field theory suggest that chiral symmetry is restored in neutron matter at a much lower density than previously expected. We study the transition to this new state using quantum Monte Carlo techniques to calculate the equation of state for uniform matter. We show that the new critical density can be unambiguously identified in terms of the energy needed to form uniform matter, and in other observables. We determine that the new critical density is about one tenth of the nuclear saturation density. We also show that our simulations can be used to test mean field models and explore the region in which chiral perturbation theory might be quantitatively reliable.' author: - 'G. Rupak' - 'A. Hime' - 'M. Tarbutt' bibliography: - 'refs.bib' title: | Chiral symmetry restoration in neutron matter from quantum Monte Carlo\ simulations --- Quantum Monte Carlo (QMC) simulations have been used to study strongly interacting matter at high densities in both the continuum [@Gandolfi:2011xu; @Gandolfi:2014ewa] and periodic box [@Gandolfi:2009fj; @Gandolfi:2011xu; @Gandolfi:2011wj; @Gandolfi:2012jv; @Gandolfi:2014vaa; @Gandolfi:2014ira; @Tripuraneni:2016qty]. They have been applied to neutron matter, which may provide a link between cold dense matter and neutron stars [@Horowitz:2005zs; @Hebeler:2009iv; @Kruger:2013kua; @Kruger:2013kua]. Several calculations in this area have been guided by recent progress in nuclear physics. For example, an effective field theory (EFT) that uses nucleon degrees of freedom can be systematically improved by constructing terms in the EFT expansion with increasing accuracy [@Gandolfi:2012jt; @Kaiser:2013bja; @Kaiser:2013mca; @Tews:2012fj; @Kogut:2012ag; @Lynn:2015jua]. In these approaches, nucleons interact via exchange of pions as the lightest degree of freedom, and short-range and higher order corrections are incorporated. A description of neutron matter can be developed and tested [@Gandolfi:2012jv], allowing one to confront theoretical predictions with experiment. Recently, chiral effective field theory has been used to determine properties of neutron matter at densities expected in neutron stars [@Holt:2016pjb; @Holt:2016poe]. This work provides an EFT description of a new phase of matter for which chiral symmetry is restored, but chiral perturbation theory (ChPT) [@Weinberg:1978kz; @Gasser:1983yg; @Gasser:1984gg] cannot be directly applied. The results of this EFT have recently been confirmed by lattice QCD simulations [@Astrakhantsev:2017nrs]. In this letter we explore the physics associated with this new phase in neutron matter. Specifically, we use QMC methods to determine the equation of state (EOS) of uniform neutron matter and study the chiral symmetry restoration transition in this EFT. We argue that the new critical density is unambiguously identified in observables, and can be identified using finite size effects. This phase transition provides an important benchmark for theories and methods in the nuclear physics regime. The structure of matter in this phase differs qualitatively from nuclear matter, where the strong force is mediated by the exchange of pions with mass $m_\pi \approx 140$ MeV. In particular, the mass of the $\pi^-$, the negatively charged partner of the pion, becomes arbitrarily large as the density of the system increases, leading to a phase where chiral symmetry is restored. The chiral symmetry transformation is no longer spontaneously broken in uniform matter. Note, there are important corrections to the EFT expansion of this phase that could change this qualitative picture. [@Holt:2016poe] The relevant scales in this new phase are the chiral scale $\Lambda_\chi\sim 1$ GeV and the Fermi momentum $k_F\sim m_\pi/2$. Therefore, the regime $k_F <\Lambda_\chi$ must be studied in mean field theory. We do so using the pionless effective field theory (EFT) proposed in Ref. [@Holt:2009ty] which is able to describe neutron matter at lower densities than chiral perturbation theory. The pionless EFT is consistent with chiral perturbation theory at energies larger than the cutoff, $m_\pi <\Lambda_\chi$. Therefore, it cannot be used to study strong interaction physics at low densities. However, since pionless EFT includes the same symmetry-breaking dynamics as ChPT, one can calculate the physical quantities of interest at low energies from the pionless EFT without knowing what the true QCD dynamics is. This is because at low energies only the low-momentum modes that lead to chiral symmetry breaking and are contained in the pionless theory are responsible for strong interactions. It is the low-energy theorems from chiral perturbation theory which are able to determine the correct physical quantities at low energies. Since the symmetry breaking in this phase is described well at low densities by pionless EFT, it can be used to study neutron matter. We have explored the range of validity of pionless EFT in detail [@GrillidiCortona:2015ldd]. In addition, we also considered the corrections to our pionless EFT from higher order terms from the pionless EFT power counting [@Holt:2009wj] and showed that such terms are not important in describing neutron matter. We will now make a brief description of the pionless EFT. The leading order terms in the effective Lagrangian of pionless EFT include operators that correspond to the contact interactions in ChPT [@Zhuang:1994dw]. We choose to use the notation of Ref. [@Holt:2009wj] and include the operators $$\begin{aligned} \mathcal{L}_{\rm LO}^{(3)}=\bar\psi_N \left[ \frac{1}{2}\left( \Delta m -C_3\tau^z \right)\vec{\tau}\cdot \vec{t}_i -\frac{1}{4}(C_1+\tilde{C}_1)\left(\vec{\sigma}\cdot\vec{t}_i\right) -\frac{1}{4}D_1\vec{\sigma}\cdot (\vec{\sigma}\cdot \vec{t}_i) \right]\psi_N,\label{eq:chiral}\end{aligned}$$ where $\psi_N$ is the nucleon field, $\vec{t}_i$ are the isospin $1/2$ operators, and $C_i, \tilde{C}_i$ and $D_1$ are parameters. The pion mass appears in the chiral Lagrangian through the mass splitting $\Delta m = m_n-m_p$. Therefore, the chiral Lagrangian should be modified by removing this splitting from the nucleon energy density. The leading chiral Lagrangian in this phase is [@Holt:2009wj] $$\mathcal{L}_{\rm LO}^{(3)} \rightarrow \mathcal{L}_{\rm LO}^{(3)}-\frac{\Delta m}{8}( \psi^T\tau^z\psi+\psi^\dagger\tau^z\psi) \label{eq:chiral_lag}$$ We use the methods developed in Ref. [@Tews:2015ufa] to solve the eigenvalue problem associated with this theory, which led to a good description of both neutron and symmetric nuclear matter. In the following, we focus on the system that describes uniform neutron matter. ![The energy density of neutron matter for a range of densities between $n=0.1$ and $2$ fm$^{-3}$ in order to identify the critical density of chiral symmetry restoration. Dashed lines are the result of chiral perturbation theory, while solid lines are results from the pionless effective field theory. The red line at $n=0.02$ fm$^{-3}$ indicates the critical density of the pionless EFT, i.e. chiral symmetry restoration. []{data-label="fig:ed"}](energy_density.eps){width="\linewidth"} In order to determine the critical density in neutron matter, we calculate the equation of state and other quantities of interest in QMC simulations. The results from simulations are input into the pionless EFT and the leading operators are fit to match the results from QMC. We now discuss the various observables of interest. The equations of motion for the neutron matter system can be solved by directly solving the eigenvalue problem of a generalized eigenvalue equation in Ref. [@Gandolfi:2014ewa]. The generalized eigenvalue equation can be recast as a multi-variable variational problem with one variable for each value of $\vec{p}$. We use a basis of plane waves $$f_{i,\vec{p}}(\vec{r})=\frac{1}{(2\pi)^{3/2}}\exp(\vec{p}\cdot \vec{r})\chi_i \label{eq:pw}$$ where $\vec{p}$ is the three-momentum, $\chi_i$ are isospin projection operators, and $i$ labels the particle-hole states. We use a variational method to solve the generalized eigenvalue equation