Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature

Quantum Electrodynamics (QED) in $1+1$ dimensions, also known as the Schwinger model Schwinger:1962tp , is a famous model of Quantum Field Theory (QFT) that has played an important role for over 60 years Lowenstein:1971fc ; Casher:1974vf ; Coleman:1975pw . It is a useful theoretical laboratory for various important phenomena, including QFT anomalies and confinement of charge. Its lattice Hamiltonian implementations Kogut:1974ag ; Banks:1975gq have connections with condensed matter and atomic physics, and, in recent years, there have been efforts to construct experimental setups for its quantum simulations (for a review, see Banuls:2019bmf ).

The model with one massless Dirac fermion of charge $g$ is exactly solvable, reducing to the non-interacting Schwinger boson of mass $M_{S}=g/\sqrt{\pi}$; this can be concisely demonstrated via the bosonization of the fermion Coleman:1975pw . The $\mathrm{U}(1)$ chiral symmetry of the massless action is broken by the Schwinger anomaly. The massive model, in addition to containing the obvious dimensionless parameter $m/g$, depends on the $\theta$ angle related to the introduction of a background electric field Coleman:1975pw . This parameter, which has periodicity $2\pi$, is somewhat analogous to the $\theta$ angle of the $3+1$ dimensional gauge theory.

Generalizations of the Schwinger model to $N_{f}>1$ flavors of fermions of charge $g$ exhibit a richer set of phenomena Coleman:1976uz . When the fermions are massless, the Schwinger model has $\mathrm{SU}(N_{f})\times\mathrm{SU}(N_{f})$ chiral symmetry. Its low-energy limit is described Gepner:1984au ; Affleck:1985wa by the $\mathrm{SU}(N_{f})_{1}$ Wess-Zumino-Witten (WZW) model, which is a Conformal Field Theory (CFT) of central charge $N_{f}-1$. The $N_{f}>1$ Schwinger model also contains a massive sector that includes the Schwinger boson. Therefore, it was hoped that the multiflavor Schwinger models may provide simple realizations of the “unparticle physics” idea Georgi:2007ek , and this motivated the papers Georgi:2019tch ; Georgi:2020jik ; Georgi:2022sdu . As in these papers, we will focus on $N_{f}=2$, where for $m=0$ the IR CFT is described by a compact scalar at the self-dual radius. While investigations of this model have a long history including Coleman:1976uz ; Steinhardt:1977tx ; Smilga:1992hx ; Hetrick:1995wq ; Smilga:1998dh ; Hosotani:1998kd ; Berruto:1999ga ; Hip:2021jgp ; Albergo:2022qfi ; Funcke:2023lli , we will present a number of new results: 1) Even in the limit of small masses, we can have spontaneous symmetry breaking of the charge conjugation symmetry, or critical behavior, or an IR trivial phase. 2) For $\theta=\pi$ and $m/g\ll 1$, there is an effective field theory description in terms of the sine-Gordon model with $\beta\approx\sqrt{8\pi}$ Coleman:1976uz ; Smilga:1998dh . We describe the $\mathrm{SU}(2)$-invariant RG trajectory, which flows from asymptotic freedom in the UV, and in the IR it produces an exponentially small mass gap $\sim e^{-Ag^{2}/m^{2}}$, with $A\approx 0.111$ as we show below. Therefore, the $N_{f}=2$ Schwinger model with $\theta=\pi$ has some qualitative similarities with QCD because it can exhibit dimensional transmutation.

We discuss the zero-temperature phase diagram as a function of $\theta$ and the masses $m_{1}$ and $m_{2}$ of the two fermion flavors, which we can restrict to be positive (some aspects of the phase structure were discussed in the past Coleman:1976uz ; Smilga:1998dh ; Georgi:2022sdu ). Our proposal is that, while for all $\theta\neq\pi$ this model has a non-degenerate vacuum, for $\theta=\pi$ the phase diagram is as in Figure 1. It contains two critical curves that pass through the origin, along which the low-energy physics is governed by the 2D Ising CFT of central charge $c=1/2$. In the shaded region of Figure 1, the charge conjugation symmetry $C$, defined below, is spontaneously broken, leading to two degenerate vacua. This phenomenon, which was recently studied in Funcke:2023lli , is reminiscent of the spontaneous breaking of $CP$ symmetry in 4D Yang-Mills theory at $\theta=\pi$ Dashen:1970et ; Creutz:1995wf ; Smilga:1998dh ; Creutz:2010ts ; Creutz:2018vgl ; Gaiotto:2017yup , and there are analogous phenomena in 2D scalar QED Komargodski:2017dmc ; Sulejmanpasic:2020lyq . We present both analytical and numerical evidence for the phase diagram in Figure 1. On the numerical side, our calculations using the Hamiltonian lattice approach are in excellent agreement with the continuum analysis. The convergence of the numerical calculations is significantly improved by including the mass shift (4) derived in Dempsey:2022nys .

Let us consider the Schwinger model with $N_{f}$ fermion flavors of masses $m_{\alpha}$, with $\alpha=1,\ldots,N_{f}$. While the $m_{\alpha}$ are, in general, complex parameters, the $\mathrm{U}(1)^{N_{f}}$ axial transformations can be used to set all $m_{\alpha}$ real with $m_{\alpha}\geq 0$. Then, the Lagrangian density is

Here, $\epsilon^{01}=1$, $\not{D}=\gamma^{\mu}(\partial_{\mu}+iA_{\mu})$, and $(\gamma^{0},\gamma^{1})=(\sigma_{3},i\sigma_{2})$ obey $\{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}=2\mathop{\rm diag}\nolimits\{1,-1\}$.

To study this model numerically, we use the Hamiltonian lattice formulation of Kogut:1974ag ; Banks:1975gq , where the spatial direction is discretized into $N$ sites, with $N$ even, while the time direction remains continuous. The two-component Dirac fermions of each flavor are staggered, with the $\gamma^{0}$ eigenstates of eigenvalue $+1$ and $-1$ being placed on even and odd sites, respectively. The lattice Hamiltonian is

Here, $a$ is the lattice spacing, $c_{n,\alpha}$ and $c_{n,\alpha}^{\dagger}$ are the annihilation and creation operators for a fermion of flavor $\alpha$ on site $n$, and $U_{n}=e^{i\phi_{n}}$ is a unitary operator living on the link between sites $n$ and $n+1$. The electric field strengths $L_{n}=-i\frac{\partial}{\partial\phi_{n}}$ are integer-valued, while $\theta\in[0,2\pi)$ comes from the $\theta$-term in the action, and $\frac{\theta}{2\pi}$ acts as a fractional background electric field. The Hamiltonian should be supplemented by the Gauss law constraint

The parameters $g$ and $\theta$ of the lattice model should be identified with the analogous parameters of the continuum model (1). As argued in Dempsey:2022nys , one should take

In Dempsey:2022nys , it was also shown that when $N_{f}$ is even and $m_{\alpha}=0$, the lattice theory is invariant under translation by one site, which corresponds to a discrete chiral symmetry in the continuum. In the leading strong coupling limit, where the hopping term is ignored, the ground state can be highly degenerate. For $N_{f}=2$ and $m_{1}=m_{2}=0$, we find that the strong coupling degeneracy is $3^{N}+1$ for $\theta=0$, while it is $2^{N}$ for $\theta=\pi$. The latter fact provides a starting point for the correspondence between the $N_{f}=2$ Schwinger model at $\theta=\pi$ and the Heisenberg antiferromagnet Hosotani:1998kd ; Berruto:1999ga .

The integrated fermion bilinear operator $\int dx\,\bar{\Psi}\Psi$ translates into the lattice operator $\sum_{n=0}^{N-1}(-1)^{n}c^{\dagger}_{n,\alpha}c_{n,\alpha}$, which is odd under the unit shift. The uniqueness of the ground state away from the strong coupling limit for $m_{\alpha}=0$, and the symmetry under the unit translation, imply that the VEV of the mass operator vanishes on a periodic lattice with an even number of sites.

When $\theta=0$ or $\pi$, for any $m_{\alpha}$, the models (1) and (2) are invariant under a charge conjugation symmetry $C$. In the continuum, $C$ acts as

where $\gamma^{5}=\gamma^{0}\gamma^{1}=\sigma_{1}$, and on the lattice it acts as Berruto:1999ga

It is this symmetry $C$ that is broken whenever there are two degenerate vacua in the phase diagram in Figure 1.

For any $\theta$ and $m_{\alpha}=m$ for all $\alpha$, the models (1) and (2) are invariant under an $\mathrm{SU}(N_{f})$ symmetry under which the fermions transform in the fundamental representation. When $m=0$, in the continuum model (1), this $\mathrm{SU}(N_{f})$ symmetry is enhanced to $\mathrm{SU}(N_{f})\times\mathrm{SU}(N_{f})$.

Let us now take $N_{f}=2$ and present continuum field theory arguments in support of the phase diagram of the 2-flavor Schwinger model in Figure 1.

We study the $N_{f}=2$ Schwinger model numerically using the lattice Hamiltonian (2) (see also Funcke:2023lli ).

While the one-flavor model can be studied efficiently via exact diagonalization Hamer:1982mx ; Dempsey:2022nys , with two flavors the number of states grows so quickly with the number $N$ of lattice sites that this becomes impractical. Instead, we employ tensor network methods, using a matrix product state (MPS) ansatz to approximate the ground state Banuls:2013jaa ; Funcke:2023lli . To optimize the MPS ansatz, we use ITensors.jl ITensor-r0.3 ; ITensor . We use open boundary conditions, since this allows us to study the behavior of much larger lattices.

The MPS form of the ground state makes it especially simple to calculate the entanglement entropy for a left-right bipartition of the open chain. Let $S_{x}(N,a)$ denote the entanglement entropy for a subsystem of the leftmost $xN$ sites in a chain of $N$ sites with lattice spacing $a$. Then, at a critical point with central charge $c$, the entropy is expected to grow like Calabrese:2009qy

At any other point, this logarithmic growth of the entropy will plateau when $N\sim\frac{\xi}{a}$, where $\xi$ is the correlation length. By combining this result with a finite-size scaling analysis, one can derive very precise estimates for the locations of critical points in the continuum theory from values of the entanglement entropy on a finite lattice Campostrini:2014lta .

In Figure 2, we show the behavior of the entanglement entropy for a fixed lattice, along with the precise estimate of the critical curve obtained via the intersection method outlined in Campostrini:2014lta . The finite-size scaling analysis confirms that this curve has $c=\frac{1}{2}$. By fitting the leading large-mass behavior of this curve, we find

and the coefficient of $g^{2}/m_{1}^{2}$ is in good agreement with the value $\frac{0.3335}{12\pi}\approx 0.0088$ predicted from (10).

We can also use lattice calculations of the entanglement entropy to estimate the growth of the correlation length for small $m_{1}=m_{2}=m$. For a fixed lattice, we can compare the dependence of the entanglement entropy on the subsystem size with (22) to obtain an estimate $c_{\text{est}}$ for the central charge. Anywhere away from a critical point, this estimate will tend to zero around $N\sim\frac{\xi}{a}$. We can thus take a fiducial cutoff for $c_{\text{est}}$, and define $\frac{\beta\xi}{a}$ as the lattice size when $c_{\text{est}}$ crosses below this cutoff, where $\beta$ is an unknown constant.

Figure 3 shows this estimate of the logarithm of the correlation length along the $\mathrm{SU}(2)$-invariant line as a function of $m^{-2}$. The linear behavior suggests a scaling of the form $\xi\sim e^{A\frac{g^{2}}{m^{2}}}$ at small $m/g$, as explained after (20). Furthermore, extrapolating the slope as $m\to 0$ to the continuum limit $a\to 0$ gives $A=0.11(1)$, in good agreement with the theoretical value in (20).

In this paper, we presented analytical and numerical evidence for the phase diagram of the $N_{f}=2$ Schwinger model at $\theta=\pi$ shown in Figure 1. The behavior we find is quite different from that at $\theta\neq\pi$: along the $\mathrm{SU}(2)$-invariant line the theory contains a nearly marginal operator which leads to logarithmic RG flow of BKT type. As a result, for $m\ll g$ the mass gap is exponentially small, $\sim e^{-Ag^{2}/m^{2}}$. Along this $\mathrm{SU}(2)$-symmetric line, Georgi Georgi:2022sdu calculated the anomalous dimensions of operators perturbatively in powers of $(m/g)^{2}$. The fact that the mass gap is exponentially small makes the theory for $m\ll g$ “nearly conformal” in a large range of energies, so that perturbative anomalous dimension calculations should be parametrically reliable. We thus hope that the calculations of Georgi:2022sdu can be checked numerically using the lattice Hamiltonian setup, but we leave this question for future work.

We find that the $\mathbb{Z}_{2}$ charge conjugation symmetry is spontaneously broken in the entire shaded region of the phase diagram in Figure 1. This region becomes exponentially narrow near $m=0$ and is bounded by 2D Ising CFTs. It is interesting to ask how the addition to the action of 4-fermion operators may change this phase diagram. We also leave this question for future work.

We are grateful to Michael Creutz, Howard Georgi, Etsuko Itou, Andrei Katsevich, Zohar Komargodski, Lenny Susskind, Yuya Tanizaki, Grisha Tarnopolsky, and Edward Witten for very useful discussions. We thank a referee for important comments and pointing out very useful references. This work was supported in part by the US National Science Foundation under Grants No. PHY-2111977 and PHY-2209997, and by the Simons Foundation Grants No. 488653 and 917464. RD was also supported in part by an NSF Graduate Research Fellowship.