Detecting the critical point through entanglement in the Schwinger model

Using quantum simulations on classical hardware, we study the phase diagram of the massive Schwinger model with a θ -term at ﬁnite chemical potential µ . We ﬁnd that the quantum critical point in the phase diagram of the model can be detected through the entanglement entropy and entanglement spectrum. As a ﬁrst step, we chart the phase diagram using conventional methods by computing the dependence of the charge and chiral condensates on the fermion mass m , coupling constant g , and the chemical potential µ . At zero density, the Schwinger model possesses a quantum critical point at θ = π and m/g (cid:39) 0 . 33. We ﬁnd that the position of this quantum critical point depends on the chemical potential. Near this quantum critical point, we observe a sharp maximum in the entanglement entropy. Moreover, we ﬁnd that the quantum critical point can be located from the entanglement spectrum by detecting the position of the gap closing point.

Introduction.-The Schwinger model [1] is quantum electrodynamics in (1 + 1) space-time dimensions.This model shares many properties with QCD, including confinement (in the massive case), and chiral symmetry breaking.For the case of massless fermions, the Schwinger model is analytically solvable and equivalent to the theory of a free massive boson field [1][2][3][4][5][6][7][8][9].However, the massive Schwinger model cannot be solved analytically, and requires a numerical approach.
Quantum simulations are potentially advantageous for this purpose, as the Hilbert space grows exponentially with the number of lattice sites.In recent years, there has been a growing interest in quantum algorithms as an efficient, and potentially superior, way to explore the dynamics of quantum field theories, including the Schwinger model .Another advantage of using quantum algorithms to simulate quantum field theories is the absence of the fermion sign problem at finite chemical potential [59][60][61][62].
In the present work, we use classical hardware to simulate the massive Schwinger model at finite temperature and chemical potential.We analyze the entanglement structure of the ground state, and demonstrate that entanglement can be used as an efficient diagnostic tool for detecting the phase transition, and in particular the presence of the critical point in the phase diagram.
In particular we find that, at the critical point, there is a level crossing in the entanglement spectrum.Also, near the critical point, the entanglement entropy for a single spatial interval follows predictions from two-dimensional conformal field theory.It is well known that entanglement is a good probe of topological order [63,64].Our results suggests that entanglement can also be used to detect the fluctuating topology as well.
In [65], we showed that the presence of the critical point in the phase diagram can be detected through a sharp growth of topological fluctuations reflected by the real-time topological susceptibility.In this work, we confirm the presence of the critical point at finite and vanishing chemical potential by a calculation of the topological susceptibility, as well as the charge susceptibility.
1 + 1-dimensional QED.-In the presence of finite chemical potential, the action of the massive Schwinger model with θ term in (1+1)-dimensional Minkowski space in temporal gauge A 0 = 0 is Here, A µ is the U (1) gauge potential, E = Ȧ1 is the corresponding electric field, ψ is a two-component fermion field, m is the fermion mass, µ is the chemical potential, and γ µ are two-dimensional γ-matrices satisfying the Clifford algebra.We use the metric convention η µν = diag(1, −1).From Eq. ( 1), we see that the massive theory with a positive mass m > 0 at θ = π is equivalent to the theory at θ = 0 with a negative mass −m.
FIG. 1: Illustration of the lattice setup.Red dots are fermions and blue dots are anti-fermions.We take the boundary condition as L0 = −Q/2 which leads to LN = +Q/2, with Q being the total charge.See text for explanation.We first discretize the continuum system to obtain the lattice Hamiltonian as follows: We discretize the spatial coordinate as z = a n, where a is a lattice constant and n is an integer.We choose to work with staggered fermions [66,67], in which a two-component Dirac √ a for odd (even) n.We further define L n = E(an)/g and U n = e −iagA1(an) , respectively, as the gauge field and gauge link operators connecting the n th and (n + 1) th sites.We define our Dirac matrices as γ 0 = Z, γ 1 = i Y , and γ 5 = γ 0 γ 1 = X, where X, Y , and Z are the Pauli matrices.The lattice Hamiltonian then reads Using open boundary condition (χ 0 = χ N +1 = 0), we can eliminate the link fields U n by the gauge transformation, [65,68]).Gauge invariance of the theory requires that physical states must respect Gauss's law, is the local charge operator.As illustrated in Fig. 1, we set the boundary electric field as Q n being the total charge.This leads to the solution In this setup, the charge conjugation (C) is equivalent to the parity reflection (P).The P parity is explicitly preserved at θ = 0, and so is C conjugation.Finally, we use the Jordan-Wigner transformation [69]

after which the Hamiltonian becomes
In our study, we diagonalize this Hamiltonian to obtain eigenstates and compute expectation values.
Phase Diagram.-The expectation value of an observable O at finite temperature is defined by O ≡ Tr(ρ th O), where the thermal density matrix is a function of temperature (T ), chemical potential, and mass.Observables of interest to us include the total charge ( Q ≡ ψ † ψ dz), chiral condensate ( ψψ ≡ ψψ dz/L), and charge susceptibility (χ In particular, we study the entanglement entropy between two subsystems Let ρ be a density operator of the entire system A ∪ B. The entanglement entropy between A and B is defined by S EE, ≡ −Tr A (ρ A log ρ A ), where ρ A is defined by tracing out the Hilbert space of B: ρ A = Tr B ρ.
The entanglement spectrum, {λ i }, is spanned by the eigenvalues of ρ A .In this study we choose ρ as the ground state of the Hamiltonian, ρ = |ψ gs ψ gs |.
In Fig. 2, we show our results for the thermal expectation values of the charge, chiral condensate, charge fluctuation, and ground-state entanglement entropy (T = 0 and = N/2) for different values of fermion mass and chemical potential at T = g/10 and θ = 0, with lattice parameters N = 10, a = 0.5/g, Under the charge conjugation µ → −µ, we confirm that the total charge is C-odd, Q → − Q , whereas the chiral condensate, second-order charge susceptibility, and entanglement entropy are C-even.
We note that at finite chemical potential, the Hamiltonian (5) can be written as H(µ) = H 0 − µ Q, where the total charge Q commutes with H 0 ≡ H(µ = 0) and, therefore, is a good quantum number.Thus, we may label a quantum state by its energy level and electric charge, |q, n , representing the n th lowest energy eigenstate that carries q units of charge.We also denote that H 0 |q, n = E (q) n |q, n and Q |q, n = q |q, n .Charge conjugation symmetry requires that E n .At vanishing chemical potential, we find that the q-dependent lowest energies follow a hierarchy of charge , increase with |q|.This is caused by the Coulomb repulsive interaction between charged fermions.At finite µ, the energy would be shifted according to the charge, H(µ) |q, n = (E from below, the ground state of the system switches from q = 0 to q = 1.By a similar argument, we find the charge of the ground state to be q in the interval (E These discrete energy levels depend on the fermion mass m, and define the lines of the first-order phase transitions in the phase diagram.These boundaries are shown by black dashed curves in all panels of Fig. 2.These boundaries separate different ground states, and we observe different values of entanglement entropy in each state.At low temperature, T = g/10, we observe a clear jump of the total charge across the boundaries, where the charge susceptibility also peaks. Motivated by the peak in the entanglement entropy in the branch around µ = 0, we investigate the phase with vanishing chemical potential in more detail.We show the entanglement entropy, the scalar condensate, and the charge and topological susceptibilities at µ = 0 in Fig. 3.The topological susceptibility is defined as χ CS,2 = ( q 2 CS − q CS 2 ), where q CS = ∂ µ K µ is the Chern-Pontryagin number density associated with the divergence of the Chern-Simons current (K µ ).In (1 + 1) dimensions, K µ = (g/2π)ε µν A ν and q CS = (g/4π)ε µν F µν = (g/2π)E = (g 2 /2π) g/10 g/5 g/3 T = FIG.3: (From top to bottom) Fermion mass dependence of chiral condensate, vector charge susceptibility, chiral susceptibility, and vacuum entanglement entropy, with θ = 0 and µ = 0.In the top three panels, red, blue, and purple curves represent T = g/3, g/5, and g/10, respectively.
We observe that entanglement entropy peaks at m * ≈ −0.11 g, where the chiral condensate also vanishes.Meanwhile, fluctuations of both vector charge and Chern-Simons numbers peak around the same mass m * at low temperatures.The value of m * is consistent with previous calculation [65], which takes the real-time topological susceptibility to detect the critical point.
The coincidence of the critical point of the phase diagram with the peak of the entanglement entropy is known for some models.For example, it is suggested that entanglement measures are suitable tools for analyzing phase transitions in spin systems [70,71].However, as stressed in [70], the discontinuity in entanglement entropy is not always related to a critical point in a phase diagram.
In the µ = 0 branches of the phase diagram in Fig. 2, we observe that entanglement entropy monotonically increases (decreases) with mass for odd (even) charge.This is due to the mass dependence of the charge density in the middle of system, at the boundary of subsystems A and B, see lower panels in Fig. 4. For finite chemical potential, we see a hint of the crystalline structure discussed in [72][73][74] in Fig. 4.
In a conformal field theory on a finite interval of length L = aN , the entanglement entropy of a subsystem with where c is the central charge.It is known that the massive Schwinger model at critical mass m * is a conformal fixed point and belongs to the universality class of the classical 2d Ising model, in which the central charge is expected to approach c = 1 in the continuum limit.In what follows we study the entanglement entropy between the subsystems A and B are defined as in Fig. 1, where the length of A is taken to be .We study the finite size effects in the entanglement entropy by varying the lattice spacing (a) and number of lattice sites (N ).For each choice of a and N , we first determine the critical mass by maximizing S EE when l = N/2 and then compute S EE,l at different values of l.We refer to Fig. 7 of the Supplemental Material for more details.By fitting the results using Eq.(7), we obtain the central charge (see Fig. 5).It is clear that the l dependence of S EE,l is well-described by (7), and the corresponding central charge is within ∼ 15% from c = 1.We suspect that the deviation from c = 1 is caused by finite size effects, as the continuum limit ag → 0 in Fig. 5 is beyond the reach of our numerical capacity.This is supported by the fact that c seems to approach unity when the lattice spacing decreases and the number of sites increases.
Finally, we study the entanglement spectrum.Suppose that the dimension of the Hilbert space of subsystem A is not larger than that of subsystem B: and the coefficients ξ α (α = 1, • • • , dimH A ) constitute the entanglement spectrum [64].
Our simulations show that the gap in the entanglement spectrum closes at the critical point, i.e. at m * −0.11g, see the upper panel of Fig. 6.We present the results at µ = 0, but we find similar band crossing points in the entanglement spectrum at finite chemical potential as well.The parity symmetry in the Schwinger model is broken if θ = 0, π due to the background electric field, which results in a non-zero gap in the entanglement spectrum.We confirm this by showing in the lower panel of Fig. 6 the entanglement spectrum at θ = π/2.
To summarize, we have performed a quantum simulation of the massive Schwinger model at finite chemical potential and temperature, and have found that the location of the critical point in the phase diagram can be determined from the entanglement entropy and the entanglement spectrum.The entanglement entropy peaks in the vicinity of the critical point, showing conformal behavior.The entanglement spectrum near the critical point exhibits the level crossing phenomenon.We speculate that the entanglement entropy and entanglement spectrum show a similar behavior near the critical points in other, more complicated field theories, including QCD.It would be interesting to confirm these findings in other theories such as holographic QCD [76,77], and to investigate whether the symmetry-resolved entanglement [78] in the Schwinger model can indicate the presence of the critical point as well.

FIG. 2 :
FIG. 2: (From left to right) Density plot of total charge, chiral condensate, charge fluctuation, and ground-state left-right entanglement entropy for different values of fermion mass and chemical potential, with N = 20, L = 4/g, and θ = 0.The entanglement entropy is for the ground state, corresponding to T = 0, whereas the other panels are for T = g/10.The first-order phase transition line (black dashed contours) are overlaid on top of every panel.See text for more details.
and the gaps between the neighbouring energy levels, E

FIG. 4 :FIG. 5 :
FIG. 4: (From left to right) Spatial distribution of chiral condensate (upper panel) and charge density (lower panel) for Q = 1, 2, and 3 states, with shown values of fermion mass.