Supplementary material for "Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach"

Supplementary Material for: Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach Martín Zapata1⋆, Federico Finkel2†and Artemio González-López2‡ 1Theory Division, Max Planck Institute of Quantum Optics, D-85748 Garching, Germany 2Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Plaza de las Ciencias 1, 28040 Madrid, Spain ⋆[email protected] , † [email protected] , ‡ [email protected] In this Supplementary Material we present a detailed computation of the fermionic density of the Rindler chain for several values of its parameter. The Rindler chain The chain1(2) with parameters Jn=J0 n N,Bn=0 was introduced in ref. [2]and referred to as the Rindler chain, since its associated spacetime metric ds2=J(x)2dt2−dx2is the Rindler metric (i.e., the metric of two-dimensional Minkowski space as perceived by a uniformly accelerated observer). We shall take, for convenience, J0=1/2, and add a linear magnetic field Bn=bn/N, so that the continuum limit of the chain’s parameters reads J(x) = x 2ℓ,B(x) = bx ℓ, (S1) with ba real parameter. To begin with, ξ(x,ϵ;b) = ℓϵ −bx x=−ξ(x,−ϵ;−b) implies, by eq. (50), that aρ(x,ϵ;b)=1−aρ(x,−ϵ,−b). (In fact, a similar relation can be proved for the discrete average occupation numbers 〈c† ncn〉.) We may therefore assume, without loss of generality, that b⩾0. In what follows we will discuss the three cases b=0 (the original Rindler chain), b=1, and b>1 (the case 0 <b<1 is qualitatively similar to b=0). 1. b=0 From fig. 1and eq. (50) it follows that there is a single depletion interval (for ϵ < 0) or saturation interval (for ϵ > 0) [0, x1(ϵ)], where x1(ϵ) = ℓ|ϵ|. The normalization constant in eq. (35b) is easily computed: A(ϵ)−2=Zℓ ℓ|ϵ| dx p4J2(x)−ϵ2=Zℓ ℓ|ϵ| dx qx2 ℓ2−ϵ2 =ℓarccosh(1/|ϵ|). 1In what follows, equation numbers in arabic refer to the body of the paper (ref. [1]). All other figure and equation numbers refer to this Supplementary Material. Local fermion density in inhomogeneous chains Supplementary Material x B(x)±2J(x) ℓ 1 −1 x B(x)±2J(x) ℓ 2 x B(x)±2J(x) b−1 b+1 ℓ Figure 1: Plot of B(x)±2J(x)for the Rindler chain with b=0 (left), b=1 (center), and b>1 (right). By eq. (36), the equation of the envelopes of the WKB single-particle eigenfunctions is y=p2arccosh(1/|ϵ|)−1/2(x2−ℓϵ2)−1/4Θ(x−ℓ|ϵ|). We shall next compute the approximation (47) to the filling fraction as a function of the Fermi energy ϵF. By the particle-hole symmetry due to the vanishing of the magnetic field (cf. remark 4.4 in the main text), we can restrict ourselves to Fermi energies ϵF⩽0, and write ν(ϵF) = 1 πℓZℓ ℓ|ϵF| dxarccos|ϵF| x/ℓ=1 πZ1 |ϵF| dsarccos|ϵF| s. The integral is easily evaluated by the standard change of variables |ϵF|/s=cosu, followed by an integration by parts, with the result ν(ϵF) = 1 πarccos(−ϵF)+ϵFarccosh(1/|ϵF|). (S2) Using the symmetry (53) we easily deduce that the above formula is actually valid for both positive and negative Fermi energies. In particular, for ϵF=0 (half-filling) we obtain ν(0) = 1/2, as expected. Finally, for ϵF⩽0 the WKB approximation to the fermion density is given by aρ(x,ϵF) = 1 πarccosℓ|ϵF| xΘ(x−ℓ|ϵF|),ϵF⩽0, while (again by eq. (53)) aρ(x,ϵF)=1−aρ(x,−ϵF)for ϵF>0. In other words, for all ϵF∈[−1,1]we have aρ(x,ϵF) = Θ(ϵF) + 1 πarccos−ℓϵF xΘ(x−ℓ|ϵF|). (S3) 2. b=1 In this case B(x)−2J(x)=0, B(x)+2J(x) = 2x ℓ, and hence, according to eqs. (32) and (33), the single-particle spectrum is contained in the interval [0,2]. In this case the WKB eigenfunctions with energy ϵvanish on 2 Local fermion density in inhomogeneous chains Supplementary Material the interval [0,ℓϵ/2]. Moreover, since B(x)−2J(x)is identically zero, the interval [0,ℓϵF/2]is a saturation interval for any Fermi energy ϵF(cf. the discussion at the end of section 4 in the main text). The normalization constant in this case is given by A(ϵ)−2=Zℓ ℓϵ/2 dx qx2 ℓ2−(ϵ−x ℓ)2 =ℓZ1 ϵ/2 ds pϵ(2s−ϵ)=ℓv t2 ϵ−1. The equation of the envelopes of the WKB eigenfunctions is accordingly y=±ℓ1−ϵ 2x−ℓϵ 2−1/4 Θx−ℓϵ 2. Since in this case ξ∗(x,ϵ)=1 inside the saturation interval [0,ℓϵ/2], the WKB approximation to the filling fraction ν(ϵF)is given by ν(ϵF) = 1 πℓ·πℓϵF 2+1 πℓZℓ ℓϵF/2 dxarccosx/ℓ −ϵF x/ℓ =ϵF 2+1 πZ1 ϵF/2 dsarccos1−ϵF s. The integral can be again evaluated by a change of variable followed by integration by parts. After straightforward simplifications we thus obtain ν(ϵF) = 1 πarccos(1−ϵF) + ÆϵF(2−ϵF). (S4) Finally, for all Fermi energies ϵF∈[0,2]the WKB approximation to the fermionic density now reads aρ(x,ϵF) = ΘℓϵF 2−x+1 πarccos1−ℓϵF xΘx−ℓϵF 2. (S5) 3. b>1 Finally, for b>1 the plot of B(x)±2J(x)is as presented in fig. 1(right); in particular, in this case the single-particle spectrum lies on the interval [0, b+1]. It is apparent from this figure that the behavior of the filling fraction ν(ϵF)and the fermionic density ρ(x,ϵF)depends on whether ϵFis less or greater than the critical value B(ℓ)−2J(ℓ) = b−1. The same is true for the normalization constant A(ϵ). For instance, for 0 ⩽ϵ < b−1 the two turning points are x1(ϵ) = ℓϵ b+1,x2(ϵ) = ℓϵ b−1, (S6) and therefore A(ϵ)−2=Zx2(ϵ) x1(ϵ) dx Çx2 ℓ2−ϵ−bx ℓ2=ℓ pb2−1Zx2(ϵ) x1(ϵ) dx qx−x1(ϵ)x2(ϵ)−x=πℓ pb2−1. In particular, the equation of the envelopes of the WKB wave functions is in this case y=±v t2 πx−x1(ϵ)x2(ϵ)−x−1/4. 3 Local fermion density in inhomogeneous chains Supplementary Material b=0 -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ϵF ν(ϵF) b=1 0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 ϵF ν(ϵF) b=2 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 ϵF ν(ϵF) Figure 2: Filling fractions ν(ϵF)for the Rindler chain with b=0,1,2 and N=400 spins (blue markers) compared to their WKB approximations (S2), (S4), and (S7) (continuous red line). In particular, note the linear growth of ν(ϵF)with b=2 over the interval 0 ⩽ϵF⩽b−1=1, in full agreement with eq. (S7a). On the other hand, for b−1⩽ϵ⩽b+1 there is a single turning point x1(ϵ), and hence A(ϵ)−2=Zℓ x1(ϵ) dx Çx2 ℓ2−ϵ−bx ℓ2=ℓ pb2−1Z1 ϵ b+1 ds qs−ϵ b+1 ϵ b−1−s =ℓ pb2−1π 2+arcsinb2−1 ϵ−b. The equation of the WKB envelopes is now y=±p2π 2+arcsinb2−1 ϵ−b−1/2x−ℓϵ b+1 ℓϵ b−1−x−1/4 . Let us turn next to the WKB approximation to the filling fraction. From fig. 1 (right) it is apparent that ξ∗(x,ϵF)=1 for 0 ⩽x⩽x1(ϵF), while ξ∗(x,ϵF)=−1 for x2(ϵF)⩽x⩽ℓwhen ϵF∈[0, b−1]. We thus have ν(ϵF) = x1(ϵF) πℓ+1 πℓZmin(x2(ϵF),ℓ) x1(ϵF) dxarccosb−ℓϵF x =ϵF π(b+1)+ϵF πZπ t0(ϵF) dttsin t (b−cos t)2, where t0(ϵ)=0 for 0 ⩽ϵ⩽b−1 and t0(ϵ)=arccos(b−ϵ)for b−1⩽ϵ⩽b+1. Integrating by parts and simplifying the boundary term we obtain ν(ϵF) = ϵF π t0(ϵF) b−cos t0(ϵF)+ϵF πZπ t0(ϵF) dt b−cos t =ϵF π t0(ϵF) b−cos t0(ϵF)+2ϵF πpb2−1arctanÇb+1 b−1tant 2 π t0(ϵF) =ϵF π t0(ϵF) b−cos t0(ϵF)+ϵF pb2−1−2ϵF πpb2−1arctanÇb+1 b−1tant0(ϵF) 2. Hence for 0 ⩽ϵF⩽b−1 we have ν(ϵF) = ϵF pb2−1, (S7a) 4 Local fermion density in inhomogeneous chains Supplementary Material while for b−1⩽ϵF⩽b+1 ν(ϵF) = ϵF pb2−1+1 πarccos(b−ϵF)−2ϵF πpb2−1arctanr(b+1)(1−b+ϵF) (b−1)(1+b−ϵF). (S7b) b=0, ν=1/4 0 100 200 300 400 0.0 0.1 0.2 0.3 0.4 0.5 n 〈cn †cn〉 0 100 200 300 400 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 n ϕn b=1, ν=1/2 0 100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 n 〈cn †cn〉 0 100 200 300 400 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 n ϕn b=2, ν=1/2 0 100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 n 〈cn †cn〉 0 100 200 300 400 -0.10 -0.05 0.00 0.05 0.10 0.15 n ϕn b=2, ν=3/4 0 100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 n 〈cn †cn〉 0 100 200 300 400 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 n ϕn Figure 3: Left column: local fermion density for the Rindler chain with N=400 spins, and several values of the parameter band the filling fraction ν(blue markers) compared to their WKB approximations (S3), (S5), and (S8) (red line). Right column: plot of the corresponding single-particle eigenfunctions φn(ϵ)with energies ϵ=Nν(in blue) compared to their WKB envelopes (red lines). 5 Local fermion density in inhomogeneous chains Supplementary Material Finally, from fig. 1(right) and eq. (50) we deduce that aρ(x,ϵF) = 1, 0 ⩽x⩽x1(ϵF) = ℓϵF b+1, for Fermi energies ϵF∈[0, b+1]. Moreover, for 0 ⩽ϵF⩽b−1 we have ρ(x,ϵF) = 1 πaarccosb−ℓϵF x,ℓϵF b+1⩽x⩽ℓϵF b−1, 0, ℓϵF b−1⩽x⩽ℓ,(S8a) while for b−1⩽ϵF⩽b+1 the depletion interval at the chain’s right end disappears, i.e., aρ(x,ϵF) = 1 πarccosb−ℓϵF x,ℓϵF b+1⩽x⩽ℓ. (S8b) As can be seen from figs. 2and 3, the agreement of the WKB approximations derived above with the numerical results is excellent across the full range of values of the parameter band the filling fraction ν(or the single-particle energy ϵ). References [1]M. Zapata, F. Finkel and A. González-López, Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach, submitted to SciPost Phys. [2]B. Mula, N. Samos Sáenz de Buruaga, G. Sierra, S. N. Santalla and J. RodríguezLaguna, Depletion in fermionic chains with inhomogeneous hoppings, Phys. Rev. B 106, 224204(10) (2022), doi:10.1103/PhysRevB.106.224204. 6