Topological Work and the Laws of Thermodynamics

Transcription

1 Topological Work and the Laws of Thermodynamics Yiheng Xu, 1 Ferdinand Evers, 2 and Charles A. Stafford 1 1 Department of Physics, University of Ariona, 1118 East Fourth Street, Tucson, AZ Institut I - Theoretische Physik Universittsstrae 31, D Regensburg Dated: December 22, 2017 We consider entropy and persistent currents induced by topological phases in multiply-connected open quantum systems. The conventional formula for the heat current is shown to be problematic for persistent currents, implying a divergent entropy current as temperature goes to ero, in contradiction to the third law. The apparent paradox is resolved through the inclusion of a topological work term in the first law corresponding to the persistent work done in establishing the topological fields. I. INTRODUCTION The conventional definition of the entropy current is problematic. I conv S II. = I E µi N T ENTROPY The grand canonical potential is given by 1 Ω = k B T 1 Dω, Φ ln 1 + e βω µ dω, 2 where Dω, Φ is the density of states see Appendix A on the Green s function formalism. Note that Dω, Φ is a periodic function of Φ with period Φ 0, defined as Φ 0 h/q. Then, the entropy of the system is S = Ω T = µ,φ Dω, Φsωdω. 3 In this equilibrium configuration, there are two alternative expressions for sω, ] sω = k B [fω ln fω + 1 fω ln1 fω 4 = 1 ω µ fω + k B ln 1 + e βω µ, 5 T where fω is the Fermi function. Several properties of S could be mentioned in the paper, 1. S is a state function, so we write S = Sµ, T, Φ for our model. 2. lim T 0 ST = 0 holds for a fully open quantum system. 3. lim T Sµ, T, Φ = const, for a system whose spectrum is bounded above, which demonstrates that the entropy at high temperatures is independent of µ and Φ. A. First Law The thermodynamic formula for the grand canonical potential is Ω = E T S µn. 6 The mean number of particles N and the average energy E are specified as A λ = ω λ Dω, Φfωdω, 7 where N = A 0 and E = A 1. Taking the total differentiation on both sides of Eq. 6, together with the persistent electric current I = Ω/ Φ and N = Ω/ µ, yields T ds = de µdn + IdΦ. 8 In the left panel of Fig. 1, the first law is exactly T ds = de µdn + IdΦ for the paths 1 4 and 2 3 while it is reduced to T ds = de µdn for the paths 1 2 and 3 4. The electric work done in the benene ring model at a fixed temperature, such as T = 20000K or T = 100K, is related to the area under one of the curves in the right panel of Fig. 1. Moreover, if we take each fermion to be either spin up or spin down, and a straight line with linear charge density τ is perpendicular 2 to and enclosed by the loop in addition to the AB-flux source 3, the first law becomes T ds = de µdn + ζ=, I 0 ζ dϕ ζ 9 see Appendix E on the AB 3 and AC 4 Effects in the first law. Here, the extensive variables, i.e., entropy S, energy E, and particle number N include contributions from the two spin orientations of the particles; I 0 and I 0 stand for the particle currents for spin up and spin down respectively. Also, ϕ and ϕ are related to the AB phase ϕ AB t = qφt/ and the characteristic AC phase ϕ AC t = µτt/ c 2 ɛ 0 by ϕ ϕ AB +ϕ AC and ϕ ϕ AB ϕ 5 AC.

2 2 B. Third Law IV. CONCLUSION Generically, the third law of thermodynamics is { lim S = 0, for non-degenerate ground states, T 0 k B ln g, for degenerate ground states 10 where g is the degeneracy of the ground states. In the benene ring model, the entropy always goes strictly to ero because Dω, Φ is everywhere defined along the real axis of ω see Appendix D. Fig. 2 demonstrates the nice convergence of the entropy. III. EQUILIBRIUM CURRENTS The persistent particle current I N and the persistent energy current I E are I λ = ω λ Dω, Φfωνω, Φdω, 11 where I = qi 0 = qi N and I E = I 1. The group frequency νω, Φ is defined in Appendix B, and is equal to the group velocity divided by the circumference for a 1D ring. At the low temperature limit, the conventional formula gives µ lim T IS conv = ω µ Dω, Φνω, Φdω. 12 T 0 That is to say, lim T 0 Iconv S ±, 13 except specific values of Φ, so the result is paradoxical with respect to the third law. In addition, Fig. 3 shows the existence of the paradox in the benene ring model. By analogy with A λ and I λ, we propose the new persistent entropy current to be Equivalently, where I Ω k B T I new S = I new S = 1 T Dω, Φsωνω, Φdω. 14 I E µi N I Ω, 15 Dω, Φνω, Φ ln 1 + e βω µ dω, 16 I Ω is the electric work induced current. The numeric analysis of the new I S is given in Fig. 4. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy under Award No. DE-SC F. Evers acknowledges support by the German Science Foundation. Appendix A: Green s Function Formalism The one-body retarded Green s function is 6 Ĝ R ω, Φ = ωî Ĥ ˆΣ R 1. A1 The operator Ĥ stands for the one-body Hamiltonian, Ĥ = p qat 2 2m + U r. A2 Suppose we represent the Wannier function of the nth band and centered at the lattice vector R as r n, R = W n r, R 7, the Hamiltonian is 7,8 Ĥ = n, R U n n, R n R + n, R + α t n e i q At α n, R, A3 R, α where U n and t n are the average potential and the hopping parameter associated with the nth band respectively 7, and α is a set of vectors pointing from R to its nearest neighbors 7. On the other hand, the self-energy ˆΣ R is written as ˆΣ R ω = i 2 ˆΓ A4 in the broad-band limit, with ˆΓ = ˆΓ. Finally, the density of states is 6 Dω, Φ = 1 {ĜR } 2πi Tr ω, Φ ĜA ω, Φ, A5 [ĜR ]. where ĜA ω, Φ = ω, Φ We define Appendix B: Circulation Frequency 1. Continuous Spectrum χω, Φ ω Dω, Φdω. B1

3 3 Using integration-by-parts technique toward the grand canonical potential leads to Ω = k B T = [ χω, Φ ω ] ln 1 fω dω χω, Φfωdω. Therefore, the persistent electric current is obtained, I = Ω Φ = µ,t 1 χω, Φ qfωdω q Φ B2 B3 B4 Notice that the circulation frequency νω, Φ can be defined implicitly via I qfωdω, Φνω, Φdω. Together with the formula for I, we have νω, Φ = B5 1 χω, Φ qdω, Φ Φ. B6 2. Discrete Spectrum Suppose Ĥ Ψ n = ε n Ψ n and P p q At, then the one-body group velocity is Also, Feynman-Hellman theorem states that ε n Φ = Ψ n Ĥ Φ Ψ n. B8 The circulation frequency defined as ν n v n /L is given by ν n = 1 q ε n Φ. Appendix C: Benene Ring Example B9 For the benene ring model, we assume that there is only one band and At = Φt/L L: the circumference of the benene molecule. The matrix elements of Ĥ are U 0, for ij = 11, 22,..., 66; te i πφ 3Φ 0, for ij = 21, 32,..., 65, 16; Ĥ ij = πφ i 3Φ te 0, for ij = 12, 23,..., 56, 61; 0, otherwise. C1 In addition, the lead is coupled to the six atomic sites of the ring uniformly, ˆΣ R = i 2 ΓÎ. C2 Therefore, with the knowledge in Appendix A, the density of states is Dω, Φ = 6 π KωQω LωP ω, Φ P ω, Φ 2 + Qω 2. C3 v n = Ψ n Ĥ P Ψ n. B7 The expressions for the four functions K, L, P and Q are given by Kω = xx 2 3y 2 t 2 x 2 y 2 3t 2 2xy 2 3x 2 y 2 t 2, C4 Lω = 2x 2 yx 2 3y 2 t 2 + y3x 2 y 2 t 2 x 2 y 2 3t 2, P ω, Φ = x 2 x 2 3y 2 3t 2 2 y 2 3x 2 y 2 3t 2 2 4t 6 cos 2 π Φ Φ 0 C5 C6 and Qω = 2xyx 2 3y 2 3t 2 3x 2 y 2 3t 2, C7 where x ω U 0 and ] y Γ/2. It can be shown that [P ω, Φ 2 + Qω 2 cannot be ero, so Dω, Φ is defined everywhere along the real axis of ω. Appendix D: AB and AC Effects in the First Law Let s set up an orthogonal Cartesian coordinate system such that a closed physical contour C, along which the spin-half, charged, non-interacting and identical fermions

4 4 are constrained to move, lies in the = 0 plane; a solenoid and a straight line with linear charge density τ are perpendicular 2 to the = 0 plane but enclosed by C. Evidently, the AB phase is 3,9 ϕ AB t = q C A d x = q Φt. D1 In terms of the AC effect 4, the corresponding phase operator is 9 ˆϕ AC = 1 E c 2 µ d x, D2 C where the magnetic moment operator satisfies µ = µ σ and σ = σ r, σ φ, σ. Together with the special configuration of the source of the electric field, we have ˆϕ AC = µτt c 2 ɛ 0 σ. D3 The two eigenstates of σ are with eigenvalue +1 and of 1, so we define the characteristic AC phase to be the eigenvalue of for ˆϕ AC, expressed as ϕ AC t = µτt c 2 ɛ 0. The first law of thermodynamics is T ds = de µdn + ζ=, qi 0 ζ dφ + D4 P de, D5 where q is the charge of each fermion, and denotes the statistical and quantum average. The electric dipole moment operator P in the lab frame is 10 P = 1 v µ. c2 D6 Note that v is the velocity operator, specified as v = ṙˆr + r φ ˆφ, D7 and the magnetic dipole moment operator is related to the Pauli matrices by µ = µ σ = µ σ r ˆr + σ φ ˆφ + σ ẑ. D8 Therefore, the r component of the dipole operator is P r = µ c 2 r φσ. D9 Together with the fact that the electric field has only an r component, E r = τt 2πɛ 0 r 1, D10 the dipole interaction Hamiltonian is given by P de = P r de r = µ c 2 r φσ 1 r 1 dτ = µ 2πɛ 0 2πɛ 0 c φσ 2 dτ. D11 Plus, we can define a circulation frequency operator as ν φ/2π, so the last equation becomes P d E = µdτ ɛ 0 c 2 νσ. D12 Now, the second version of the first law is obtained, T ds = de µdn + I σ dφ+ µdτ ɛ σ 0 c 2 νσ. D13 To solve for the trace, let s take { n } to be the eigenstates of ˆν, i.e., ˆν n = ν n n. Then, the spatial state n and the spin state σ collectively specify the one-body state, expressed as nσ = n. D14 σ n Notice that we may construct νσ in the basis of nσ, leading to the expectation value νσ = nσ ˆνˆσ nσ = nσ N n=1 n ˆν n σ n σ n ˆσ D15 where N represents the total number of circulating fermions. As a consequence of N = N + N, the trace can be split into the spin-up part and the spin-down part, νσ = N m=1 N m ˆν m k ˆν k = I 0 I 0, D16 k=1 where I 0 stands for the particle current. Finally, the third version of the first law of thermodynamics is T ds = de µdn + I σ dφ + µdτ ɛ σ 0 c 2 I 0 I 0. D17 Together with the AB and AC phase, the first law can be yielded. Appendix E: Canonical Ensemble For a thermodynamically and quantum-mechanically closed N-particle system occupying a state r, the total energy is E r = N i=1 ε n i, leading to the partition function the probability function and the free energy Z = r e βer, P r = 1 Z e βer. F = k B T ln Z. E1 E2, E3

5 5 1. Third Law 2. Conventional Entropy Current The entropy S is The persistent particle and energy currents are I λ = N P r ε λ n i ν ni. E6 r r i=1 S = F T. Φ E4 Next, the chemical potential is µ = F N, T F N 1, T. E7 We assume no degeneracy on the Nth largest one-body energy level, ε 1 ε 2 ε N 1 < ε N < ε N+1, E8 The low temperature limit becomes so the conventional formula gives lim T IS conv = T 0 N i=1 ν i ε i ε N. E9 That is to say, lim S = k B ln ground-state degeneracy. E5 T 0 lim T 0 Iconv S ±. E10 1 C. A. Stafford, D. Baeriswyl, and J. Bürki, Phys. Rev. Lett. 79, X.-G. He and B. H. McKellar, Physics Letters B 256, Y. Aharonov and D. Bohm, Phys. Rev. 115, Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, H.-P. Eckle, H. Johannesson, and C. A. Stafford, Phys. Rev. Lett. 87, S. Datta, Electronic transport in mesoscopic systems, Vol. 3. Cambridge University Press, Cambridge, New York, 1995 pp M. P. Marder, Condensed matter physics, 2nd ed. Wiley, Hoboken, N.J, 2010 pp , and N. Byers and C. N. Yang, Phys. Rev. Lett. 7, E. N. Bogachek and U. Landman, Phys. Rev. B 50, S. M. Al-Jaber, X. Zhu, and W. C. Henneberger, European Journal of Physics 12,

6 6 FIG. 1: Left panel: The qualitative plot of the two possible thermodynamic paths, i.e, and 1 4 3, which start from the common initial state 1 with the high temperature T H and the ero flux and end up with the final state 3 with the low temperature T C and non-ero flux. The arrows stand for the evolution of the thermodynamic states. Right panel: The persistent electric current of the system as a function of the magnetic flux at the chemical potential µ = 3eV and the coupling strength Γ = 2.7eV for two different temperatures in the benene ring model. FIG. 2: Left panel: The entropy of the system as a function of the temperature at the chemical potential µ = 0eV and the coupling strength Γ = 2.7eV for two different flux values in the benene ring model. Right panel: The entropy of the system as a function of the temperature at the chemical potential µ = 3eV and the coupling strength Γ = 2.7eV for two different flux values in the benene ring model.

7 7 FIG. 3: Left panel: The heat current of the system as a function of the magnetic flux at the chemical potential µ = 1eV and the coupling strength Γ = 2.7eV for different temperatures. Right panel: The heat current of the system as a function of the magnetic flux at the chemical potential µ = 3eV and the coupling strength Γ = 2.7eV for different temperatures. FIG. 4: The persistent entropy current of the system as a function of the magnetic flux at the chemical potential µ = 3eV and the coupling strength Γ = 2.7eV for different temperatures.