Phase Diagram of One-Dimensional Bosons in an Array of Local Nonlinear Potentials at Zero Temperature

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1 Commun. Theor. Phys. (Beijing, China) 36 (001) pp c International Academic Publishers Vol. 36, No. 3, September 15, 001 Phase Diagram of One-Dimensional Bosons in an Array of Local Nonlinear Potentials at Zero Temperature SHI Yun-Long, 1, ZHANG Yu-Mei, 1 CHEN Hong 1 and WU Xiang 1 1 Pohl Institute of Solid State Physics, Tongji University, Shanghai 0009, China Department of Physics, Yanbei Normal Institute, Datong , Shanxi Province, China (Received August 4, 000; Revised February 7, 001) Abstract The Gaussian wave functional method is applied to a boson system with an array of local nonlinear potentials cos[βφ(nr)] to study the phase diagram of its ground state. The stable ground state is determined by the renormalized mass µ which is a function of the parameter γ = β /4π, the strength of potential α and the potential concentration c. In different cases γ < γ 1, γ 1 < γ < γ and γ > γ, µ can have different multiplicities, the phase diagram in parameter space is thus depicted. The value γ = γ 1 depends on the concentration c, for c 0, it coincides with that of the single impurity model; while γ = coincides with the conclusion of the continuous model. PACS numbers: 7.10.Fk, Qm, d Key words: one-dimensional electrons, doping, phase transition 1 Introduction The phase transition problem in one-dimensional (1D) quantum sine-gordon system at zero temperature attracts much attention. It is known that there are mainly two kinds of 1D sine-gordon models. One is the global sine-gordon model where the nonlinear potential exists in the whole space. The global sine-gordon model has been used to describe various 1D quantum systems and two-dimensional (D) classical systems, e.g., the S = 1 XXZ and S = 1 Heisenberg spin chain, [1,] the interacting model of 1D electron gases, [3] the 1D Kondo lattice model, [4] and the D classical Coulomb gas or D classical X-Y model. [5,6] For 1D quantum systems, the nonlinear potential comes from large momentum scattering occurring globally in systems, such as the backward scattering due to Coulomb interaction in 1D electron systems and the Kondo scattering due to many magnetic impurities in Kondo lattice model. The other is the local sine-gordon model where the nonlinear potential is introduced only at one space point, x = 0 for example. The local sine-gordon model has been used to describe single impurity behavior in 1D strongly systems, e.g., single impurity scattering in Luttinger liquid. [7 9] The global sine-gordon model is characterized by a renormalized mass µ g, which is the gap of low-energy excitation. The past works have shown that when the renormalized mass µ g 0, the system experiences a phase transition from the strong correlation to the weak correlation and the phase boundary is at β = 8π [10 1] (β is connected to the electron-electron interaction in 1D electron system). While for the local sine-gordon model, there also exists a renormalized local mass µ l. When the renormalized mass µ l 0, the low-energy behavior of the system will similarly changes from the local strong correlation to the weak correlation, but the phase boundary is at β = 4π. [7 9,13] Because impurities or defects are usually present in materials, the 1D space will no longer be homogeneous for the electron dynamics. It is known that 1D fermion model can be mapped into 1D boson model. [14] With the increase of the impurities (or defects) concentration, the local sine-gordon system can also transit to the continuous quantum sine-gordon system. The value of impurity concentration will directly affect the potential scattering action of system and the coherency. For the multi-impurity model, we believe that there also exists a renormalized mass µ, which can determine the division of the phase regions. The phase boundaries of the system will be shifted with the change of doped concentration. Therefore, it is desirable to investigate physical implication of the system s phase diagram which varies with impurity concentration from the single-impurity behavior until to the many- (or continuous-)impurity behavior and to study the low-energy physics of the multi-impurity model. In this paper the behavior of the ground state of the boson model with a periodic array of local nonlinear potentials is studied by using the Gaussian wave functional method. The model can be regarded as originating from the Kondo lattice model in a special case. The latter is proposed for densely doped magnetic impurities or pure compounds. [4,15] The renormalized local mass equation which depends on the impurity concentration is derived, and the phase diagram in the parameter plane is depicted The project supported by National Natural Science Foundation of China, Natural Science Foundation of Shanxi Province, and the Foundation for Young Leaders of Disciplines in Science from Education Commission of Shanxi Province of China

2 376 SHI Yun-Long, ZHANG Yu-Mei, CHEN Hong and WU Xiang Vol. 36 by using the mass equation for different impurity concentrations, and also the relation between lattice impurity model and the two extreme models is described. Model and Variational Treatment We study the boson model with a periodic array of local nonlinear potential. Its Hamiltonian is H = 1 [Π (x) + ( φ(x)) ]dx + α β V (x)[1 cos βφ(x)]dx (1) with V (x) = n=0,±1, δ(x nr), () where R stands for the spacing between neighboring potentials and α denotes the strength of the periodic scattering barrier. The parameter β can be related to the scale of electron-electron interaction in the fermion model. [16] This model can be derived from the 1D Kondo lattice model or the orbital Kondo array model when the longitudinal exchange takes a special value. [4,15] The details of the variational procedures have been given in Ref. [17]. Here we simply sketch the results. We will simulate the ground state and the low-lying excitations of Eq. (1) by the following solvable model Hamiltonian with the same spatial structure, H M = 1 {Π (x) + [ φ(x)] + µv (x)[φ(x)] }dx. (3) Naturally, µ can be viewed as a renormalized mass, similar to what has been done in the conventional sine-gordon model [1] and the local sine-gordon model, [13] and µ will be determined variationally. The Hamiltonian (3) takes a bilinear form, its ground state is a Gaussian wave functional, Φ 0 (µ) { = N 1/ exp 1 } dxdx φ(x)k 1 (x,x )φ(x ), (4) in which the kernel is defined as dyk(x,y)k 1 (y,x ) = δ(x x ), where K(x,y) can be expressed by the eigenmode of Eq. (3) as K(x,y) = k 1 ε k ψ k (x)ψ k (y). (5) The expectation value of the original Hamiltonian (1) in the trial ground state (4) can easily be evaluated as E(µ) = Φ 0 (µ) H Φ 0 (µ) = Φ 0 (µ) H M Φ 0 (µ) + Φ 0 (µ) H H M Φ 0 (µ) = E 0 (µ) 1 4 µnk α β N e β K/4, (6) where E 0 (µ) denotes the zero point energy of H M and N is the total number of the potential point in the whole space L. K can be written as K = K(nR,nR) = K(0,0) = 1 π ΛR 0 dz z + (µr)z coth z + (µr) /4, (7) where Λ is a large momentum cutoff. Now the variational procedures for the Gaussian wave functional can be applied. [18,19] The minimal energy condition d E(µ) = 0 (8) dµ can be evaluated by using the following relation d dµ E 0(µ) = Φ 0 (µ) d dµ H M Φ 0 (µ) = 1 NK, (9) 4 yielding the mass equation ( µ = α exp 1 ) 4 β K = α exp { β 4π ΛR 0 dz }.(10) z + (µr)z coth z (µr) Correspondingly, the zero point energy E 0 (µ) can be integrated by Eq. (9), therefore E(µ) in Eq. (6) can be written as E(µ) = N πr + ΛR 0 1 ln{ µr + z coth z z + (µr)z coth z (µr) }dz 1 4 µnk α β N e β K/4 + C. (11) Thus the ground state energy E(µ) in Eq. (6) appears as an explicit function of µ which will be solved from the mass equation (10). 3 Renormalized Mass and Phase Diagram The renormalized mass µ is directly determined by the impurity scattering intensity α, the parameter γ(= β /4π) and the impurity concentration c(= (ΛR) 1 ). The phase diagram in the parameter plane is in turn determined by the renormalized mass of the stable ground state. In fact, the ground state is only referred to the case of E(µ) being a local minimum, so the stability condition is constrained by d E ( dµ = 1 4 N dk ) dk µβ 0, (1) 4 dµ dµ

3 No. 3 Phase Diagram of One-Dimensional Bosons in an Array of Local Nonlinear Potentials at 377 and the critical condition then follows dk µβ 4 dµ = 0. (13) From Eq. (10) one can draw constant mass curves in (α/λ) γ plane. The stability boundary of this family of curve can be determined by Eqs (10) and (13). In Fig. 1 we show these boundaries for c = by numerical calculation, but the constant mass curve is not included for simplicity of the figure. The parameter plane (α/λ) γ is divided into six regions. In different regions the constant mass curves have different overlappings corresponding to non-uniqueness of µ under the condition of the extremum of the system energy. Thus with the decrease of the renormalized mass, the constant mass curves show a two-stage behavior, they moves at first around γ = γ 1 and eventually approach to the phase boundary γ = which clearly suggests the crossover from the model of single impurity to that of continuous impurity. Correspondingly, the critical condition (13) becomes 1 µ ) [ 1 + 8( Λ 4 + c γ ] µ ( 4 (1 c) Λ + 1 γ ) c = 0. (16) From Eq. (16) we find that according to the situation of solution for µ three different cases γ < γ 1, γ 1 < γ < γ and γ > γ can be distinguished with γ 1 = 1 + c(1 3c) + (1 c) c (1 c), γ =. (17) The first separation line at γ = γ 1 depends on the impurity concentration c, with decrease of c it shifts to left. When c tends to zero (corresponding to approaching the exact single impurity model), γ 1 consistent with the known results. [8] Here γ coincides with the point of the second-order phase transition of vanishing renormalized mass µ of the continuous sine-gordon model. [13] In Fig. 1 equation (10) is solved numerically for µ, one finds that the above approximation in Eqs (14) (17) is qualitatively correct. (i) When γ < γ 1, in the whole interval of α/λ (corresponding to the region A of Fig. 1), µ has only one solution corresponding to the minimum of the energy E(µ), which can be demonstrated clearly by the curves E µ (Fig. ) for constant scattering strength by numerically solving Eq. (11) for c = and γ = 0.5. In this region the mass is always nonzero. Fig. 1 Phase diagram based on constant mass curves for c = The first and second vertical lines correspond to γ = γ 1, γ respectively. Before the numerical calculation being carried out, the phase boundary can also be determined by using the approximate analytic results from the critical condition of Eq. (13). To this end we adopt { z coth z 1, z 1, = (14) z, z 1. Thus, expression of K in Eq. (7) now reduces to K = 1 π (ΛR + µr/) ln + O(µR). (15) µr + (µr) /4 Fig. Energy E as a function of renormalized mass µ for c = 0.001, γ = 0.5, α/λ = 1, 10 and 100. (ii) When γ >, for any impurity concentration there exists one critical value α. When α < α, only the state corresponding to µ = 0 is stable, which can be shown by the curve E µ for α/λ = 1 (Fig. 3) by numerically

4 378 SHI Yun-Long, ZHANG Yu-Mei, CHEN Hong and WU Xiang Vol. 36 solving Eq. (11) for c = 0.001, γ =.5. This region (region F of Fig. 1) corresponds to the zero mass region of the continuous model. When α > α, µ has two nonzero solutions corresponding to the extremum of the energy E(µ). The larger value µ L of µ corresponding to the stable state, while the smaller value µ S of µ to the unstable state, and µ = 0 is also a stable state. Therefore, the region E of Fig. 1 contains an equilibrium line E(µ L ) = E(0) of the first-order phase transition (dashed line in the region E). Above this line, µ = 0 represents the metastable state, below the phase boundary, µ = 0 is the stable state. This situation is shown by the E µ curves in Fig. 3 for α/λ = 100 and The boundary between regions E and F in Fig. 1 is given by the critical value α determined by the combination of Eqs (10) and (13). Obviously, this phase transition belongs to the first order, namely µ 0 uncontinuously on the boundary. Meanwhile the phase boundary between regions B and F corresponds to the second-order phase transition, namely µ 0 continuously on the γ = boundary. energy minimum, as shown by the α/λ = 100 curve in Fig. 4. This region (region C of Fig. 1) corresponds to the rarefied impurity behavior. When α 1 < α < α (region D of Fig. 1), every α value has three different nonzero µ values corresponding to the extremum of the energy E(µ). Among them the largest µ and the smallest µ correspond to the minimum of energy, representing the stable state or metastable state of the system, while the middle value corresponds to the maximum of energy, or unstable state, as shown by the α/λ = 40 curve in Fig. 4. The multiplicity of µ value is demonstrated by the (µ/λ) (α/λ) curve in Fig. 5 for γ = 1.5 and different values of c. The critical α 1 and α are given by the combination of Eqs (10) and (13), they establish the boundaries between regions C, D and B, D in Fig. 1. Using the approximation (16) we can estimate the right terminals of the boundaries, which is at α/λ 4/c for C-D boundary and at α/λ 3(1 c) for B-D boundary. Based on the above conclusion, the parameter plane between γ 1 and γ = can be divided into three regions: the region of the dense impurity behavior, i.e. the liquid-like region (region B), the region of rarefied impurity behaviour, i.e. gas-like (region C) and the coexisting region (region D). Fig. 3 Energy E as a function of renormalized mass µ for c = 0.001, γ =.5, α/λ = 1, 100 and (iii) When γ 1 < γ < γ, for any impurity concentration c there exist two critical values α 1 and α. When α < α 1, µ has only one nonzero value corresponding to the energy minimum, which is shown by the α/λ = 1 curve in Fig. 4 by numerically solving Eq. (11) for c = and γ = 1.5. This region (region B of Fig. 1) coincides with the massive region of the continuous model for small µ, therefore it corresponds to the dense impurity behavior. When α > α, µ has also only one nonzero value corresponding to the Fig. 4 Energy E as a function of renormalized mass µ for c = 0.001, γ = 1.5, α/λ = 1, 40 and 100. In Fig. 5, the rule of equal area can be used to determine which one among three µ values in the mixed regions corresponds to the stable state. For the example of c = 0.01 curve, when α > α 0, µ corresponds to the stable state of the system, while µ corresponds to the metastable state, thus the model shows the dense impurity behavior. If the system is positioned at the rarefied

5 No. 3 Phase Diagram of One-Dimensional Bosons in an Array of Local Nonlinear Potentials at 379 impurity region, with the decrease of α it will pass P point, cross the PMQ mixed region, then jump to Q point of the dense impurity region, which is similar to the gas-liquid transition. The exact positions of P and Q can be determined by the condition that the two areas circled by the PMQ line and PMQ curve are equal. Fig. 5 Renormalized mass µ/λ as a function of α/λ for γ = 1.5 and different concentrations. The dashed lines 1,, 3 correspond to α = α 1, α 0, α respectively. In fact, using the minimal energy condition d E(µ) = 0, dµ it can be derived from Eq. (6) that lnα E(µ) = N µ. (18) β If P and Q points exist simultaneously and both satisfy the stable condition of the system, then E P (µ ) = E Q (µ ) should hold, it follows that EQ E P µ d lnα = 0. (19) In this way the exact positions of P and Q can be fixed to determine the corresponding value of α/λ. Therefore, there should exist an equilibrium line E(µ ) = E(µ ) of the first-order phase transition in region D (the dashed line in region D). On the equilibrium line both phases are stable. Below this phase line E(µ ) < E(µ ) and above the line E(µ ) < E(µ ). The energy extremum and the renormalized mass value of the system of the corresponding position above the phase boundary are shown by the α/λ = 40 curve of Fig Conclusion and Discussion In this paper we have discussed the phase transition of the boson system with a periodic array of local nonlinear potentials using the Gaussian wave functional method. The results show that the ground state and low-lying states of the impurity lattice system can be described by the renormalized mass µ in the α/λ γ space. According to the number and type of stable solution for µ, we can divide the parameter plane into several regions. In regions A, B and C there is only one nonzero value for µ respectively. The upper-left part extends continuously to the limit of the single impurity model, while the lower-right part approaches to the continuous impurity model. The regions C and B are not directly connected, but a continuous change of µ is possible via region A. Both solutions in regions B and C extend into region D where two stable solutions exist, and an equilibrium line appears, on which they are equally stable. Beyond this line one solution is stable and the other is metastable. The equilibrium line extends over γ = to region E, where there are also two stable solutions for µ, however, here one solution is µ = 0. Finally, in region F only µ = 0 is a stable solution. From the above conclusion, it can be seen that the ground state energy has different analytic forms in different regions of the parameter plane. When the parameters change, the change of the energy over the phase boundary is discontinuous or not smooth, namely the system experiences a quantum phase transition at 0 K and the change of the correlation function (i.e. the derivative of the energy) over the phase boundary is not continuous. Therefore, the low energy behavior of the model can be divided into three phases. One is the condensed phase, corresponding to the vanishing renormalized mass µ = 0 (similar to the dipole moment phase of D Coulomb gas at low temperature. [11] It is well known that D Coulomb gas at finite temperature has a one to one correspondence with the sine-gordon model at 0 K [11] ). It is the only stable state in the region F. The second is the gas-like phase, its behavior extends smoothly to that of single impurity model. In region C it is the only stable region. The third is the liquid-like phase which is similar to the case of continuous sine-gordon model, in region B it is the only stable state. In region D these two phases with different nonzero µ coexist resulting in an equilibrium line, i.e. a region for the first-order phase transition. The equilibrium line ends at γ = γ 1 which turns out to be a critical point, since these two phases can continuously change to each other via region A. On the other hand, the intercept of the first-order transition line in regions D and F and

6 380 SHI Yun-Long, ZHANG Yu-Mei, CHEN Hong and WU Xiang Vol. 36 the second-order transition line between regions B and F appears as a triple point, where all three phases coexist. We notice that when concentration c tends to zero γ 1 1, the constant mass curves at first asymptotically coincide with those of the single impurity model. However, when µ decreases to µ/λ c or µ R 1, these curves deviate, passing by the boundary at γ = 1 they at last approach γ =. This two-stage phenomenon is somewhat different from the behavior of the single impurity model, which cannot be realized in reality. On the other hand, when c increases, region C expands and region B diminishes, but the whole region D shrinks, which means that the difference between the solutions for µ in region B and region C becomes less significant. When c 1/4, γ 1 γ = and the region D completely disappears, the points b and d both lower and coalesce to one point, the picture of the continuum model is thus restored. The above two limiting cases both show a one scale asymptotics. For the impurity lattice model, however, there exist in general two scales µ and R 1. Only when they are well separated, i.e., µr 1 or µr 1, can the asymptotics of the correlation be divided into two regimes, each controlled by one of them. Otherwise the asymptotics cannot be simply described, it may even change discontinuously when the parameters change. In summary, we have applied the Gaussian wave functional method to study the phase transition of the bosons with an array of local nonlinear potentials cos[ βφ(nr)]. We have demonstrated that the stable ground state is determined by the renormalized mass µ which is a function of the parameter γ = β /4π, the strength of potential α and the potential concentration c. In different cases γ < γ 1, γ 1 < γ < γ and γ > γ, µ can have different multiplicities, the phase diagram in parameter space is thus depicted. The value γ = γ 1 depends on the concentration c, for c 0, it coincides with that of the single impurity model, while γ = coincides with the conclusion of the continuous model. References [1] J.D. Johnson, S. Krinsky and B.M. McCoy, Phys. Rev. A8 (1973) 56; A. Luther, Phys. Rev. B14 (1976) 153. [] Hong CHEN, Lu YU and Zhao-Bin SU, Phys. Rev. B48 (1993) 169. [3] J. Voit, Rep. Prog. Phys. 57 (1994) 977. [4] V.J. Emery and S.A. Kivelson, Phys. Rev. Lett. 71 (1993) [5] P. Minnhagen, Phys. Rev. B3 (1985) [6] Guang-Ming ZHANG, Hong CHEN and Xiang WU, Phys. Rev. B48 (1993) [7] A.O. Gogolin, Phys. Rev. Lett. 71 (1993) 995. [8] C.L. Kane and M.P.A. Fisher, Phys. Rev. B46 (199) [9] L.I. Glazman, J.M. Rusin and B.I. Shklovskii, Phys. Rev. B45 (199) [10] S. Coleman, Phys. Rev. D11 (1975) 088. [11] P. Minnhagen, Rev. Mod. Phys. 59 (1987) [1] Bo-Wei XU and Yu-Mei ZHANG, J. Phys. A5 (199) L1039. [13] Bo-Wei XU and Yu-Mei ZHANG, J. Phys. A9 (1996) [14] A.O. Gogolin, A.A. Nersesyan and A.M. Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press, Cambridge (1998). [15] O. Zachar, S.A. Kivelson and V.J. Emery, Phys. Rev. Lett. 77 (1996) 134. [16] V.J. Emery, Highly Conducting One-Dimensional Solids, eds J.T. Devreese, R.P. Evrard and V.E. van Doren, Plenum, New York (1979). [17] Yu-Mei ZHANG, Yun-Long SHI and Hong CHEN, Phys. Rev. B57 (1998) [18] T. Barnes and G.I. Ghandour, Phys. Rev. D (1980) 94. [19] P.M. Stevenson, Phys. Rev. D30 (1984) 171.