Two Magnetic Impurities in a Spin Chain

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Notice that the Kondo temperature Tk ∼ N cos−1 (πc/γ) when c → ic. This is similar to ... B 56, 300 (1997). [9] A.A. Zvyagin ... 100A, 108 (1984). [14] K.J.B. Lee ...
Two Magnetic Impurities in a Spin Chain Zhan-Ning Hu∗ and Fu-Cho Pu† Institute of Physics and Center for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100080, China

Abstract

arXiv:cond-mat/9807263v1 [cond-mat.str-el] 18 Jul 1998

In this letter, the Kondo magnetic effect is studied for the XXZ spin chain where the impurities are coupled to the edges of the system. The Hamiltonian of our model can be constructed from the transfer matrix. It is exactly solvable and the exchange constants between the bulk and the boundary impurities are arbitrary. The finite size corrections to the ground state energy are obtained for the magnetic impurities and the boundary condition. The specific heat and the susceptibility contributed by the impurities are derived and the Kondo temperature is given explicitly by the use of the Luttinger-Fermi liquid picture and the Bethe ansatz method. PACS: 75.10.Jm, 71.10.-w, 75.20.Hr, 72.15.Qm, 71.27.+a, 75.30.Hx Keywords: Spin chain; Impurity model; Kondo problem; transfer matrix; Bethe ansatz equation; Finite size correction; Specific heat; Susceptibility; Kondo temperature

∗ †

E-mail: [email protected] E-mail: [email protected]

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It is well known that the spin dynamics of the Kondo problem is equivalent to the dynamics of the spin chain with the magnetic impurities [1]. Much progress has been made recently on the impurity models based on the methods of renormalization-group techniques [2], conformal field theory [3–5] and the integrability investigations. The quantum inverse scattering method and the Bethe ansatz technique have been the powerful tools to study the integrable impurities within the framework of the quantum spin chains. Magnetic impurities play an important role in the strongly correlated electron systems, especially for one dimensional systems. And the impurity usually destroys the integrability when it is embedded into the ‘pure’ quantum chain. Then it is a challenging problem to deal with the impurity effects in the strongly correlated electron systems for the quantum integrable models without losing the integrability. Now several integrable impurity models have been found [6–9] besides the magnetic impurities in the noninteracting metals such as the Anderson impurity model and the multichannel Kondo problem [1,10–12]. As a quantum mechanical model of magnetism, the Heisenberg spin chain turned out to be very fruitful and the integrable impurity problem for this model was first considered in Ref.[13] and generalized to arbitrary spin in Ref. [14]. Besides the natural understanding of completely integrable quantum systems, the quantum method of the inverse problem provides us also a very useful technique to construct the impurity model which preserves integrability. Indeed, this scheme has been adapted in Ref. [6-8]. But, for the periodic boundary systems, some unphysical terms must be present in the Hamiltonian to maintain the integrability, though they may be irrelevant when the impurities are embedded in the system. Notice that the fact that the impurities cut the one-dimensional system into the ‘piece’ when they are introduced and (then) the open boundary systems are formed with the impurities at the ends of the systems. So the integrable impurity models [15–18] can be studied with the use of open boundary conditions. In recent years, various integrable models on the open chains with boundary terms have been investigated (See, for example, references [19-22]). In Ref. [23] the open Heisenberg chain with impurities has been discussed and the integrability is given in Ref. [24] when the two impurities are coupled to the XXZ chain. In the present letter, we will discuss the effects of the magnetic impurities in the XXZ spin chain and the finite size correction contributed by the impurities. Our starting point is the following Hamiltonian H=

N −1 X

y x z σjx σj+1 + σjy σj+1 + cosh ησjz σj+1

j=1

  cosh η z z +J σ1x dxa + σ1y dya + σ1 da cosh c   cosh η z z y y x x σN db +J σN db + σN db + cosh c

(1)

 where the constants J, c and η satisfy the relation J = sinh2 η cosh c/ sinh2 η − sinh2 c . The Pauli operators σjx,y,z act non-trivially only in the jth quantum vector space and dx,y,z , dx,y,z are the spins of magnetic impurities with a b s = 1/2 . This system can been constructed from the transfer matrix [25] n o (2) U (λ) = T rτ K + (λ) Tτ (λ) K − (λ) Tbτ (λ)

where

Tτ (λ) = Lτ b (λ + c) Lτ N (λ) · · · Lτ 2 (λ) Lτ 1 (λ) ,

The L operator is given by

Tbτ (λ) = Lτ 1 (λ) Lτ 2 (λ) · · · Lτ N (λ) Lτ b (λ − c) . Lτ n (λ) =

4 X

wj στj ⊗ σnj

j=1

with w1 = w2 =

1 sinh η, 2

w4 − w3 = sinh λ, 2

w4 + w3 = sinh (λ + η) for the subindex n = 1, 2, · · · , N , a, b being the quantum vector spaces and the impurity sites at both of the ends. The symbol T r denotes the trace in the auxiliary space τ in C 2 . The σ j are the Pauli operators for j = 1, 2, 3 with σj4 being the identity operator. The boundary reflection matrices take their forms as K + (λ) = I, K − (λ) = Lτ a (λ + c) Lτ a (λ − c) where c is an arbitrary constant. Then the Hamiltonian (1) can be obtained from the logarithmic derivative of the transfer matrix U (λ) when the spectral parameter λ → 0 . By the use of the standard Bethe ansatz method, the eigenvalue of the Hamiltonian is M X

sinh η cosh (2λj ) − cosh η j=1   2J cosh η + N −1+ cosh c

E = 4 sinh η

for the Hamiltonian (1) with the Bethe ansatz equation   cosh λj − η2 sinh2 λj + c + η2   cosh λj + η2 sinh2 λj + c − η2 ·

sinh2 λj − c + sinh2 λj − c −

=−

(

η 2  η 2

 )2N +1 sinh λj + η2  sinh λj − η2

M Y sinh (λj − λk + η) sinh (λj + λk + η) . sinh (λj − λk − η) sinh (λj + λk − η)

(3)

k=1

where j = 1, 2, · · · , M with M being the number of down spins. Let [26,27] h i  γ γ = 2 tan−1 cot tanh λ , Φ λ, 2 2 2iγ 1 − e . e2iΓ = iγ e − e−iγ By taking the logarithm of the Bethe ansatz equation (3), we have that   γ γ + 2Φ λj − c, 2Φ λj + c, 2 2 γ + (2N + 1) Φ λj , − Φ (λj , Γ) 2 M X = 4πIj + {Φ (λj − λk , γ) + Φ (λj + λk , γ)}

(4)

k=1

where η = iγ . Ij is an integer or the half-odd integer. Under the thermodynamic limits we can write down the Bethe ansatz equation as the form Z ∞ dµσ (µ) Φ′ (λ − µ, γ) + 2πσ (λ) −∞

where

 γ 1 N = Φ′ λ, + σ (λ) 2 2N

3

(5)

  γ γ σ N (λ) = 2Φ′ λ + c, + 2Φ′ λ − c, 2  2 γ ′ ′ −Φ (λ, Γ) + Φ λ, . 2

The σ (λ) is the distributed function. The σ N (λ) is the effect of the open boundary condition and the magnetic impurities. It will give us the finite size corrections introduced by the two impurities coupled to both of the ends of the system. Fourier transformation on the equation (5) gives us the distributed function as the form σ (λ) =

with

1 1 1 + · πλ 2N (2π)2 2γ cosh γ Z ∞ sinh (πω) σ eN (ω) dωe−i2λω · 2 sinh (πω − γω) cosh (γω) −∞

σ eN (ω) =

4π sinh (πω − γω) · sinh (πω)   sinh (πω − 2Γω) · 1 + 4 cos (2cω) − sinh (πω − γω)

where the second term with the factor 1/ (2N ) denotes the finite size correction due to open boundary impurities. Therefore, the distributed function has the form 1 1 σf (λ) + 2N 2γ cosh πλ γ

σ (λ) = with the finite correction: σf (λ) =

−1 2π + +

Z



dωe−i2λω

−∞

1

γ cosh π(λ+c) γ

+

sinh (πω − 2Γω) sinh (πω − γω) cosh (γω) 1 γ cosh π(λ−c) γ

1 . 2γ cosh πλ γ

It is owing to the magnetic impurities coupled to the ends of the spin chain. The ground state energy per site for our system is Z ∞ E dλ 2 = 2 sin γ N −∞ cosh (πλ) [cos γ − cosh (2γλ)]  2J 1 −1 + cos γ + N cosh c  Z ∞ σf (λ) sin2 γ . +2 dλ cos γ − cosh (2λ) −∞ It is obvious that the ground state energy corresponding to the bulk is exactly same as the periodic case [28]. Its finite correction is, based on the boundary condition, Z ∞ sinh [(π − 2Γ) ω] Eb = |sin γ| dω −1 sinh (πω) cosh (γω) −∞ Z ∞ sin2 γ . + dλ cosh (πλ) [cos γ − cosh (2γλ)] −∞ The contribution given by the magnetic impurities has the following form:

4

Ei =

Z

4 sin2 γ cosh (πλ) [cos γ − cosh (2γλ − 2c)]





−∞

+

2 sin2 γ . sinh2 c + sin2 γ

Obviously, when c = 0, our model degenerates to the XXZ spin chain with the free boundary condition. In the following section, we shall study further the physical properties of the impurity model using the LuttingerFermi theory. We can regard 2πIj /N in equation (4) as the momentum of the jth quasi-particle. Similarly as in the periodic boundary case, the energy E takes its minimized value when λ → ∞ . Then the Fermi velocity of the system can be defined as ε′ (λ) vF = d 2πσ (λ) λ→∞ where the symbol εd denotes the dressed energy of the model (See reference [29]). Now we set σf = σi + σb with the relations 1

σi (λ) =

σb (λ) =

γ cosh

−1 2π +

Z

π(λ+c) γ





−∞

+

1 γ cosh π(λ−c) γ

,

e−i2λω sinh (πω − 2Γω) sinh (πω − γω) cosh (γω)

1 , 2γ cosh πλ γ

describing the finite corrections of the distributed function contributed by the magnetic impurities and open boundary condition, respectively. The state density of the system at the Fermi surface can be expressed as [30–32,15]  1 σi (λ) 1 1+ N (∞) = 2πvF 2N σ0 (λ)  1 σb (λ) . + 2N σ0 (λ) λ→∞

Therefore, the low temperature specific heat contributed by the magnetic impurities has the expression: Ci =

πc 1 γ cosh γ T. 3N sin γ

where T denotes the temperature of the system. The correction to the susceptibility is   4 πc χ0 χi = cosh N γ where χ0 is the susceptibility in the bulk. The Kondo temperature for our system is given by Tk =

1 N 2π cosh πc γ

since the Kondo temperature is nothing but the effective Fermi temperature in the local Fermi-liquid theory [33]. Notice that the Kondo temperature Tk ∼ N cos−1 (πc/γ) when c → ic. This is similar to the result for the XXX Heisenberg spin chain and supports also the result obtained earlier in Ref. [34], that there is a crossover from an exponential law to an algebraic law when the coupling constant between the host spins and the impurities changes for the Kondo temperature. In the above discussion, the impurity contributions are obtained in the thermodynamic limits since the Bethe ansatz equations can be written down as the integral equations which may be solved by a Fourier transformation. It is an interesting problem to discuss the impurity effects for the finite lattice case. In fact, in Ref. [35] de Vega and Woynarovich have given a method to calculate the leading-order finite size corrections to the ground state energy of 5

the model which is soluble by the Bethe ansatz. The situations for the XXZ Heisenberg chain and the Hubbard chain are also studied [36,37,29] by using the finite size scaling technique [38–40]. The impurity effects in different sectors of the present model and its critical properties are also the interesting subjects for further discussions. To conclude, we have studied the solvable magnetic impurity model within the framework of open boundary XXZ Heisenberg chain where the impurities are coupled to the ends of the system. The Hamiltonian of the system with impurities can been derived from the transfer matrix. The finite size correction to the ground state energy is obtained for the contribution of the magnetic impurities. With the help of the Luttinger-Fermi description we get the specific heat and the susceptibility due to the impurities. The Kondo temperature is also given explicitly.

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