Proceedings TH2002 Supplement (2003) 419 – 430
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/040419-12 $ 1.50+0.20/0
Proceedings TH2002
Integrable Models of Disordered Impurities in
Quantum Spin and Correlated Electron Chains
A.A. Zvyagin and B.I. Verkin
Abstract. Exact solutions for several models of random ensembles of magnetic impurities in critical quantum antiferromagnetic chains and multichannel Kondo impurities are presented. We obtain analytic expressions for temperature and magnetic
field dependencies of many important low-energy characteristics of those models.
We show that the only low energy parameter which gets renormalized due to the
disorder is the effective crossover scale connected with each impurity; others are
universal.
The study of the behavior of magnetic impurities coupled to paramagnetic
hosts remains one of the most interesting problems of the many-body physics. The
Kondo effect [1, 2], which describes the exchange interaction between the spin of a
magnetic impurity and those of itinerant electrons, is the best known example in
which modern theoretical methods have manifested their strength. In the last few
years the interest in the non-Fermi-liquid (NFL) behavior of magnetic systems and
metallic alloys has grown considerably. A large class of conducting non-magnetic
materials does not behave as usual Fermi liquids (FL) at low temperatures, T .
One of the best known examples of such behavior is the Kondo effect for multi
(n) channel electron systems: For an impurity spin S < n/2 a NFL critical behavior results [3]. However, for the most of dirty metals and alloys, in which the
NFL behavior was observed (see Refs. [4, 5] and references therein), the magnetic
susceptibility, χ, and low-T specific heat, c, usually manifest logarithmic or weak
power law behavior with T . The resistivity linearly decreases with T with a large
residual part. It is different from the predictions of the theory of the overscreened
Kondo effect. This, together with the alloy nature of mentioned materials, suggests that a disorder (a random distribution of localized f -electrons or a random
coupling to the conducting electron host) can play the main role in the low-T NFL
character of such systems. The change in interactions between impurity sites and
host spins can be considered as a modification of the characteristic energy scale,
the Kondo temperature TK . At this scale the behavior of a magnetic impurity
manifests the crossover from the strong coupling regime (for T, h, TK , h is an
external magnetic field) to the weak coupling regime TK h, T . The impurity
spin behaves asymptotically free in the weak coupling case, and it is screened by
host spins in the strong coupling case. Random distributions of magnetic characteristics of impurities can be connected either with the randomness of exchange
couplings of itinerant electrons with local moments, or with the randomness of
densities of conduction electron states: Both of them renormalize TK . Another
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interesting topic of research, related to the above, is the behavior of random magnetic impurities in one-dimensional (1D) antiferromagnetic (AF) spin chains. It
is known that the physics of a single magnetic impurity in a 1D AF Heisenberg
spin S = 12 chain and that of a single Kondo impurity in a 3D free electron host
are described by similar Bethe ansatz theories [1, 2, 3, 6, 7]. Hence, for a random
distribution of magnetic impurities one can suppose that low dimensionality is not
essential for the Kondo screening. The absence of the magnetic ordering in NFL
Kondo systems [4, 5] also confirms this assumption.
The goal of this presentation is to find exact solutions for the ground state and
thermodynamic characteristics of disordered ensembles (for various distributions
of impurity-host couplings) of magnetic impurities in magnetically uniaxial spin-S
chains in the critical region, i.e., in the domain of values of the magnetic anisotropy,
where excitations of the homogeneous host are gapless; of random ensembles of
multichannel Kondo impurities, coupled locally to the free electron host with the
“easy-plane” magnetic anisotropy of that coupling and to the correlated electron
model. The magnetic anisotropy parameter (where it exists) is assumed to be
homogeneous for host and impurity spins.
The Hamiltonian
of the integrable spin-S chain with spin-S impurities has
the form H = j 2Hj,j+1 + Himp + Himp−imp (the host exchange constant J is
set to 2). In general, the form of the Hamiltonian H is very complicated [8]; it
depends on S, S , the anisotropy parameter γ, and on parameters θi , which define
exchange couplings between each impurity and the host. For the simplest case of
a spin S impurity introduced into the spin 12 Heisenberg chain (γ = 0) we have
j+1 . The impurities’ part of the Hamiltonian has the form (for the
j S
Hj,j+1 = S
j-th impurity situated between sites m and m + 1 of the host) [6, 9, 10, 6]
1
Himp = [θj2 + (S + )2 ]−1 (Hm,imp + Himp,m+1 ) + {Hm,imp , Himp,m+1 } −
2
1
(1)
( + 3S (S + 1))Hm,m+1 − iθj [Hm,imp , Himp,m+1 ] ,
4
where [., .] ({., .}) denotes the (anti)commutator. The case θj = 0 and S = 12
corresponds to the inclusion of an additional site to the host. For θj → ∞ an
impurity spin is totally decoupled from the host. If two impurities are situated
between two neighboring host sites, they can interact directly, without destroying
integrability, e.g., for the
case the impurity-impurity part of the Hamil isotropic
j
Hj,j+1 . These impurity-impurity couplings can
tonian is Himp−imp = j θj2 Jimp
model the Ruderman-Kittel-Kasuya-Yosida interaction between impurities in concentrated metallic alloys. Here we shall study the case of a small number of such
neighboring impurities.
Any number of described above impurities can be independently incorporated
into the host. Each of them will be characterized by its own coupling to the host,
i.e., by its own θj . The lattice Hamiltonian has additional terms, which renormalize the coupling between the neighboring sites of the host, and three-spin terms.
However it was shown [9, 8] that in the long-wave limit such a lattice form of
Vol. 4, 2003
Integrable Models of Disordered Impurities
421
the impurity Hamiltonian yields the well-known form of the contact impurity-host
interaction similar to the one of the usual Kondo problem [1, 2]. The contact impurity coupling in this (conformal) limit is also determined by the same constant
θj . All considered here impurities are elastic scatterers, i.e., each excitation only
changes its phase when scatters off each impurity, but not reflected. The same
property holds for a standard Kondo impurity in a free electron host [1, 2]. However we study lattice models too, hence all two-particle scattering processes, e.g.,
from one Fermi point to the other (backscattering) are exactly taken into account
for them.
Eigenvalues and eigenstates of the Hamiltonian are parameterized by quantum numbers, (rapidities), {uj }M
j=1 , where M is related to the z-projection of the
total spin as S z = (S + S cimp )L − M , where L is the total length of the chain
(including impurity sites) and cimp is the concentration of impurities. Notice, that
the Hamiltonian commutes with the z-projection of the total spin, hence M enumerates all possible states. We consider not very large concentrations of impurities.
Those rapidities are the solutions of the following Bethe equations
L
e2s (uj − θf ) = −
f =1
M
e2 (uj − uk ) ,
(2)
k=1
where j = 1, ..., M , en (x) = sinh γ(x + in)/ sinh γ(x − in), s = S for the host
sites (with θf = 0) and s = S for impurity sites, where θf can be nonzero). The
energy of the state with the z-projection
of the total spin, characterized by M , is
M
equal to E0 ≡ Le0 = −i[sin(2γS)/4S] j=1 (d/duj ) ln eS (uj ), valid for the cases,
in which lengths of clusters of neighboring impurities, interacting with each other,
are small.
The behavior of the ensemble of multichannel (with 2S channels) Kondo
impurities, each of which are connected to the free electron gas via own “easyplane”-anisotropic local exchange interaction, with the Hamiltonian [2]
2S 1
+
δ(x − xj )
∆j [(S )zj ]2 + Jjpar (S )zj ×
HK =
2 j
k,l,σ
l=1
†
†
perp
+ †
− †
(cx,l,↑ cx,l,↑ − cx,l,↓ cx,l,↓ ) + Jj [(S )j cx,l,↓ cx,l,↑ + (S )j cx,l,↑ cx,l,↓ ] , (3)
k c†k,l,σ ck,l,σ
where c†k,l,σ creates an electron of the channel l with the spin σ and impurities
are situated at sites xj , can be also described within the Bethe ansatz scheme. In
the scaling limit for small magnetic anisotropy one has Jjpar = 2γ/θj ρ, Jjperp =
Jjpar (1−δj /3), and ∆j = −Jjpar δj /3, where δj = (θj2 /2)+(γ 2 /8) and ρ is the density
of states of conduction electrons at the Fermi level. In this case the low-energy spin
behavior (which is the most important one for Kondo impurities) is determined
by the solution of Eqs. (2), while the energy is determined via EK ≡ Le0K =
M
−i[sin(2γS)/4S] j=1 ln es (uj ). The condition of the applicability of the Bethe
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ansatz scheme for ensembles of disordered Kondo impurities is the presence of large
enough numbers of impurities with equal exchange constants, while those constants
for other impurities can be randomly distributed. For small enough concentration
of impurities, the probability to have long clusters of impurities, connected by the
direct impurity-impurity interactions is small and in the thermodynamic limit L →
∞ one can neglect the contribution of such clusters. In this case the contribution of
each impurity (or of each small cluster of directly connected impurities) is additive,
and we can solve the problem for each impurity (cluster), determined by the local
exchange coupling constant, related to θj , and, then, introducing the distribution
of θj over the system, average the answers for thermodynamic characteristics. Such
an additivity property is the consequence of the exact integrability of the problem,
and is connected with the structure of the considered Hamiltonians. It turns out,
however, that in the long-wave limit the “triangular” structure of the impurity-host
interaction actually produces the local contact impurity-host interaction [10, 9] and
the “fine-tuning” structure of couplings between magnetic impurities and the host
becomes nonessential.
In the absence of the magnetic field the ground state energy of the impurity
is
π sin(2γS)
e0 (θj ) = −
4γS
dωe
i
πωθj
γ
2
π
sinh[ πω
γ min(S, S )] sinh[( 2γ − πmax(S, S ))ω]
sinh(πω) sinh( πω
2 )
,
(4)
and the total ground state energy is equal to E0 = j e0 (πθj /γ), where the sum
is taken over all sites (for sites without impurities we put e0 (0)|S =S ). For the
Kondo impurities Eq. (4) can be used with the overall multiplier 12 . The coupling
j
) is determined by the constant θj . The same
of the impurity to the host (Jimp
constant determines the effective Kondo temperature of the impurity in a spin
chain via TjK ∝ exp(−π|θj |). The random distribution of θj can be described
by some distribution function P (θj ), for example, for a single impurity we have
P (θj ) = δ(θj − θ). Then for energies higher than the crossover Kondo scale one
has an asymptotically free impurity spin S , while for lower energies the impurity
spin is underscreened for S > S (with the Curie-like behavior of the remnant
effective spin S − S), totally screened for S = S (with the usual marginal FL-like
behavior, persisting with the finite χ and linear T -dependence of the specific heat
at low T , and hence finite Wilson ratio in the ground state), and overscreened
for S < S, with the critical NFL behavior of a single spin [3]. It is similar to
the findings of the theory of a Kondo impurity in a free electron matrix [2, 1].
In other words, θj measures the shift off the Kondo resonance (higher values of
|θj | correspond to lower values on the Kondo scale) of the impurity level with the
host spin excitations, similar to the standard picture of the Kondo effect in the
free electron host. For example, the ground state energy per site is equal to (we
Vol. 4, 2003
Integrable Models of Disordered Impurities
423
consider small enough γ < π/2S) for S = S
e0 (θj , h) = e0 (θj ) −
2S 2 γ(π − 2γS)h2
Ah2+2γ(π−2γS)
−
− ... ,
3
2π sin(2γS)TjK
TjK
(5)
is the Fermi
for h TjK , where for small γ, TjK = v exp(−π|θj |/γ) (v = π sin(2γS)
2γS
velocity of low-lying excitations) plays the role of the “local” Kondo temperature,
and A is some non-universal constant, cf. Refs. [11, 12, 13]. Each single magnetic
impurity is totally compensated for h ≤ TjK . The susceptibility of a single impurity
is finite as h → 0, and is renormalized by a factor of TjK with respect to the host
susceptibility. The total ground state energy is the sum of energies of all impurities
and host spins (latters with θj = 0, i.e., TjK = v). We shall show below that the
strong disorder in the distribution of the local Kondo temperatures can lead to the
divergent magnetic susceptibility for h → 0, i.e., to the NFL behavior. For S > S,
for h TjK we get (cf. Refs. [11, 12, 13])
(S − S)πh
− Ch
e0 (θj , h) = e0 (θj ) −
(π − 2γS)
h
TjK
2γ/(π−2γS)
+ ... ,
(6)
where C is the non-universal constant. We can see that for h → 0 the magnetization
is finite for a single impurity, and spins of single impurities are underscreened to
the value S − S by host low-lying excitations. Finally, for S < S and h TjK
we have
1/S
h
e0 (θj , h) = e0 (θj ) − C h
+ ... ,
(7)
TjK
for S > 1, where C is the non-universal constant, and for S = 1, S =
e0 (θj , h) = e0 (θj ) −
2γ(π − 2γ)h2
ln(TjK /h) + . . . .
4π 3 sin(2γ)TjK
1
2
we have
(8)
Hence, for S < S spins of single impurities are overscreened, which produces the
NFL behavior.
For our 1D inhomogeneous quantum spin system at finite T we study the
associated 2D classical vertex model instead of the direct treatment of the 1D
quantum system. The thermodynamics of ensembles of impurities for any T and
h is described by the solution of the following set of nonlinear integral equations
(details of the derivation are given in Ref. [8]) for the “energy density” functions
of spinons a, ā, A = 1 + a, Ā = 1 + ā, and functions yn and Yn , which depend on
the spectral parameter x:
ln y1 (x) = k (x − y) ln Y2 (y)dy ,
ln yj (x) = k (x − y) ln[Yj−1 (y)Yj+1 (y)]dy , 2 ≤ j ≤ 2S − 1 ,
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Proceedings TH2002
k (x − y) ln Y2S −2 (y) + k (x − y + i) ln A(y) +
k(x − y − i) ln Ā(y) dy = ln y2S −1 (x) ,
k(x − y) ln Ā(y) − k(x − y − iπ + i) ln A(y) +
v
πh
−
,
k (x − y + i) ln Y2S −1 (y) dy = ln a(x) +
T cosh x 2(π − γ)T
k(x − y) ln A(y) − k(x − y + iπ − i) ln Ā(y) +
k (x − y − i) ln Y2S −1 (y) dy = ln ā(x) +
πh
v
+
T cosh x 2(π − γ)T
(9)
with an infinitesimal > 0 and kernel functions
1
k(x) =
2π
2
dω
sinh[( πγS − (2S + 1)π)ω] cos(xω)
and
k (x) =
π−γ
2 cosh( πω
2 ) sinh( γ Sπω)
1
2π
dω
cos(xω)
.
2 cosh( πω
2 )
(10)
(11)
The free energy per site f is given by
f (x) = e0 (x) −
T ln A(y)dy
,
2π ±
cosh(x − y ± i)
(12)
where e0 is the ground state energy. The free energy of the total chain with impurities is F = j f [(πθj /γ) + iπ(S − S)], where the sum is taken over all the
sites (for sites without impurities we get f (0)). The free energy per impurity of
the multichannel Kondo problem of the ensemble of disordered impurities fK is
given by
T sinh(x − y ± i) ln A(y)dy
fK (x) = e0K (x) −
,
(13)
2π ±
cosh2 (x − y ± i)
where e0K is the ground state energy. Notice that for S < S one has to put
ln Y2S into Eqs. (12)-(13) instead of ln AĀ. It turns our that for large arguments,
which are important for low-energy characteristics, the behaviors of the kernels of
Eqs. (12) and (13) are similar. The difference is important for energies of order of
the values of local exchange constants and higher. Notice that the set Eqs. (12)-(13)
is finite, unlike the infinite set of nonlinear integral equations, which describe the
thermodynamics of the Kondo impurity in the framework of the standard “string
hypothesis” [1, 2].
We can also find the exact solution for the supersymmetric t−J chain with
the finite concentration of disordered magnetic impurities [14]. The Hamiltonian
Vol. 4, 2003
Integrable Models of Disordered Impurities
425
has the form, similar to the integrable disordered spin 12 Heisenberg AF chain, but
with
m − 1 nl nm ) ,
l S
Hl,m = −t
P (c†l,σ cm,σ + H.c.)P + J(S
(14)
4
σ
l = c† Ŝσ,σ cl,σ , nl =
where c†l,σ creates an electron with spin σ at the site l, S
l,σ
†
n
,
n
=
c
c
and
P
=
(1−n
)(1−n
)
exclude
double
occupations
l,σ
l,−σ
m,−σ
σ l,σ
l,σ l,σ
at each site. In the supersymmetric point we put J = 2t = 2. The nonlinear integral
equations, which describe the thermodynamics of the model are:
h
µ
2π
Ψa (x + i) + +
−
ln a(x) =
T
T
2T
Ψa (y) ln(1 + ā)|x−y+2i dy − Ψa (y) ln(1 + c)|x−y+i dy ,
h
µ
2π ∗
Ψā (x − i) + −
−
ln ā(x) =
T
T
2T
Ψ∗ā (y) ln(1 + a)|x−y−2i dy − Ψ∗ā (y) ln(1 + c)|x−y−i ,
2µ
2π
Ψc +
− Ψc (y) ln(1 + c)(x − y)dy −
ln c(x) =
T
T
Ψa (y) ln(1 + ā)|x−y+i − Ψā (y) ln(1 + a)|x−y−i ,
(15)
where 2πΨa (x) = [x(x − i)]−1 , 2πΨā (x) = [x(x + i)]−1 , πΨc (x) = [x2 + 1]−1
and µ is the chemical potential. The free energy per site is equal to f (θj ) = 2µ −
T ln c(θj ). At low T we have f (θj ) ≈ e0 (θj )−(πT 2 /6)[(1/vc(θj ))+(1/vs (θj ))]+. . .,
where vc,s (θj ) are velocities of charge and spin low-lying excitations at Fermi
points, shifted by θj . Central charges of the semidirect product of charge and spin
conformal algebras are equal to 1 and do not depend on θj : They are universal.
Only velocities of low-lying charge and spin excitations get renormalized by the
disorder. For low densities of electrons (µ T ) for h = 0 we have f (θj ) ≈ ej0 (θj ) −
2
T ln(1 + 2e−1/T (θj +1) ).
In the high density regime µ T (ln c ≈ ln(1 + c)) we get
f (x) = e0 (x) + µ − T dy ln(1 + a)(1 + ā)/ cosh π(x − y), similar to the integrable
spin 12 disordered chain. Hence, for the dense limit at low T the dependence on θj
enters only as TjK , and the charge sector is irrelevant for low energies.
For S > S a single impurity is underscreened by low-lying excitations of the
host. The magnetic susceptibility χj of such an impurity is divergent at h = 0
for T → 0. The specific heat cj exhibits the Schottky anomaly, related to the
undercompensated spin of the impurity. The entropy of a single impurity at T =
h = 0 becomes nonzero, Sj = ln[1 + 2π(S − S)/(π − 2γS)]. A finite magnetic field
lifts the degeneracy and the remnant entropy becomes zero. Naturally, the total
low-T magnetic susceptibility of any disordered ensemble of such impurities is also
divergent at low T . On the other hand, for S < S the spins of low-lying excitations
overscreen the spin of a single magnetic impurity. This yields the critical behavior,
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Proceedings TH2002
which reveals itself in the divergences of the T → 0 magnetic susceptibility of a
single magnetic impurity and of the low-T Sommerfeld coefficient of the specific
heat cj /T for h = 0. In this case one has a remnant T = h = 0 entropy of each
impurity Sj = ln(sin[π(2S + 1)/(2S + 2)]/ sin[π/(2S + 2)]), which is removed by
a finite h that lifts the spin degeneracy of the system. Solving Eqs. (9)-(13) we
get that at low T one has cj ∝ χj ∼ (T /TjK )2/(S+1) for S > 1, and TjK cj /T ∝
TjK χj ∼ ln(TjK /T ) at h = 0. The total low-T χj and the Sommerfeld coefficient
of any disordered ensemble of such impurities are also divergent at low T . In both
cases the disorder in distributions of impurity-host couplings, or local exchanges
between Kondo impurities and conduction electrons, does not yield any qualitative
changes, introducing only specific additional features of the NFL behavior of the
system, which is already present for a single magnetic impurity. More interesting
situation is in the case S = S. Here the solution of Eqs. (9)-(13) can be obtained
analytically [8]. The low-T behavior of the free energy per site (for h = 0) is given
by
πSγT 2
3S 3
f (θj ) = e0 (θj ) −
1+
+ ...
(16)
2(S + 1) sin(2γS)TjK
[ln(αTjK /T )]3
In the presence of a weak magnetic field h T we calculate the temperature
corrections to the free energy per site
πSγT 2
h2
f (θj ) = ej0 (θj , h) −
−
2(S + 1) sin(2γS)TjK
4πTjK
ln | ln(αTjK /T )|
1
+
× 1+
+ O(T 2 ) . (17)
(2S + 1) ln(αTjK /T ) (2S + 1)2 ln2 (αTjK /T )
For a single impurity P (TjK ) = δ(TjK − TK ) we immediately recover the known
Kondo behavior of the asymptotically free spin (characteristic for a Kondo impurity in a free electron host [1, 2] and for a single impurity in a Heisenberg AF
chain [6]). For the homogeneous case we put θj = 0 (i.e., TjK → v where v is
the Fermi velocity of spinons). The central charge of the conformal field theory
is 3S/(S + 1) and does not depend on the parameter of an impurity θj . Also,
conformal dimensions do not depend on the parameters of the disorder. The only
one parameter gets renormalized in the disordered case — the Fermi velocity of
the U(1)-symmetric low-lying excitations (Kondo scale plays the role of a “local
Fermi velocity” for an impurity [9]).
Our models permit to average over a distribution of θj (or “local” Fermi
velocities) because of the factorization of the free energy of the system. This is the
consequence of the integrability of our models (i.e., of the only elastic scattering
off impurities). Note that θj -dependence, present in the low-energy characteristics,
results only in the universal scales, TjK , (it is not so for higher energies, but
the latters are irrelevant for the low T disorder-driven divergences). Hence, for
low energies we can use distributions of TjK , which are also more appropriate
Vol. 4, 2003
Integrable Models of Disordered Impurities
427
in the connection to experiments. That is why the main features of low-energy
characteristics of considered models are determined by the distributions of the
effective Kondo scales for impurities.
Let us consider the strong disorder distribution, which starts with the term
P (TjK ) ∝ G−λ (TjK )λ−1 (λ < 1), valid till some energy scale G for the lowest
values of TjK (that distribution pertains to some real disordered quantum spin
chains and heavy fermion alloys). Now we can calculate the low-T behavior of
the average magnetic susceptibility and the Sommerfeld coefficient of the specific heat as (the lower limit of the integral over the distribution of TjK gives a
regular contribution) χ ∝ c/T ∼ G−λ T λ−1 . These formulas definitely manifest the low-T divergences of χ and c/T and the strong renormalization in
the disordered system as compared to the homogeneous situation. The ground
state average magnetization reveals M z ∼ (h/G)λ behavior, also different from
the homogeneous case. For the important marginal case λ = 1, logarithmic T divergences appear. Here one has the distribution P (TjK = 0) = P0 = 0, valid
till G. Then averaging the low-T part of χand Sommerfeld coefficient we obtain χ ∝ c/T ∼ −(P0 /2π)[ln(G/T ) + ln ln(αG/T ) + . . .]. We again see the
zero-T divergences of χ and c/T (more weak, though, comparing to the previous case). We can also calculate the low-h ground state magnetization: M z ∼
hP0 [− ln(h/G) − ln(ln(h/C G)) + . . .. The weak power law or logarithmic dependence pertains to the Griffiths singularities in the proximity of the critical point.
For these distributions of TjK the Wilson ratio at T = 0 is equal to 2π 2 /3, characteristic for a FL-like situation. Our above mentioned results for low T are valid also
for random ensembles of S = n/2 (where n is the number of channels) multichannel Kondo impurities with the local anisotropic, generally speaking, interaction of
the latters with conduction electrons, because at low T the difference between the
energy of the spin chain and the spin subsystem of the Kondo impurity in a metal
is small.
We illustrate these analytic results by numerical calculations for the solutions of Eqs. (9)-(13) (for accurate numerical calculations see Ref. [15]). In Fig. 1
the T -dependencies for the magnetic susceptibility and the Sommerfeld coefficient
for the most usual AF isotropic spin- 21 chain are depicted. The solid lines show
finite values of the low-T χ and c/T in the homogeneous case. However, dotted lines present answers for the distribution of θj (which, in turn, corresponds
to the distribution of impurity-host exchange constants) with a strong disorder.
The latter means that wings of distributions are large enough, comparing to maxima of distributions. The dotted line corresponds to the Lorentzian distribution
with P (θj ) = [(2θj /γ)2 + π 2 ]−1 . One can see the qualitative difference between
the behavior of S = S strongly disordered magnetic impurities comparing to the
homogeneous chain. The magnetic susceptibility and the Sommerfeld coefficient
diverge at T → 0 for strongly randomly distributed parameters of impurity-host
couplings (notice that in Ref. [9] we have shown that at low T the only Tjk determine the scaling behavior of local impurities). The low-T asymptotics of the
Lorentzian disorder are χ ∼ [T ln(1/T )]−1 and c ∼ [ln(1/T )]−2 . The changes
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A.A. Zvyagin and B.I. Verkin
Proceedings TH2002
6.0
1.0
2c/T
Susceptibility
4.0
0.5
2.0
0.0
0.0
0.5
1.0
T/2
1.5
2.0
0.0
0.0
0.5
1.0
T/2
1.5
2.0
Figure 1: Left: Magnetic susceptibility at h = 0 for the isotropic spin- 21 antiferromagnetic chain with 12 magnetic impurities. The exchange constant of the host is
2. Solid line shows the homogeneous chain; the long-dashed line — the Gaussian
distribution; the dotted line — the Lorentzian distribution of θj ; the dashed line
— the homogeneous chain for h = 0.2 (actually coincides with the solid line); the
dot-dashed line — the Lorentzian distribution for h = 0.2. Right: the same for the
Sommerfeld coefficient.
due to the nonzero magnetic anisotropy of the “easy-plane” type are only qualitative, because such an anisotropy does not produce gaps for low-energy excitations,
and, hence, the system remains in the critical regime. On the other hand, a weak
disorder does not produce such qualitative changes in the behavior of ensembles of
impurities. By a weak disorder we mean a narrow distribution of θj . Long-dashed
lines of Fig. 1 depict the T -behavior of the ensemble impurities with the weak
Gaussian distribution of θj (close to a single impurity distribution). Such a narrow distribution does not yield divergences of the low-T magnetic susceptibility
and Sommerfeld coefficient of the specific heat. The reason for such a different
behavior of wide and narrow distributions (or strong and weak disorders, respectively), is clear. At low energies the local Kondo temperature defines the crossover
scale for the behavior of a magnetic impurity. For the case S = S, a single magnetic impurity is screened by low-lying excitations of the host for T < TjK , and is
not screened for T > TjK (with the Curie-like behavior of the unscreened remnant
spin). For ensembles of magnetic impurities with a weak disorder, T is larger than
the average Kondo temperature of the ensemble of impurities, and, hence, the total magnetic susceptibility and the Sommerfeld coefficient are finite for T → 0.
Contrary, for a strong disorder, many local Kondo temperatures are less than T .
Those impurities remain unscreened by low-lying spin excitations of the host, and,
hence, the total magnetic susceptibility and the Sommerfeld coefficient become divergent for T → 0. Finally we show how the magnetic field lifts the degeneracy. In
Vol. 4, 2003
Integrable Models of Disordered Impurities
429
Fig. 1 the T behavior of magnetic susceptibilities and Sommerfeld coefficients for
the isotropic cases for the homogeneous chain and Lorentzian distribution but for
the nonzero magnetic field are depicted (cf. curves for h = 0). Such a field removes
divergences in the low-T susceptibilities and Sommerfeld coefficients for models
with a strong disorder. For the homogeneous AF spin chain a weak magnetic field
does not yield any qualitative changes in the T behavior (actually the changes for
such a small field are not seen in Fig. 1 compared to h = 0 in this case).
For higher values of spins the changes, comparing to the case S = S = 12
are only quantative. For example, the values of χ and c become larger for larger
spin values. However, there are no drastic changes in the behavior of disordered
ensembles of impurities, in comparison with the above. This is natural, because
only low-lying excitations (which have Dirac seas in the ground state) are responsible for the Kondo-like screening of spins of impurities, while other excitations
(which quasienergies are described by yp and Yp ), are more higher-energetic. In
other words, spinons, which describe the SU(2) (or U(1)) symmetries of the system (or the Gaussian model) are essential for the process of screening of magnetic
impurities, while excitations, which describe the sl2 symmetry of the restricted
solid-on-solid sector [16] (or parafermions [17]) of models with S ≥ 1, do not play
a qualitative role in that process.
In conclusion, in this presentation we have considered the class of exactly solvable models of magnetic impurities in critical quantum antiferromagnetic chains
and multichannel Kondo impurities. We have studied their ground state properties and constructed the finite set of nonlinear integral equations, which exactly
describe thermodynamics of the models. We have obtained several analytic lowenergy expressions for temperature- and magnetic field- dependencies of important characteristics of considered models. We have shown that the only low-energy
parameter, which gets renormalized due to the disorder, is the “local velocity”
of low-lying excitations (or the effective crossover scale connected with each impurity); others appear to be universal. Note that the finite size corrections to
the ground state behavior of considered disordered systems can be obtained from
above formulas by replacing, e.g., (G/T ) → L.
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Proceedings TH2002
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A.A. Zvyagin
Max Planck Institute for Chemical Physics of Solids
D-01187 Dresden, Germany
B.I. Verkin
Institute for Low Temperature Physics
and Engineering of the NAS of Ukraine,
Kharkov, 61103, Ukraine