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On the entangled-disentangled transition in magnetic flux liquids
PIHSICA
Physica C 185-189 (1991) 1427-1428 North-Holland
ON T H E ENTANGLED-DISENTANGLED T R A N S I T I O N IN MAGNETIC FLUX LIQUIDS E.I. KORNILOV
Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Head Post O~ce, P.O. Box 79, Moscow I01000, USSR The lattice models of thermally disordered flux line (FL) system are studied by the random walk (RW) method. A finite-size analysis of a possible transition between entangled and disentangled phases is performed.
Nelson has suggested 1,2 a sharp transition between
and end at odd points r l , r 2 , . . . ,rN of the z-axis. We
the entangled and disentangled phases with incre~ing
ascribe the statistical weight x to a step along a lattice
slab thickness, L, parallel to the external field H. This
edge, and the weight y to that along a diagonal. Thus
treatment was based on an analogy between the disor-
we shall distinguish the FL length unit parallel to the
dered flux system and the 2D bosons at T = 0. How-
external field H from the one whose direction deviates
ever, the FLs' flexibility and their nonintersection may
from H. In this representation we actually replace the
be naturally described by the r a n d o m walks (RW) mo-
long-range potential between vortices by the hard-core
dels 3. The purpose of present paper is to compare the
one in the form Y(r) = {c~, if r < ~, 0 otherwise}.
boson approximation of FL system 1,2 with the fermion
The role of a coherence length, ~(T), is played by the
one 4.
thickness of the lattice edges. The lattice space unit
Consider a 2D M x L lattice turned by rr/4 so that,
of the z - a x i s makes sense of an intervortex distance
the diagonals coincide with the z - and t - coordinate
at the regular Abrikosov regime above a strong enough
axes. To each elementary cell we add a diagonal pa-
field dcnowd as H ' . Therefore it is convenient to intro-
rallel to the t - a x i s . We assume the periodic boundary
duce a dimensionless field p = H/H* which measures
conditions (PBC) in both directions. We represent the
the disordered phase. We discuss now the Nelson hypothesis 1,2 that "fluc-
single FL as a trace of the random walker, beginning at point r, belonging to an odd coordinate of the z - a x i s ,
tuations in high-T materials lead to a new entangled
and going over the lattice edges so that each step in-
flux phase in a magnetic field." Tkis phase should ap-
creases the t - c o o r d i n a t e of the walker. Analogously,
pear with the increasing of the sample thickness L[[H
we consider a simple cubic lattice turned in space so
when the single-particle boson wave functions are over-
that a principal diagonal is vertical. Adding the very
lapped in the ground state.
diagonal to each cubic cell and introducing the vertical
has usually a manifestation in thermodynamic functions
time-like axis, we get a 3D RW lattice model that was
such as susceptibility or specific heat. To explain the
used previously s to study the thermodynamics of long
properties of the disordered FLs phase we have calcu-
polymers.
late(l the logarithm of the partition flmction for the fi-
Any change of regimes
In 2D the RW returns to the origin point after pas-
nite L. For tile sake of simplicity we use now the choice
sing through the lattice due to the PBC along the t - a x i s .
of step weights with y = 0. This choice gives no loss of
The RW is not self-intersecting under the above men-
the essential process under examination; what is more,
tioned rules. We also demand that there exist no in-
it makes the random walkers more mobile.
tersecting walks in any sets of N walks, which begin
obtain 4 the following expression
0921-4534/91/$0~.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.
One can
E.I. Komilov / Entangled.disentangled transition in magnetic flux liquids
1428
~f [2...
~zD(~) = (2~)D-'LJo
0.4r
//,/°-4
2~
"'/o dD-'~ 1"11+
+(~,~D_,(~,))LI.
I
(I)
I~
4
o.,
Here AI = 2cos@ and A2 = (e i¢1 q-e i~ We.i#z+i~ ) which
are the RW "structure factors" of our models.
An
isothermal susceptibility per lattice site K and a specific heat C are defined as 6 p=x
O mx
, K=
: -g~=° °
( h z ~, ,
and C = K \ - - ~ /
p
1
(2)
O.S[ calculated along the x fugacity axis (y = 0). The shapes of susceptibility are collected in Figure.
b)
I I
The curves
have a feebly marked peak. This fact may mean the manifestation of regimes changing from nearly nonin-
0.4
teracting FLs at low density to dense packing behavior at another density limit. The susceptibility peak position p,n~ and the slab thickness L obey the law (3)
pma~, cx A" L -'~
0
for both dimensions. We have estimated 4 the values A3D = 0.59, o'3• -- 0.44. and A2D - C.70, a2D = 0.46.
The closeness of this numerical values is surprise because a vast topological difference exists between the 2D and 3D FLs systems: there is no entanglement in the 2D case whereas the entanglement is always present in the thermodynamic limit L ---+ oo in the 3D case.
P
FIGURE The dependence of the susceptibility per lattice site K on the FLs density p, a) for 2D; b) for 3D lattice models. The labels near the curves correspond to the following values of the longitudinal slab size L: 1-oe, 2-4, 3-8, 4-16, 5-32, 6-64, 7-128, 8-256. Inset: Two sets of corresponding draft plots of the specific heat C per site for both 2D and 3D cases.
Hence we conclude that in the framework of our model the peculiarities of both 2D and 3D FLs systems arise rather from the "usual" FLs collisions than from their entanglement. These speculations are not quite rigorous because
REFERENCES 1. D.R. Nelson, Phys. Rev. Lett. 60 (1988) 1973.
they are based on the approximating solution in 3D.
2. D.R. Nelson and S.H. Seung, Phys. Rev. B 39 (1989) 9153
The possible experimental test of the validity of either
3. K. Ziegler, Europhys. Lett. 9 (1989) 277
fh~
fe~-r2"12u;~nv*~r,,T-f,,-.,';.-,.-,.-~,l.;.,-.,,-.3,4 ~. , l . ~
t. ....
1.2
consist in determination of the exponent ~ in (3). As it follows from formulas (3.12-13) in 2 the quantum fluctuations become important for bosons when L < p. Thus the corresponding boson exponent aB = 1 1 strongly differs from our estimation aF ~_ 5.
4. E.i. Kornilov and A.A. Litvin: Z. Phys. B Condensed Matter (in press) (1991) 5. E.I. Kornilov and V.B. Priezzhev: Z. Phys. B Condensed Matter 54 (1984) 351 G. D.S. Gaunt: Phys. Roy. 197 (1969) 174
Physica C 185-189 (1991) 1427-1428 North-Holland
ON T H E ENTANGLED-DISENTANGLED T R A N S I T I O N IN MAGNETIC FLUX LIQUIDS E.I. KORNILOV
Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Head Post O~ce, P.O. Box 79, Moscow I01000, USSR The lattice models of thermally disordered flux line (FL) system are studied by the random walk (RW) method. A finite-size analysis of a possible transition between entangled and disentangled phases is performed.
Nelson has suggested 1,2 a sharp transition between
and end at odd points r l , r 2 , . . . ,rN of the z-axis. We
the entangled and disentangled phases with incre~ing
ascribe the statistical weight x to a step along a lattice
slab thickness, L, parallel to the external field H. This
edge, and the weight y to that along a diagonal. Thus
treatment was based on an analogy between the disor-
we shall distinguish the FL length unit parallel to the
dered flux system and the 2D bosons at T = 0. How-
external field H from the one whose direction deviates
ever, the FLs' flexibility and their nonintersection may
from H. In this representation we actually replace the
be naturally described by the r a n d o m walks (RW) mo-
long-range potential between vortices by the hard-core
dels 3. The purpose of present paper is to compare the
one in the form Y(r) = {c~, if r < ~, 0 otherwise}.
boson approximation of FL system 1,2 with the fermion
The role of a coherence length, ~(T), is played by the
one 4.
thickness of the lattice edges. The lattice space unit
Consider a 2D M x L lattice turned by rr/4 so that,
of the z - a x i s makes sense of an intervortex distance
the diagonals coincide with the z - and t - coordinate
at the regular Abrikosov regime above a strong enough
axes. To each elementary cell we add a diagonal pa-
field dcnowd as H ' . Therefore it is convenient to intro-
rallel to the t - a x i s . We assume the periodic boundary
duce a dimensionless field p = H/H* which measures
conditions (PBC) in both directions. We represent the
the disordered phase. We discuss now the Nelson hypothesis 1,2 that "fluc-
single FL as a trace of the random walker, beginning at point r, belonging to an odd coordinate of the z - a x i s ,
tuations in high-T materials lead to a new entangled
and going over the lattice edges so that each step in-
flux phase in a magnetic field." Tkis phase should ap-
creases the t - c o o r d i n a t e of the walker. Analogously,
pear with the increasing of the sample thickness L[[H
we consider a simple cubic lattice turned in space so
when the single-particle boson wave functions are over-
that a principal diagonal is vertical. Adding the very
lapped in the ground state.
diagonal to each cubic cell and introducing the vertical
has usually a manifestation in thermodynamic functions
time-like axis, we get a 3D RW lattice model that was
such as susceptibility or specific heat. To explain the
used previously s to study the thermodynamics of long
properties of the disordered FLs phase we have calcu-
polymers.
late(l the logarithm of the partition flmction for the fi-
Any change of regimes
In 2D the RW returns to the origin point after pas-
nite L. For tile sake of simplicity we use now the choice
sing through the lattice due to the PBC along the t - a x i s .
of step weights with y = 0. This choice gives no loss of
The RW is not self-intersecting under the above men-
the essential process under examination; what is more,
tioned rules. We also demand that there exist no in-
it makes the random walkers more mobile.
tersecting walks in any sets of N walks, which begin
obtain 4 the following expression
0921-4534/91/$0~.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.
One can
E.I. Komilov / Entangled.disentangled transition in magnetic flux liquids
1428
~f [2...
~zD(~) = (2~)D-'LJo
0.4r
//,/°-4
2~
"'/o dD-'~ 1"11+
+(~,~D_,(~,))LI.
I
(I)
I~
4
o.,
Here AI = 2cos@ and A2 = (e i¢1 q-e i~ We.i#z+i~ ) which
are the RW "structure factors" of our models.
An
isothermal susceptibility per lattice site K and a specific heat C are defined as 6 p=x
O mx
, K=
: -g~=° °
( h z ~, ,
and C = K \ - - ~ /
p
1
(2)
O.S[ calculated along the x fugacity axis (y = 0). The shapes of susceptibility are collected in Figure.
b)
I I
The curves
have a feebly marked peak. This fact may mean the manifestation of regimes changing from nearly nonin-
0.4
teracting FLs at low density to dense packing behavior at another density limit. The susceptibility peak position p,n~ and the slab thickness L obey the law (3)
pma~, cx A" L -'~
0
for both dimensions. We have estimated 4 the values A3D = 0.59, o'3• -- 0.44. and A2D - C.70, a2D = 0.46.
The closeness of this numerical values is surprise because a vast topological difference exists between the 2D and 3D FLs systems: there is no entanglement in the 2D case whereas the entanglement is always present in the thermodynamic limit L ---+ oo in the 3D case.
P
FIGURE The dependence of the susceptibility per lattice site K on the FLs density p, a) for 2D; b) for 3D lattice models. The labels near the curves correspond to the following values of the longitudinal slab size L: 1-oe, 2-4, 3-8, 4-16, 5-32, 6-64, 7-128, 8-256. Inset: Two sets of corresponding draft plots of the specific heat C per site for both 2D and 3D cases.
Hence we conclude that in the framework of our model the peculiarities of both 2D and 3D FLs systems arise rather from the "usual" FLs collisions than from their entanglement. These speculations are not quite rigorous because
REFERENCES 1. D.R. Nelson, Phys. Rev. Lett. 60 (1988) 1973.
they are based on the approximating solution in 3D.
2. D.R. Nelson and S.H. Seung, Phys. Rev. B 39 (1989) 9153
The possible experimental test of the validity of either
3. K. Ziegler, Europhys. Lett. 9 (1989) 277
fh~
fe~-r2"12u;~nv*~r,,T-f,,-.,';.-,.-,.-~,l.;.,-.,,-.3,4 ~. , l . ~
t. ....
1.2
consist in determination of the exponent ~ in (3). As it follows from formulas (3.12-13) in 2 the quantum fluctuations become important for bosons when L < p. Thus the corresponding boson exponent aB = 1 1 strongly differs from our estimation aF ~_ 5.
4. E.i. Kornilov and A.A. Litvin: Z. Phys. B Condensed Matter (in press) (1991) 5. E.I. Kornilov and V.B. Priezzhev: Z. Phys. B Condensed Matter 54 (1984) 351 G. D.S. Gaunt: Phys. Roy. 197 (1969) 174