Boundary critical phenomena of the random transverse Ising model in D>=2 dimensions

0.29 MB·English

by  István A. Kovács

Boundary critical phenomena of the random transverse Ising model in D ≥ 2
dimensions
Istv´an A. Kov´acs1, and Ferenc Igl´oi1,2,
∗ †
1Wigner Research Centre, Institute for Solid State Physics and Optics, H-1525 Budapest, P.O.Box 49, Hungary
2Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary
(Dated: January 22, 2013)
Usingthestrongdisorderrenormalizationgroupmethodwestudynumericallythecriticalbehavior
of therandom transverse Ising model at a free surface, at acorner and at an edge in D=2, 3 and
3 4-dimensionallattices. Thesurfacemagnetizationexponentsarefoundtobe: xs=1.60(2), 2.65(15)
1 and 3.7(1) in D=2, 3 and 4, respectively, which do not depend on the form of disorder. We have
0 also studied critical magnetization profiles in slab, pyramid and wedge geometries with fixed-free
2 boundaryconditions and analyzed their scaling behavior.

n
a
J I. INTRODUCTION another, more simple approximation methods have been
1 developedandappliedto the RTIM25–31. Oneofthose25
2 is based on the quantum cavity approach32, which is
The quantum Ising model with random couplings
foundtoreproducesomeoftheexactresultsin1D.How-
and/orwith randomtransversefields (RTIM)is the pro-
] ever, in the Bethe lattice with an effective dimensional-
n totype of disordered quantum magnets having discrete
n symmetry. This model has a zero-temperature quantum ity of Deff = 2 the method has predicted conventional
- phasetransition,the propertiesofwhichhavebeenstud- random critical behavior instead of IDFP scaling. The
s
quantum cavity method is shown to be equivalent to a
i ied by a special strong disorder renormalization group
d (SDRG) method1. In this method the strongest local linearizedtransfermatrix approach27. Ifnolinearization
.
t termsoftheHamiltonianaresuccessivelyeliminatedand is performed (this is the so called non-linear transfer ap-
a
proach) than the method has lead to IDFP behavior for
m atthesametimenewtermsaregeneratedperturbatively
between remaining degrees of freedom2. In one dimen- D ≥ 2, too27. Also approximate renormalization group
- schemes have been suggested28–31, during which the or-
d sion(1D)wherethetopologyofthelatticestaysinvariant
n underthetransformationtheSDRGequationshavebeen der of the RG steps is changed in such a way that the
o solvedanalyticallyinthevicinityofthequantumcritical proliferationofnewcouplingsisavoided. Thesemethods
c point3. In this case the phase transition is shown to be have reproduced some exact 1D results and also provide
[ controlled by a so called infinite disorder fixed point4 IDFP behavior for D ≥ 2, in agreement with the stan-
2 (IDFP), in which disorder fluctuations are completely dard SDRG method.
v dominant over quantum fluctuations and therefore the Most of the results about the critical behavior of the
3
renormalization steps are asymptotically exact for large RTIM have been calculated for bulk quantities. For ex-
2
scales. IndeedtheSDRGresultsin1Dareconsistentwith ample the order-parameter of the RTIM is the average
6
2 findingsofotheranalytical5,6 andnumericalmethods7–9. magnetizationanditsvalueinthebulk,mb,hasthescal-
. In higher dimensional lattices the topology of the ing behavior mb ∼L−x, where L is the linear size of the
1
lattice is changed during the SDRG steps, therefore system and x is the scaling exponent of the bulk magne-
1
2 the SDRG method has to be implemented numerically. tization. Real systems, however, have finite extent and
1 The first numerical calculations have been performed they are limited by boundaries. At a free surface the
: in 2D10–15 and more recently an efficient numerical scaling behavior of the average surface magnetization,
v
i algorithm16–18 of the present authors made possible to ms, involves a new exponent33–35, xs. Due to missing
X extend the calculations17–20 to 3D and 4D, as well as to bondsatthesurfacethereisweakerorder,thereforegen-
r Erdo˝s-R´enyi random graphs, which are infinite dimen- erally xs > x. For the 1D RTIM several properties of
a sional lattices. In all dimensions the phase transition is the surface magnetization (the distribution function, av-
found to be controlled by an IDFP, which justifies that erage and typical behavior, etc) is exactly known3,5,9,36.
the SDRG method provides asymptotically exact results Forexamplethesurfacescalingexponent,xs =1/2,isre-
for large systems in higher dimensions, too. Quantum lated to the persistence properties of1D randomwalks9.
MonteCarlosimulationsforthe 2DRTIMareconsistent For higher dimensional RTIM less attention is paid
with the SDRG results21. Similarly, simulation results to the calculation of the surface magnetization: we are
for the random contact process22,23 - which is expected aware of one recent work27, in which the surface magne-
to be in the same universality class24 as the RTIM - are tization exponent has been calculated by the non-linear
in agreement with the SDRG results in 2D and in 3D. transferapproach. The obtainedvalues arex =1.2and
s
In D > 1 dimensions during the SDRG iterations a 1.34, in 2D and 3D, respectively, which are to be com-
largenumberofnewcouplingsaregeneratedbetweenre- paredwith the SDRG results for the bulk magnetization
motesites,whichmakesthenumericalimplementationof exponent16–18: x=0.98 and 1.84,in 2D and 3D, respec-
the method rather cumbersome. To avoid this problem tively. Since in 3D the surface magnetization exponent
2
of the non-linear transfer approach is smaller, than the (These are located at l = 1, thus m = 1). For the sur-
1
expected correct value of the bulk magnetization expo- face magnetization a slab of size L×ND 1 (L < N) is
−
nent we conclude that the non-linear transfer approach used, and in the short direction we use fixed-free b.c.,
underestimates the values of x . Therefore there is a ne- while in the other (D−1)-directions periodic b.c. is ap-
s
cessity to obtain more accurate estimates for the surface plied. Thesurfacemagnetizationisgivenbym =m ,
s l=L
critical properties of the RTIM. which scales at the critical point as ms ∼L−xs.
In this paper we study the boundary critical behavior Thecornermagnetization ismeasuredin2Datthefree
of the RTIM in higher dimensionalsystems for D =2, 3 corner of a half square and in 3D at the free corner of a
and 4 by the SDRG method. In the calculation we use cube,havingtheshapeofapyramid. (Inthefollowingwe
the numerical algorithm, which has been developed in use the term pyramid in the 2D case, too.) The spins at
Refs.17,18 and has been used to study the bulk critical thebaseofthe pyramidarefixed,while atothersurfaces
behavior of the systems. Besides the surface magneti- free b.c.-s are used. The magnetization profile, m is
l
zation exponent, x , we calculate local magnetization measured between the base (l = 1) and the corner (l =
s
exponents37, which are associated with corners (in 2D L = √DL), and the corner magnetization is given by
c 2
and 3D) as well as with edges (in 3D). We also calcu- m = m . This scales at the critical point as m ∼
late critical magnetization profiles, when spins are fixed L−cxc witlh=Ltche corner exponent, xc. c
at some surfaces of the system and study their scaling Edgemagnetization iscalculatedin3Datthefreeedge
properties. oflengthN ofasquarecolumnofsizeL×L×N,L<N.
The structure of the paper is the following. The The square column is cut at the square diagonal plane,
essence of the SDRG method and its application to the thushavetheshapeofawedge,andthespinsatthebase
calculation of the boundary magnetization is described of the wedge are fixed. In the long direction periodic
in Sec.II. Ourresults for 2D, 3Dand 4Dlattices arepre- b.c. is used, while at the other two symmetric surfaces
sented in Sec.III and discussed in the final section. free b.c. is applied. The magnetization profile, m is
l
measured between the base of the wedge (l =1) and the
freeedge(l =L = √2L). m is translationallyinvariant
II. SDRG CALCULATION OF THE LOCAL e 2 l
along the long direction and the edge magnetization is
MAGNETIZATION
given by m = m . This scales at the critical point
e l=Le
as me ∼L−xe, with the edge exponent, xe. The applied
Here we consider the boundary critical properties of
geometries in 3D are shown in Fig. 1.
the RTIM defined by the Hamiltonian:
H=−XJijσixσjx−Xhiσiz (1)
ij i
in terms of the σx,z Pauli operators at site i of a D- N N L
i
dimensional cubic lattice. The J > 0 nearest-neighbor
ij N L L
couplings and the h > 0 transverse fields are indepen-
i L L L
dent random numbers taken from the distributions p(J)
andq(h), respectively. Inthis paper weuse twodifferent
disorder distributions, which have already been used in
FIG. 1: (Color online) The three geometries used in the
Refs.16–18 for the calculationofthe bulk properties. The
calculationofthelocalmagnetizationin3D.Leftpanel: slab;
advantage of using these distributions is, that the loca- middle panel: wedge; right panel: pyramid. Spins at the
tion of the critical points have already been determined. shadedplanesarefixedandthearrowsindicatethedirections
In the bulk calculations we have observed that the two in which thedensity profiles are measured.
distributions lead to identical critical exponents, within
the errorof the numericalmethod. Here we assume that We note that the corners and edges we consider here
universality holds to local critical properties, too, and have the specific opening angle: π/2. The local critical
check this assumption numerically. In both type of dis- exponentsaregenerallyangledependent37,butwedonot
orderthecouplingsaretakenfromauniformdistribution: study this problem in the present paper.
p(J)=Θ(J)Θ(1−J), where Θ(x) is the Heaviside step- To calculate the local magnetization in the different
function. For the box-h model the transverse fields have geometries we have used the SDRG method, which is an
a box-likedistribution q(h)= 1 Θ(h)Θ(h −h), whereas iterativeprocedureworkinginthe energyspace. Ateach
hb b
for the fixed-h model the transverse fields are constant: step the largest local term of the Hamiltonian, either a
q(h)=δ(h−h ). coupling, J , or a transverse field, h , is decimated and
f ij i
In the calculations of different local magnetizations new terms are generated between the remaining sites in
(surface,edgeandcorner)wehaveuseddifferentfinitege- a perturbation calculation. For coupling decimation the
ometries, in which fixed spin boundary conditions (b.c.) twosites,iandj withoriginalmagneticmoments,µ and
i
have been used at given planes and the magnetization µ ,aremergedtoanewclusterwithaneffectivemoment
j
profile,m ,ismeasuredperpendiculartothefixedplanes. µ = µ +µ , which is placed in an effective transverse
l ′ij i j
3
field of strength: h =h h /J . In transverse field dec- the typical aspect ratios of slabs and wedges as well as
′ij i j ij
imation the site i is eliminated and its nearest-neighbor the typical number of disorder realizations are collected
sites, say j and k will be connected by an effective cou- in Table I. Since only a small fraction of samples con-
pling: J = max{J J /h ,J }. In this last step the tains such a correlation cluster, C, which have also sites
j′k ji ik i jk
so called maximum rule is applied, the use of which is at the free extremity of the system (surface, edge or cor-
justified at an IDFP. ner) one should consider a large number of realizations.
We apply the numerical algorithm of the SDRG Forsurfacesandedgesinagivensamplethereareseveral
method in Refs.17,18, which has been used to locate the end-point positions, for which we perform the averages.
criticalpoint of the system (for the two forms of the dis- For corners, however, there is just one end-point in a
order)andto calculatethe bulkcriticalexponentsatthe sample, therefore one should take even larger number of
IDFPfordifferentdimensions,D =2, 3and4(whichare realizations. In the following we present our numerical
disorderindependentandlistedinTableII).Tocalculate results obtained in different dimensions.
thecriticalmagnetizationprofile,m ,werenormalizethe
l
systemuptothelasteffectivesiteandconsidertheeffec-
TABLE I: Details of the numerical calculation of the local
tive cluster, C, which contains the fixed sites at l =1. If
the system has an IDFP, then all spins of C are strongly magnetization. Lmax: largest linearsize;N/L: typicalaspect
ratio; N#: typicalnumberof realizations.
correlated: in leading order all these spins point to the
same direction as at l =1, whereas other sites (not con- slab pyramid wedge
tained in C) have negligible contribution to the longitu- Lmax N/L N# Lmax N# Lmax N/L N#
dinal magnetization. Let us denote by nl the number of 2D 512 4. 106 256 107
sitesinC atpositionl andthe numberofequivalentsites 3D 64 2. 106 64 108 64 2. 107
byn˜ (itisNd−1,N and1intheslab,wedgeandpyramid 4D 32 1.5 105
geometry, respectively). The average value of the local
magnetization is then given by: m = [n /n˜] , where
l l av
[...] stands for the average over disorder realizations.
av
At the critical point the asymptotic form of the mag-
netization profile is given by scaling considerations. Ac- A. Calculations in 2D
cordingtoFisherandde Gennes38 the decayofthemag-
netization from the fixed surface is given by: 1. Surface magnetization
ml ∼l−xb, x=xb, 1≪l≪L, (2)
thus it includes the bulk magnetization exponent. Close
to the freeendpoint(surface,corneroredge)the magne-
tization profile has a different power-law decay33: 1.1
-1
10
ml′ ∼(l′)xαb, 1≪l′ =Lα−l+1≪Lα , (3) r 1
with x =x −x and α relatesto the type of endpoint: 0.9
αb α
m
so,btcaionrined(eapnedndLesnt≡esLt)im. aTtheessfeorretlhaetilooncsalwmilalgbneetuizseadtiotno 10-3 0 0.2 0.4λ 0.6 0.8 1
exponents. The two scaling relations in Eqs.(2) and (3)
can be incorporated into an interpolation formula:
A fixed
ml = Lx [sin(πλ)]x[cos(πλ/2)]xα , (4) 10-5 box
0 0.2 0.4 0.6 0.8 1
with λ = l . This relation is exact for 1D conformally λ
Lα
invariantquantumsystems39. AlthoughtheRTIMisnot
conformally invariant, in 1D Eq.(4) is found to be an
excellent approximation8,40. In the following in the slab
FIG. 2: (Color online) Magnetization profiles in 2D in
geometry (α = s) we shall check the accuracy of Eq.(4)
the slab geometry for fixed-free b.c.-s in a system of width
in higher dimensions, too. L = 512 for box-h and fixed-h randomness. The interpola-
tion formula in Eq.(4) is represented by dashed lines. In the
inset the ratio of the magnetization profile and the interpo-
III. RESULTS lation formula in Eq.(4) is shown for x = 0.982, xs = 1.6,
Afixed =1.20 and Abox =0.282.
We have calculated the magnetization profiles in the
three geometries described in Sec.II in different dimen- Themagnetizationprofilesinthe slabgeometrycalcu-
sions: 2≤D ≤4 by the SDRG method using two differ- lated by the two types of disorder are shown in Fig.2 as
ent forms of disorder. The largest sizes of the systems, a function of the relative position: λ = l/L, see Eq.(4).
4
Here we use a finite-size shift of l =O(1) at the bound-
0
1.72
aries. As already observed in the calculation of the bulk
magnetization the typical correlationclusters for fixed-h -1 1.68
10
disorder contain approximately 6-times more sites, than
xs 1.64
for box-h disorder. As a consequence the magnetization
profiles are also comparatively larger for fixed-h disor- 1.6
m
der. As seen inFig.2 the magnetizationis monotonously -3 1.56
10 0 0.02 0.04 0.06
decreasing and the variation is very fast near the two
1/L
end-points, which are then analyzed in log-log plots in
Figs.3 and 4, respectively. fixed
box
10-5 xsb=0.65
1 10 100
10-1 l’+l’0
m FIG. 4: (Color online) Magnetization profiles near the free
-3 boundary in 2D for the slab geometry for the two type of
10
fixed randomness with L = 512. In both cases the decay is char-
fixed, corner acterized by the exponent, xsb = 0.65(2). In the inset the
box finite-size estimates for the surface magnetization exponent
box, corner arepresented. Theextrapolated(disorderindependent)value
10-5 x=0.98 is given in Table II.
1 10 100
l+l
0
formula, in which the exponents in Table II have been
used. As seen in this figure the interpolation formula
FIG. 3: (Color online) Magnetization profiles near thefixed
boundary in 2D for the slab and the pyramid geometries for represents a good approximation, but the agreement is
the fixes-h and box-h randomness with L = 512 and 256, not perfect, the largest discrepancy is about 10%.
respectively. In all cases the decay is characterized by the
same exponent, xb =0.98(1), which according to the Fisher-
deGennesresult in Eq.(2) is equivalenttothebulkmagneti-
zation exponent,see Table II.
Close to the fixed boundary the magnetization pro-
2. Corner magnetization
files are shown in Fig.3, together with the similar pro-
files in the pyramid geometry, which will be analyzed
in Sec.IIIA2. The magnetization profiles for the two
The calculations are performed in the pyramid geom-
disorder have the same power-law decay and the de-
etry and the critical magnetization profile close to the
cay exponent is estimated from the largest systems as
fixed plane is shown in Fig.2 in a log-log plot for the
x = 0.98(1). This is to be compared with the value of
b two different initial disorder. As discussed in Sec.IIIA1
the bulk magnetization exponent x = 0.982(15), which
in this figure also the profiles in the slab geometry are
has been calculated in Ref.16 by finite size scaling. We
presented and the two types of profiles are very close to
canthusconcludethatthe Fisher-deGennesscalingpre-
eachother: theyareindistinguishablewithintheerrorof
diction in Eq.(2) is well satisfied.
the calculation. Thus in agreement with scaling theory
Alsoatthefree-boundarytheprofileshaveapower-law
the decay of the profile in the pyramid geometry is in
variation (see Fig.4) and the corresponding exponent is
a power-law form with a decay exponent, x = x. The
b
estimated from the largest system as: x =0.65(2). We
sb magnetizationprofile at the other end, i.e. starting from
havealsocalculatedthesurfacemagnetizationexponent,
thecornerisshowninFig.5andthecorrespondingdecay
x , from the finite-size scaling behavior of the surface
s exponentofthemagnetization,x ,ispresentedinTable
cb
magnetization, m . Two-point estimates for x are pre-
s s II.
sentedintheinsetofFig. 4,whichhavethesamelimiting
value for large L for the two type of disorder, which is Fromfinite-sizescalingthecornermagnetizationexpo-
presentedinTable II.Comparingx with x −x we can nent, x , is calculated by two-point fit and the effective,
sb s c
concludethatthescalingpredictioninEq.(3)issatisfied. size-dependent exponents are presented in the inset of
Wehavealsocheckedtheaccuracyoftheinterpolation Fig.5 for the two different type of disorder. The extrap-
formulainEq.(4)andintheinsetofFig.2wehaveplotted olated value which is disorder independent is given in
the ratio of the measured profile and the interpolation Table II.
5
exponent, see Table II. Near the free surface the profiles
2.5
forthe twotypes ofdisorderareshownin Fig. 7 andthe
2.4
10-1 2.3 estimated decay exponent, xsb, is presented in Table II.
xc 2.2
2.1
m
2
3
0 0.05 0.1 0.15
10-3 1/L 10-1 2.9
2.8
xs
fixed 2.7
box m 2.6
-5 xcb=1.35 10-3 2.5
10 0 0.04 0.08 0.12
1 10 100
1/L
l’+l’
0
-5
10 fixed
box
FIG.5: (Coloronline)Magnetizationprofilesinthepyramid xsb=0.84
geometrynearthecornerin2Dforthetwotypeofrandomness 1 5 10 50
withL=256. Inbothcasesthedecayischaracterizedbythe l’+l’0
exponent,xcb=1.35(10). Intheinsetthefinite-sizeestimates
for the corner magnetization exponent are presented41. The
extrapolated (disorder independent) value is given in Table
FIG. 7: (Color online) Magnetization profiles near the free
II.
boundary in 3D for the slab geometry for the two types of
randomness with L = 64. In both cases the decay is char-
acterized by the exponent, xsb = 0.84(7). In the inset the
B. Calculations in 3D
finite-size estimates for the surface magnetization exponent
arepresented. Theextrapolated(disorderindependent)value
1. Surface magnetization is given in Table II.
The surface magnetization exponent is estimated
1.2 through finite-size scaling and the effective, size-
10-1 1.1 dependent values are shown in the inset of Fig. 7 for
r 1 the two types of disorder. The extrapolated exponent is
disorderindependent andgivenin Table II.We conclude
0.9
m that the scaling relation in Eq.(3) is satisfied within the
0.8
10-3 0 0.2 0.4 0.6 0.8 1 error of the calculation.
λ
We havecheckedthe accuracyofthe interpolationfor-
mula in Eq.(4) and the ratio of the measuredprofile and
-5 the interpolation formula is shown in the inset of Fig.
10 fixed
6. Also in this case Eq.(4) is a good approximation, the
box
maximaldiscrepancyis somewhatlarger,thanin the 2D
0 0.2 0.4 0.6 0.8 1 case, see in Fig. 2.
λ
FIG. 6: (Color online) Magnetization profiles in 3D in the
slabgeometryforfixed-freeb.c.-sinasystemofwidthL=64
2. Edge magnetization
forbox-handfixed-hrandomness. Theinterpolationformula
inEq.(4)isrepresentedbydashedlines. Intheinsettheratio
of the magnetization profile and the interpolation formula in
We have measured the magnetization profile in the
Eq.(4) is shown for x = 1.84, xs = 2.65, Afixed = 1.78 and
wedge geometry and here we analyze its behavior close
Abox=0.316.
to the free edge, see Fig. 8.
The magnetization profiles in the slab geometry are Theestimateddecayexponent,x ,ispresentedinTa-
eb
shown in Fig. 6 for the two types of disorder. Close to bleIItogetherwiththeextrapolatedvalueoftheedgeex-
the fixedboundary the exponent associatedto the decay ponent, x , for which the finite-size estimates are shown
e
oftheprofileisestimatedasx =1.855(20)whichagrees intheinsetofFig. 8. Inthiscase,toothescalingrelation
b
with the finite-size estimate of the bulk magnetization in Eq.(3) is satisfied.
6
C. Calculations in 4D
3.6
-1
10 In 4D the available system sizes are limited, see Table
3.4
e I,thereforewecouldonlystudythemagnetizationprofile
x
3.2 in the slab geometry, which is shown in Fig. 10.
m
-3 3
10
0 0.04 0.08 0.12
1/L 1.4
-1
10
1.2
fixed
10-5 box r 1
x =1.75
eb 0.8
-3
10
1 3 10 30 0 0.2 0.4 0.6 0.8 1
l’+l’0 m λ
-5
10
FIG. 8: (Color online) Magnetization profiles near the free fixed
edgein3Dinthewedgegeometryforthetwotypeofrandom- box
nesswithL=64. Inbothcasesthedecayischaracterizedby
0 0.2 0.4 0.6 0.8 1
the exponent, xeb = 1.75(15). In the inset the finite-size es- λ
timatesfortheedgemagnetization exponentarepresented41.
Theextrapolated(disorderindependent)valueisgiveninTa-
ble II.
FIG. 10: (Color online) Magnetization profiles in 4D in the
slabgeometryforfixed-freeb.c.-sinasystemofwidthL=32
4.5 forbox-handfixed-hrandomness. Theinterpolationformula
inEq.(4)isrepresentedbydashedlines. Intheinsettheratio
10-1 4 of the magnetization profile and the interpolation formula is
xc shown for x=2.72, xs=3.7, Afixed =4.19 and Abox=0.625
m 3.5 .
10-3 3 0 0.04 0.08 0.12 Close to the fixed surface the decay exponent is cal-
1/L culated as xb = 2.72(10), which agrees well with the
finite-size estimate of the bulk magnetization exponent,
fixed see Table II. The magnetization profile close to the free
10-5 box surface is shown in Fig. 11.
x =2.65
cb Estimates of the decay exponent, xsb, and that of the
3 10 30 surfacemagnetizationexponent, xs, whichare presented
l’+l’0 inTableII containsomewhatlargererrors,thaninlower
dimensionalcalculations. Howeverthescalingrelationin
Eq.(3)issatisfiedinthiscase,too. Alsotheinterpolation
FIG. 9: (Color online) Magnetization profiles near the free formulainEq.(4)isagoodapproximationascanbeseen
cornerin3Dinthepyramidgeometryforthetwotypeofran- in the inset of Fig. 10.
domnesswithL=64. Inbothcasesthedecayischaracterized
by the exponent, xcb = 2.65(25). In the inset the finite-size
estimates for the edge magnetization exponent are presented
IV. DISCUSSION
up to L = 3241. The extrapolated (disorder independent)
valueis given in Table II.
InthispaperwehaveusedtheSDRGmethodtocalcu-
late the magnetization profiles of the random transverse
Ising model in 2D, 3D and 4D in different geometries:
3. Corner magnetization
slab, corner and wedge having a fixed surface. At the
criticalpoint decayexponents arecalculatedboth at the
Wecloseourstudyin3Dbycalculatingthemagnetiza- fixed end and at the free end of the profiles. These ex-
tionprofileinthe pyramidgeometry: the resultisshown ponents, which are presented in Table II are found to be
in Fig. 9 close to the free corner. (In the inset finite-size disorder independent, at least for strong enough disor-
estimates of the corner exponent are presented.) Esti- der, for which the critical properties of the system are
mates of the decay exponent, x , and the corner ex- controlled by an IDFP. From finite-size scaling studies
cb
ponent, x , are presented in Table II, which satisfy the of the local magnetization at the free ends of the profile
c
scaling relation in Eq.(3). local(surface,cornerandedge)criticalexponentsarecal-
7
tic exponent in the IDPF1,4, which has been calculated
in Refs.16–18. Thus by calculating mtyp by some other
-1 4 s
10 means one can obtain independent estimates for the ex-
ponent ψ.
xs 3.8
Concerningthedimensionaldependenceoftheaverage
m
10-3 3.6 surface magnetizationexponent, we write it in the form:
x (D) = D +p(D), where D = D −1 is the dimen-
0 0.04 0.08 0.12 s s s
1/L sionofthe surfaceandp(D) is anumber closeto 1/2. In
-5 1D p(1) = 1/2 is shown to be the persistence exponent
10
of the random walk9 and we propose here an analogous
fixed
bxox= 0.85 explanationforD >1,too. Letusdenotebyµsthenum-
sb ber of surface points of the correlation cluster, C which
1 2 5 10 20
starts at the fixed boundary. We have checked, that µ
l’+l’ s
0 has an exponential distribution: P(µ ) ∼ exp(−µ /µ˜),
s s
with µ˜ ≈ B(N/L)D 1, c.f. in 2D we have B = 0.5 and
−
B = 1.0, for fixed-h and box-h disorder, respectively.
FIG. 11: (Color online) Magnetization profiles near the free Consequently the surface points of C are grown from
boundary in 4D for the slab geometry for the two type of
uncorrelated domains. The average number of surface
randomness with L = 32. In both cases the decay is char- points in an area LD 1 then scales as [µ ] ∼ L p(D).
acterized by the exponent, xsb = 0.85(15). In the inset the As explained in Ref.18−(see Fig. 5 there) sthaevcorre−lation
finite-size estimates for the surface magnetization exponent
cluster is embedded into a connected subgraph, which
arepresented. Theextrapolated(disorderindependent)value
contains all the decimated sites (the results of both h-
is given in Table II.
and J-decimations) and which is related to a low-energy
excitation of the system. The number of points in the
culated, see Table II. For all types of profiles considered connected subgraph is L˜ ∼ LDf, where the fractal di-
here the scaling relations in Eqs.(2) and (3) are satisfied mension, Df,is closeto one. If wereplacethe connected
and the interpolation formula in Eq.(4) is found to be a subgraphwithalinear chainwithL˜ sites thenwe obtain
good approximation in the slab geometry. Our results, from the random walk result: p(D)≈D /2. Indeed our
f
concerningthepropertiesoftheaveragelocalmagnetiza- numerical estimates19 of D and the surface magnetiza-
f
tion of the RTIM are rather complete, these are compa- tionscalingdimensionsinTableIIareinagreementwith
rablewiththeexistingresultsinthenon-randomsystem. this relation. In higher dimensions we expect that the
By the SDRG method the average local magnetiza- structure of the SDRG transformation, in particular the
tion, m , is obtained as the ratio of such rare realiza- topology of the connected clusters follows the trend ob-
l
tions, in which the correlation cluster contains the given servedinthispaper,thusthesurfacemagnetizationexpo-
site. By this method the typical value of the local mag- nentgenerallyobeystherelation: x (D+1)−x (D)≈1.
s s
netization could be estimated by the strength of the Based on this result we expect that a simplified SDRG
effective coupling, J , which is generated between the procedure, such as a modified version of those used in
1′l
fixed surface and the site. For surface spins it scales as Refs.25–31 can be constructed, which captures the main
mtyp ∼ J ∼ exp(−ALψ), where ψ is a characteris- resultsaboutthesurfacecriticalpropertiesoftheRTIM.
s 1′L
TABLE II: Estimates of the critical exponents obtained by
finite-sizescaling: x,xs, xc, xe and theexponentsassociated
withthedecayoftheprofile: xb,xsb,xcb,xeb. (xistakenfrom
Refs.16–18andtheexactresultsin1DarefromRefs.3,5,9). For
the pure system (see c.f. in35) xs =0.5 and 1.27 for 1D and
2D, respectively and for D 3 the mean-field result holds:
≥
xs=(D+1)/2. The corner exponent in 2D is xc =2.06.
bulk surface corner edge
x xb xs xsb xc xcb xe xeb
1D (3 √5)/4 0.5
−
2D 0.982(15) 0.98(1) 1.60(2) 0.65(2) 2.3(1) 1.35(10)
3D 1.840(15) 1.855(20) 2.65(15) 0.84(7) 4.2(2) 2.65(25) 3.50(15) 1.75(15)
4D 2.72(12) 2.72(10) 3.7(1) 0.85(15)
8
Our results about the local critical behavior of the the different directions. One can also consider the sur-
RTIM are relevant to other random quantum magnets facecriticalbehaviorinthe presenceofenhancedsurface
havingdiscretesymmetry,wementiontherandomquan- couplings, in which case the so called extraordinary and
tumPotts42,clockandAshkin-Tellermodels43. Alsothe surface transitions33–35 could be studied, too.
surface,cornerand/oredgeexponentsoftherandomcon-
tactprocessareexpectedtobegivenbytheRTIMvalues
in Table II. To check this conjecture one should repeat
recent Monte Carlo simulations about this model22,23.
Acknowledgments
Our studies of the local critical behavior can be ex-
tendedindifferentdirections. Forexample,onecanmea-
surethecornerandedgemagnetizationexponentsatdif- This work has been supported by the Hungarian Na-
ferentopeninganglesoronecanconsideranisotropicsys- tionalResearchFundundergrantNoOTKAK75324and
tems, in which the distribution of disorder is different in K77629.
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