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the realistic world is affected by randomness . unlike non - random systems , those with randomness sometimes show rich and complicated phenomena . one of the interesting issues for such random systems is spin glass .
many studies mainly by the mean - field analysis have been successful to elucidate various concepts for understanding spin glasses @xcite .
one of the current issues in spin glasses is their nature in finite dimensions below the upper critical dimension .
unfortunately , for finite dimensions , we often rely on numerical simulations , because there are few ways to analytically study spin glasses in finite dimensions .
we need long equilibration times for the numerical simulations for spin glasses and average over many realizations of random systems to make error bars small enough .
it is thus difficult to give conclusive understanding on nature of spin glasses in finite dimensions .
to establish reliable analytical theories of spin glasses has been one of the most challenging problems for years .
a part of successful analyses to elucidate properties on spin glasses is by the use of the gauge symmetry . by use of the gauge symmetry ,
one can obtain the exact value of the internal energy , evaluate the upper bound for the specific heat , and obtain some correlation inequalities in a subspace known as the nishimori line hn81,rev3 .
this gauge symmetry also enables us to rewrite the free energy along the nishimori line as the entropy for the distribution of frustration @xcite .
many aspects on spin glasses are essentially related with frustration .
therefore we expect the possibility that a basis of establishment of a systematic approach to spin glasses would be in the gauge symmetry .
a recent related development with the gauge symmetry is the _ conjecture _ to predict the location of the multicritical point , which is the special point lying on the intersection between phase boundaries and the nishimori line as in fig .
[ fig1 ] @xcite .     ising model on two - dimensional lattice ( left panel ) and on higher dimensions ( right panel ) .
the vertical axis expresses the temperature @xmath6 , and the horizontal line denotes the concentration @xmath7 of the antiferromagnetic interactions .
the multicritical point is described by the black point ( mcp ) .
the nishimori line is described by the dashed line . for higher dimensions , not only the ferromagnetic ( fm ) and paramagnetic phases ( pm ) but also the spin glass phase ( sg ) exists.,width=283 ]
the predictions by the conjecture have shown agreement with numerical estimations roughly with precision to the third digit .
the conjecture has opened a way of a general scheme for determination of the location of the multicritical points for spin glasses on any self - dual lattices and mutually dual pairs of lattices .
nevertheless it has been found that the conjecture on several mutually dual pairs of hierarchical lattices does not always give predictions in agreement with estimations by the renormalization group analysis @xcite .
such discrepancies are not negligible , because the renormalization group analysis on hierarchical lattices gives exact solutions . to construct a more reliable technique ,
we have improved the technique leading to the location of the multicritical point by combining the concepts of the renormalization group analysis with the duality @xcite .
the improvement for hierarchical lattices has greatly succeeded as seen in the literature , because the discrepancies between the predictions by the improved technique and the exact estimations by the renormalization group analysis actually decrease .
reconsidering the improved technique on the hierarchical lattices , in this paper , we apply the improved technique to the regular lattices such as the square , triangular , and hexagonal lattices .
the present paper is organized as follows . in sec .
ii , we introduce the conventional conjecture and take a look at several published predictions .
in addition , the problem on the conjecture is pointed out here .
the improved technique on the regular lattices is proposed after review of the case on the hierarchical lattices in sec .
iii , and formulated in sec .
iv . the improved version shows a very close relationship with the entropy of the distribution of frustration as shown in this section . in sec .
v , we carry out the explicit calculations by the improved technique for the regular lattices . moreover we have to carefully evaluate the performance of the improved technique , comparing their predictions with the existing results .
we examine the correspondence with the domany s exact result domany1,domany2 of the slope of the critical point on the phase diagram .
the conclusion is in the last section of the present paper , sec .
it will be useful to review the analysis by the duality , for the conjecture is established by combination of the duality and the replica method nn , mnn .
the duality is one of the tools to identify the transition points for various types of classical spin systems such as the ising model , and the potts model by use of a symmetry embedded in the partition function @xcite .
we take the non - random ising model on the square lattice as an example .
the partition function is given as @xmath8 where @xmath9 is the ising spin taking @xmath10 and the product with the subscript @xmath11 is over the nearest neighboring sites .
we can regard this partition function as the multi - variable function of components of the edge boltzmann factor . in this case
, the edge boltzmann factor is @xmath12 ) .
we consider that the dependence on @xmath13 emerges through this edge boltzmann factors .
we set two components as @xmath14 and @xmath15 for convenience .
the component @xmath16 is often called the principal boltzmann factor , which is defined by the edge boltzmann factor for the state with all edge spins parallel the principal boltzmann factor is an important component throughout this paper .
the duality is carried out by the fourier transformation for this edge boltzmann factor defined on each bond of the lattice @xcite .
the two - component fourier transformation gives the dual edge boltzmann factors as , @xmath17 as a result , we establish the relation between the partition functions with different components as , @xmath18 we here extract two principal boltzmann factors @xmath19 and @xmath20 to measure the energy from the state with edge spins on each bond being parallel as , @xmath21 where @xmath22 stands for the number of bonds , and @xmath23 and @xmath24 are called the relative boltzmann factors defined as @xmath25 and @xmath26 . each partition function
is now reduced to a single - variable function of @xmath23 and @xmath24 , whose explicit forms are , @xmath27 being very well known , the duality relation can be given as @xmath28 to think of the dual partition function as one of another ising model with the edge boltzmann factor @xmath29 .
we can identify the critical point as a fixed point of the duality as @xmath30 , under the assumption of a unique transition . on this critical point , an appealing equation is satisfied @xmath31 .
we consider the case of the random - bond ising model and review the conventional conjecture @xcite .
the hamiltonian is defined by @xmath32 where @xmath33 denotes the quenched random coupling and the summation is over the nearest neighboring sites . though various types of distribution for @xmath33 can be considered , we here restrict ourselves to the @xmath0 ising model for convenience . the distribution function for the @xmath0 ising model
is given by @xmath34 where @xmath35 is defined by @xmath36 .
the nishimori line is given by the condition @xmath37 and is described by the dashed line in each phase diagram of fig .
[ fig1 ] .
we apply the replica method to the @xmath0 ising model on the nishimori line as , @xmath38_{\mathrm{av}}\ ] ] where @xmath39 stands for the replica number and the angular brackets denote the configurational average .
we apply the duality argument as reviewed above to this @xmath39-replicated @xmath0 ising model .
the duality gives the relationship of the partition functions with different components of the edge boltzmann factor , @xmath40 where the subscript of @xmath41 and @xmath42 denotes the number of antiparallel - spin pairs among the @xmath39 replicas .
two principal boltzmann factors @xmath19 and @xmath43 are given as @xcite , @xmath44_{\mathrm{av } } , \\
x^*_0(\beta ) & = & \left [ 2^{\frac{n}{2 } } \cosh^n\beta j_{ij } \right]_{\mathrm{% av}}.\end{aligned}\ ] ] we extract these principal boltzmann factors similarly to the case of the non - random bond ising model , @xmath45 we remark that the partition function for the @xmath39-replicated @xmath0 ising model is a multi - variable function of the edge boltzmann factors yet differently from the case of the non - random bond ising model .
we describe schematically the relationship two reduced partition functions @xmath46 as the curves of the relative boltzmann factors @xmath47 ( the thin curve going through the multicritical point @xmath2 ) and @xmath48 ( the dashed line ) as in fig .
[ trajectory ] .
we now consider the relationship between these curves by the projections on the two - dimensional plane @xmath49 for convenience .     ising model . ,
width=283 ]    as the temperature changes from @xmath50 to @xmath51 , the representative point @xmath52 moves toward the point p ( the high - temperature limit ) along the thin line in fig .
[ trajectory ] .
then the corresponding dual point @xmath53 moves along the dashed line in the opposite direction from p to f ( the low - temperature limit ) .
these features have been shown rigorously and imply the existence of the duality relation for the temperature @xcite .
if two curves describing change of @xmath54 and @xmath48 become completely coincident with each other , we can obtain a relation @xmath55 . solving this relation ,
we obtain the duality relation for the temperature @xmath56 .
the well - known duality relation @xmath28 for the non - random ising model can be indeed derived from the relation @xmath57 .
unfortunately the thin curve @xmath54 does not coincide with the dashed curve @xmath48 for the replicated @xmath0 ising model on the nishimori line as shown schematically in fig .
[ trajectory ] . in the case for the replicated @xmath58 ising model
, we can neither find a duality relation for the temperature explicitly , nor identify the multicritical point as the fixed point of duality .
we thus have provided a hypothesis on determination of the multicritical point @xcite .
we assume that the equation @xmath59 is also satisfied at the multicritical point , similarly to the non - random bond ising model at the critical point , for any @xmath39-replicated systems including the quenched system ( @xmath60 ) @xcite .
validity of this hypothesis can be rigorously shown for @xmath61 and @xmath62 , and for @xmath63 , and has been numerically confirmed for the @xmath64 case of the replicated @xmath0 ising model on the square lattice within error bars of the numerical simulation @xcite .
relying on these facts , we assume that we can predict the location of the multicritical point for the @xmath0 ising model on the square lattice by the single equation @xmath65 .
the quenched limit @xmath60 for this equation yields @xcite , @xmath66 the solution to this equation is @xmath67 .
we can also predict the location of the multicritical point for other types of randomness .
for instance , that of the gaussian ising model with the mean @xmath68 and the variance @xmath69 is given as @xmath70 @xcite .
.comparisons among the results derived by the conventional conjecture and the existing numerical results .
sq denotes the square lattice , tr means the triangular lattice , and hex means the hexagonal lattice . [ cols="<,<,<",options="header " , ]     the estimated values for the square lattice shows convergence to the exact solution @xmath71 @xcite , similarly to the case for the @xmath0 ising model .
in addition , we remark that the improved technique works very well for the critical points of the bond - diluted @xmath72-state potts model , and @xmath72-state villain model @xcite . from these points of view
, we conclude that the improved technique is also a systematic approach leading to the precise locations of the critical points in broader classes of the random spin systems .
we proposed an improved technique applicable to the square , triangular , hexagonal lattices , and derived the precise locations of the multicritical points for the @xmath0 ising model , the gaussian ising model , and the potts spin glass on the square lattice , as well as the @xmath0 ising model on the triangular lattice and the hexagonal lattice .
this improved technique is still approximation for the location of the multicritical point .
however we can enhance the precision of the approximation by the summation over spins in the cluster taken from the considered lattice , if we need the precise location of the critical points in a random spin system .
this would open a way to analytically derive the location of the critical points in random spin systems with very high precision .
unfortunately , in the low - temperature region under the nishimori line , the improved technique can not give satisfactory answers yet .
we solve this problem in the low - temperature region under the nishimori line , and have to examine the validity of some hypotheses on the improved technique .    in this paper , we restrict ourselves to the random spin systems in two - dimensional systems .
however we can apply the duality to other dimensional systems .
for example , the duality can transform the random - bond ising model on the three - dimensional cubic lattice into the random - plaquette gauge model on the three - dimensional cubic lattice .
the random - plaquette gauge model is an attractive one in terms of the quantum toric code dklp , whp .
an accuracy threshold to correct error of the quantum toric code corresponds to the location of the multicritical point on the random - plaquette gauge model with the random couplings following the @xmath0 distribution function on the three - dimensional cubic lattice .
the conventional conjecture relates this threshold with the location of the multicritical point of the @xmath0 ising model on the three - dimensional cubic lattice @xcite .
the improved technique also can not directly derive such an accuracy threshold , but can make more precise relationship between the locations of the multicritical points on the random - bond ising model and the random - plaquette gauge model .    as another direction of studies in the future
, we should clarify the physical meaning of the equation consisting of the entropy of the distribution of frustration , which determines the location of the multicritical point .
the author greatly acknowledges the fruitful discussion with prof .
h. nishimori , prof .
a. n. berker and dr .
m. hinczewski , and sincerely thanks prof .
y. okabe for sending unpublished numerical data of the bond - diluted potts model and the bond - diluted villain model , and prof .
m. picco , who informed him of a his recent result of the multicritical point on the square lattice .
he would like to also thank prof .
s. l. a. de queiroz for a valuable comment and dr .
k. takahashi for reading the manuscript and giving stimulating comments .
this work was partially supported by the ministry of education , science , sports and culture , grant - in - aid for young scientists ( b ) no .
20740218 , and for scientific research on the priority area deepening and expansion of statistical mechanical informatics ( dex - smi ) , and by crest , jst .
we present an analysis leading to precise locations of the multicritical points for spin glasses on regular lattices . the conventional technique for determination of the location of the multicritical point was previously derived using a hypothesis emerging from duality and the replica method . in the present study , we propose a systematic technique , by an improved technique , giving more precise locations of the multicritical points on the square , triangular , and hexagonal lattices by carefully examining relationship between two partition functions related with each other by the duality . 
 we can find that the multicritical points of the @xmath0 ising model are located at @xmath1 on the square lattice , where @xmath2 means the probability of @xmath3 , at @xmath4 on the triangular lattice , and at @xmath5 on the hexagonal lattice . 
 these results are in excellent agreement with recent numerical estimations .