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We study the continuum limit of the quantum nonlinear sigma model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (1002 x 8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point gc that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group beta function close to the critical point. Using the beta function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve xi /a sigma =f(T/T sigma ). Even though the lattice spacing vanishes, a finite unit of length a sigma and a temperature scale T sigma is generated via dimensional transmutation. Assuming that the nonlinear sigma model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units a sigma and T sigma to the lattice spacing aH and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales a sigma ,T sigma and aH, J is obtained, and f(T/T sigma ) can be accurately approximated by an exponential of T sigma /T below gc. We obtain a reasonable fit to the neutron scattering data of the insulator La2CuO4 by taking J=1270 K, a value close to that reported by Raman scattering experiments.
©1989 The American Physical Society.
PACS: 75.10.Jm 74.20.-z
It is the discovery of the copper oxide superconductors^{1}that intensified the study of certain theoretical models such as the antiferromagnetic (AF) Heisenberg model. The belief that the superconductivity mechanism in these materials is related to the strong correlations among purely electronic degrees of freedom^{2}as well as neutron scattering experiments^{3}which bring out strong two dimensional spin correlations have given credit to the spin cf12 AF Heisenberg model. Starting from one of the simplest models to take into account electron correlations in a nearly half filled band such as the Hubbard model, the AF Heisenberg model can be obtained at half filling by taking the strong coupling limit.^{4}In that formulation, the Heisenberg model
H hat = J ^ sum from { langle i , j rangle } ^ S vec_{i}cdot S vec_{j , }(1.1)
describes interactions (that originate from virtual electron hopping processes) between the conduction band electrons localized in the Wannier states around nearest neighbor unit cells of the copper oxide plane. Here S vec_{i_}is the spin cf12 operator of the ith cell.
Recently we simulated^{5,6}the two dimensional spin cf12 AF Heisenberg model using Handscomb's quantum Monte Carlo method. We calculated the correlation length and we found that it increases very rapidly with decreasing temperature. Our results are consistent^{6}with neutron scattering experiments. It is, however, difficult to find an efficient quantum Monte Carlo algorithm to study large systems and approach low temperatures.
It is believed^{7}that the long wavelength limit of the two dimensional (2D) quantum Heisenberg model (1.1) is equivalent to the quantum nonlinear sigma model in two space one Euclidean time dimensions. Namely, if a relationship between the parameters of the two models is appropriately established, the two models are equivalent in the parameter range where the correlation length is much larger than the lattice spacing. More recently there was a Soviet proposal^{8}that in the derivation of the nonlinear sigma model from the AF Heisenberg model one has to add a topological term in order to make the models equivalent. The consequences which such topological terms might have in the development of the theory of superconductivity in copper oxides, as well as the mathematical beauty which dresses such theories were part of the reason for the excitement about this direction of research. Later, however, the necessity of such topological terms became less clear, and in fact today the equivalence between the two models without any additional terms is still outstanding.^{9}
The nonlinear sigma model in 2+1 dimensions has recently been studied by Chakravarty, Halperin, and Nelson (CHN).^{10}Using one loop perturbative renormalization group approach, CHN relate the nonlinear sigma model to the spin cf12 AF Heisenberg model at low temperatures and give a good fit to the neutron scattering data^{3}taken on La_{2_CuO}sub 4.
Recently^{11}we have simulated the nonlinear sigma model in two space one Euclidean time dimensions and at finite physical temperature T using the Monte Carlo (MC) method. We found that we can make contact between the parameters of the spin cf12 Heisenberg model and the nonlinear sigma model by comparing the behavior of the correlation lengths at low temperatures. We also found good agreement with the results of the neutron scattering data^{3}by taking J=1270 K. The goal of the present paper is twofold. We offer more details about the simulation of Ref. 11 with new results obtained in the saddle point approximation (SPA), and we also compare the results of the two different calculational schemes. In Sec. II of this paper we formulate the nonlinear sigma model on the lattice, and in Secs. III and IV we describe the MC and SPA methods and compare them. In Sec. V we take the continuum limit of the nonlinear sigma model in 2+1 dimensions. We determine how the coupling constant of the theory (spin stiffness) should depend on the lattice spacing so that the results for the correlation length are independent of the cutoff. We calculate the model's renormalization group beta function around the three dimensional critical point that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). Using the beta function we rescale the calculated correlation lengths at various couplings and temperatures and find that they all collapse on the same curve xi / a_{sigma}= f ( T / T_{sigma})_. Even though the lattice spacing vanishes, a finite unit of length a_{sigma_}and a temperature scale T_{sigma_}are generated via dimensional transmutation. In Sec. VI, the two parameters a_{sigma_}and T_{sigma_}of the sigma model are related to the lattice spacing a_{H_}and the AF coupling J of the Heisenberg model. The numerical relationship between a_{sigma_,}T_{sigma_,}and a_{H_,}J is obtained by fitting xi (T)= f ( T / T_{sigma}) a_{sigma_}to the correlation length xi (T)= xi_{H}(T/J)_a_{H_}obtained from the simulations of the spin cf12 AF Heisenberg model. As a consequence of the assumption that the two models are equivalent at low T, we find that the spin cf12 AF Heisenberg model must order at T =0. We find that we need J=1270 K to fit the neutron scattering data^{3}for the spin correlation length of the insulator La_{2_CuO}sub 4.
The nonlinear sigma model in two space one Euclidean time dimensions is defined as^{7,10}
S_{roman}eff = rho_{0}over { 2 hbar c } int_{0^{{}beta}hbar c } d tau int d x ^ d y [ ( partial_{x}OMEGA vec )^{2}+ ( partial_{y}OMEGA vec )^{2}+ ( partial_{tau}OMEGA vec )^{2}] , L
where OMEGA vec is a three component vector field living on a unit sphere (2.2) sum from { alpha = 1 } to n ^ OMEGA_{a^{2}=1 . }(2.1)
c is the bare (unrenormalized) spin wave velocity, beta = ( 1 / K_{B}T )_, and in the case of our interests n=3. Transforming this problem on the 2+1 dimensional lattice we obtain
S_{roman}eff = - 1 over 2g sum from { x vec } ^ sum from { mu = 1 } to 3 ^ OMEGA vec ( x vec ) cdot [ OMEGA vec ( x vec + e hat_{mu}) + OMEGA vec ( x vec - e hat_{mu}) ] , L (2.3)
where x vec covers the 2+1 dimensional space time lattice of lattice spacing a and size N^{2}N_{beta_,}i.e., x_{1_,}x_{2}= 1_, 2 ,...,N, and x_{3}=1_, 2,..., N_{beta_,}
beta hbar c = N_{beta}a , (2.4)
and g = hbar c / rho_{0}a_. We have to impose a periodic boundary condition in the Euclidean time direction, i.e., OMEGA vec ( x vec + N_{beta}e hat_{3})_= OMEGA vec ( x vec ). In this model, the average of the field OMEGA vec is proportional to the average stagggered magnetization and could describe the dynamics of the spins within one isolated CuO_{2_}layer.
The generating functional Z [J] defined as
Z [J] == int ^ prod from { x vec , alpha } ^ d OMEGA_{alpha}( x vec ) exp ( - S_{roman}eff + sum from { x vec , alpha } ^ J_{alpha}( x vec ) OMEGA_{alpha}( x vec ) right ) L (2.5)
gives us the n-point correlation functions. For example, the two point function is given by
left G_{{alpha}, alpha prime } ( x vec , y vec ) = { int prod_{{}x vec prime , alpha } d OMEGA_{alpha}( x vec prime ) OMEGA_{alpha}( x vec ) OMEGA_{{}alpha prime } ( y vec ) exp ( - S_{roman}eff ) } over { int prod_{{}x vec prime , alpha } d OMEGA_{alpha}( x vec prime ) exp ( - S_{roman}eff ) } = 1 over { Z [ J ] } { delta^{2}Z [J] } over { delta J_{alpha}( x vec ) delta J_{{}alpha prime } ( y vec ) } right |_{J=0 . }(2.6)
From the two point function we can calculate the correlation length in lattice units xi_{roman}latt_ as a function of g and for various values of N_{beta_}and N. We have to take the limit N -> inf and keep the time dimension finite so that Eq. (2.4) is satisfied. If, therefore, N is large enough so that
xi_{roman}latt << N , (2.7)
the correlation length is only a function of N_{beta_}and g, and in physical units is given by
xi = xi_{roman}latt ( g , N_{beta}) a . (2.8)
For continuum limit behavior and for eliminating finite size effects xi_{roman}latt_ must satisfy
1 << xi_{roman}latt << N . (2.9)
We used the heat bath algorithm because in simulations of the classical O (n) models^{12}it seems superior over the Metropolis algorithm. In this method the field OMEGA vec at the site x vec is updated as follows. We calculate the sum over the fields at the neighboring sites,
omega vec ( x vec ) == sum from { mu = 1 } to 3 ^ OMEGA vec ( x vec + e hat_{mu}) + OMEGA vec ( x vec - e hat_{mu}) , (3.1)
and denote the polar and the azimuthal angles by theta and phi with respect to a local coordinate system having the z axis parallel to omega vec ( x vec ). The angle theta is drawn from the distribution
P ( theta ) = exp ( 1 over 2g omega ( x vec ) cos theta right ) (3.2)
and the phi from a uniform distribution in the interval (0, 2pi). Finally we find the coordinates of OMEGA vec ( x vec ) with respect to a global coordinate system.
We extract the correlation length from the projected correlation function
G_{p}( r vec , r vec prime ) == langle s vec ( r vec ) cdot s vec ( r vec prime ) rangle , (3.3a)
where
s vec ( r vec ) == 1 over N_{beta}^ sum from { x_{3}= 1 } to { N_{beta}} ^ OMEGA vec ( x vec ) , (3.3b)
and r vec is only the space part of the vector x vec. The average in Eq. (3.3a) is taken with respect to the distribution e^{{}- S_{roman}eff }_. In our simulation we used periodic boundary conditions in the space boundaries also. Typically we used 5000 Monte Carlo steps over the entire lattice for thermalization and about 10^000 for measurements. In Fig. 1 the square root of the expectation value of the staggered magnetization squared (i.e., OMEGA^{2)}is plotted as a function of g for N = N_{beta_=4,}8, 16, and 32. We see that the 3D critical point which is associated with spontaneous staggered magnetization is around g=1.45. In this paper we determine g_{c_}accurately using finite size scaling analysis.
In Fig. 2 we give the correlation function for g=1.6, N=100, and N_{beta_=6.}For periodic boundary conditions the correlation length is extracted by fitting the long distance behavior of the correlation function with
G_{p}( x - x prime ) = A ( mark e^{{}- m_{roman}latt | x - x prime | }
lineup + e^{{}- m_{roman}latt ( N - | x - x prime | } ) , (3.4)
i.e., with a B ^ cosh [ m_{roman}latt_( N/2 - | x - x prime | )], where N is the size of the one space direction. The solid line in Fig. 2 is the result of the fit. Notice that the correlation function on the logarithmic scale drops as a straight line by several orders of magnitude up to the point at which its value is as small as the error. The value of g in this case is above the critical point g_{c_=1.45,}namely, in the disorder phase where the n (n is the number of spin components and in our case n=3) modes of the theory have the same mass. In Fig. 3 we give a typical example of the correlation function below g_{c_}for g=1 and the same size lattice. The two unknown parameters A and the mass m_{roman}latt_= xi_{roman}latt^{-1_}can be determined using only two neighboring points of the correlation function. In Fig. 4 we give the mass as a function of the distance of the first of these two points from the origin. We notice that the mass drops significantly over a distance of about 10 lattice sites and beyond that it stays constant. In the region g < g_{c_}there are two masses in the theory; namely, there are n-1 modes which correspond to the Goldstone mode excitations and they become massless in the 3D theory (beta -> inf ). They are related to the radial motion of the average field and they give an exponentially small mass with the size of the finite beta hbar c. There is also a massive mode associated with fluctuations in the magnitude (radial component) of the average field. Notice that even though the local field lives on a unit sphere and, therefore, does not have radial fluctuations, the average field over a large volume of the system can have fluctuating direction as well as magnitude. In this paper we study the smallest mass, i.e., the larger correlation length which dominates the behavior of the correlation function at large distances [imagine that G ( r ) = A e^{{}- m_{1}r }_+ B e^{{}- m_{2}r }_ and because m_{1}> m_{2_,}G ( r) = B e^{{}- m_{2}r }_ at large distances].
The theory (2.3) with the field OMEGA vec satisfying the constraint (2.2) can be obtained from the following:
S_{roman}eff = sum from { x vec } ( mark - 1 over 2g ^ sum from { mu = 1 } to 3 ^ OMEGA vec ( x vec ) cdot [ OMEGA vec ( x vec + e hat_{mu}) + OMEGA vec ( x vec - e hat_{mu}) ]
left lineup + lambda (OMEGA^{2}( x vec ) - 1 )^{2}right) , (4.1)
in the limit lambda -> inf. The additional term in that limit gives delta ( size -2 sum_{{}alpha = 1 }^{n_OMEGA}sub a^{2}- 1 ), which is the constraint (2.2). Hence, we can study the theory (4.1) and choose to take the limit lambda -> inf at the end of the calculation. Using the identity
sqrt { alpha / pi } int d rho ( x vec ) e^{{}- alpha [ rho ( x vec ) - i OMEGA^{2}( x vec ) ]^{2}} =1 , (4.2)
we can introduce the auxiliary scalar field rho ( x vec ) in the generating functional Z [J] on every site of the lattice. Using Eq. (4.1) for S_{roman}eff_ we can cancel the lambda ( OMEGA^{2})^{2}by choosing alpha = lambda. Finally, shifting the field OMEGA vec ( x vec ) by a nonfluctuating vector field C vec ( x vec ),
OMEGA vec ( x vec ) = phi vec ( x vec ) + C vec ( x vec ) (4.3a)
and choosing C vec ( x vec ) such that the coefficient of the term linear in phi vec ( x vec ) vanishes, i.e.,
1 over g ^ sum from mu [ C vec ( x vec + e hat_{mu}) + C vec ( x vec - e hat_{mu}) ] L
+ 4 lambda [ 1 + i rho ( x vec ) ] C vec ( x vec ) + J vec ( x vec ) = 0 , R (4.3b)
we find
C vec ( x vec ) = g ^ sum from { y vec } ^ K^{-1}( x vec , y vec ) J vec ( y vec ) . (4.3c)
Here K^{-1}is the inverse of the matrix K which has matrix elements given by
K ( x vec , y vec ) = mark - sum from { mu =1 } to 3 ( delta_{{}x vec , y vec + e hat_{mu}} + delta_{{}x vec , y vec - e hat_{mu}} - 2 delta_{{}x vec , y vec } )
lineup - delta_{{}x vec , y vec } lcurl 6 + 4 lambda g [ 1 + i rho ( x vec ) ] rcurl , (4.3d)
and it is diagonal in the internal space of the components of the field. After some straightforward algebra we obtain
Z [J] = C prime int ^ prod from { x vec } ^ d rho ( x vec ) ^ prod from { x vec , alpha } mark d phi_{alpha}( x vec ) exp [ - sum from { x vec } ( 1 over 2g phi vec ( x vec ) ^ sum from { y vec } ^ K ( x vec , y vec ) phi vec ( y vec ) + lambda rho^{2}( x vec ) right ) right ]
lineup times exp ( g over 2 ^ sum from { x vec , y vec , alpha } ^ J_{alpha}( x vec ) K^{-1}( x vec , y vec ) J_{alpha}( y vec ) right ) . (4.4)
The exponent of (4.4) is now quadratic in the phi vec field and hence it can be integrated out to obtain
Z [J] = C prime prime int ^ prod from { x vec } ^ mark d rho ( x vec ) e^{{}- S prime } L
lineup times exp ( g over 2 ^ sum from { x vec , y vec , alpha } ^ J_{alpha}( x vec ) K^{-1}( x vec , y vec ) J_{alpha}( y vec ) right ) ,
and (4.6) S prime = lambda ^ sum from { x vec } ^ rho^{2}( x vec ) + n over 2 Tr[ ln (K) ] . (4.5)
We have, therefore, succeeded to eliminate the vector field OMEGA vec at the expense of another one component field rho ( x vec ). The above expression can be the framework of a systematic expansion in 1/n for large n. Here we restrict ourselves to the saddle point approximation which is the semiclassical approximation and also the zeroth order in a 1/n expansion. This is in very close analogy to the Wentzel Kramers Brillouin (WKB) semiclassical approximation which is also the leading order in an expansion in powers of hbar. Several nontrivial phenomena may be understood in terms of the semiclassical approximation only.
From the saddle point equation
{ delta S prime } over { delta rho ( x vec ) } = 0 , (4.7a)
for a translationally invariant solution [rho (x)=const], we obtain
rho - n g i 1 over V ^ Tr ( K^{-1}) = 0 , (4.7b)
where
Tr ( K^{-1}) = sum from { p vec } 1 over { 2 SIGMA_{mu}( 1 - cos p_{mu}) - lcurl 6 + 4 lambda g [ 1 + i rho ] rcurl } , L
where p_{mu}= n_{mu}2 pi / N_{mu_,}mu=1,2,3, and N_{1}= N_{2}= N_, N_{3}= N_{beta_,}and n_{mu_=0,1,2,...,}N_{mu}- 1_. The volume is V = N^{2}N_{beta_.}We define (4.8) - m_{0^{2}==}6 + 4 lambda g ( 1 + i rho ) . (4.7c)
Then, m_{0^{2_}is}the solution of
1 over { 4 n lambda g^{2}} ( m_{0^{2}+}4 lambda g + 6 ) = 1 over V ^ sum from { p vec } ^ 1 over { 2 SIGMA_{mu}( 1 - cos p_{mu}) + m_{0^{2}} .
}
The two point correlation function (second derivative of Z [J] with respect to J in the saddle point approximation is given by g K^{-1}( x vec , y vec ) and in momentum space is given by (4.10) G ( p ) = g over { 2 SIGMA_{mu}( 1 - cos p_{mu}) + m_{0^{2}} . }(4.9)
To find the correlation length we transform the correlation function in Minkowski space and look for poles of the form p_{1}= p_{2}= 0_, p_{3}= m_. The correlation length is given by xi_{roman}latt = m^{-1_}with m given as a solution of 2 ( 1 - cosh m ) + m_{0^{2}=0_.}We obtain
xi_{roman}latt^{roman}SPA = 1 over { 2 ^ sinh^{-1}( m_{0}/ 2 ) } , (4.11a)
and if m_{0}<< 1_ then
xi_{roman}latt apeq m_{0^{-1 . }(4.11b)
}Taking the limit lambda -> inf we obtain
1 over ng = 1 over V ^ sum from { p vec } ^ 1 over { 2 SIGMA_{mu}( 1 - cos p_{mu}) + m_{0^{2}} . }(4.12)
The above equation can be solved for m_{0_}numerically for any value of g and N_{mu_}and find xi_{roman}latt_ using (4.11). We will come back to the full solution of (4.12), but let us first give an approximate solution. Keeping N_{beta_}finite we take the limit N -> inf and after splitting off the zero mode (p_{3}= 0 )_ on the right hand side, we obtain
1 over ng = I_{0}+ I_{1 , }(4.13a)
I_{0}= 1 over N_{beta}int_{{}- pi }^{pi}{ d^{2}p } over { ( 2 pi )^{2}} 1 over { 2 SIGMA_{{}mu = 1,2 } ( 1 - cos p_{mu}) + m_{0^{2}} , }(4.13b)
I_{1}= 1 over N_{beta}^ sum from { n_{3}= 1 } to { N_{beta}} ^ int_{{}- pi }^{pi}{ d^{2}p } over { ( 2 pi )^{2}} 1 over { 2 [ 1 - cos ( 2 pi n_{3}/ N_{beta}) + 2 SIGMA_{{}mu = 1,2 } ( 1 - cos p_{mu}) + m_{0^{2}} . }(4.13c)
Let k be some small momentum cutoff, so that for | p | < k we have
SIGMA_{{}mu = 1, 2 } cos ( p_{mu}) apeq 2 - p^{2}/2 .
We may write
I_{0}apeq mark 2 over N_{beta}int_{0^{k}{}d^{2}p } over { ( 2 pi )^{2}} 1 over { p^{2}+ m_{0^{2}} }L
lineup + 2 over N_{beta}int_{k^{pi}{}d^{2}p } over { ( 2 pi )^{2}} 1 over { 2 SIGMA_{{}mu = 1,2 } ( 1 - cos p mu ) + m_{0^{2}} . }(4.14)
Assuming that there is a range of g where m_{0}-> 0_ as N_{beta}-> inf_, we obtain
I_{0}apeq - 1 over { 2 pi N_{beta}} ln ( m_{0^{2})}+ I_{2 , }L (4.15a)
I_{2}= mark 1 over { 2 pi N_{beta}} ln ( k^{2}+ m_{0^{2}) }L
lineup + 2 over N_{beta}int_{k^{pi}{}d^{2}p } over { ( 2 pi )^{2}} 1 over { 2 SIGMA_{{}mu = 1,2 } ( 1 - cos p_{mu}) + m_{0^{2}} . }(4.15b)
Therefore,
m_{0^{2}=}e^{{}- 2 pi f N_{beta}} , (4.16a)
f = 1 over ng - ( I_{1}+ I_{2}) , (4.16b)
and in order to satisfy our assumption m_{0}( N_{beta}-> inf ) =0_, we must have
g < g_{c , }(4.17a)
g_{c}= 1 over { ( I_{1}+ I_{2}) n } . (4.17b)
Hence, the correlation length
xi ( T -> 0 ) = a ^ exp ( { pi hbar c f } over { a K_{B}T } right ) (4.18)
in the SPA approximation for g < g_{c_.}
Given values for N, N_{beta_,}and g we can solve Eq. (4.12) numerically for m_{0_}and obtain xi_{roman}latt_. In Fig. 5 we compare the MC data for 50^{2}times 4 with the solution of the SPA Eq. (4.12) for the same parameter values and lattice size. We note that the overall behavior is similar, but there are significant differences. For small correlation lengths and in the regions where we have strong finite size effects the SPA and the MC results agree reasonably well. But for larger correlation lengths with small finite size effects the results are very different. In Fig. 6 we give the correlation length obtained from the Monte Carlo calculation for lattices 50^{2}times 4 and 100^{2}times 4. We notice the correlation lengths feel strong finite size effects when xi_{roman}latt > 20_ which is somewhat smaller than half the size of the smaller lattice (because of the periodic boundary conditions). In the same figure we plotted the results of SPA for the same lattices. We see that the finite size effects begin at correlation lengths of about the same size.
There are two directions to improve upon the saddle point solution: (a) We can take into account small fluctuations around the saddle point by writing
rho ( x vec ) = rho_{0}+ chi ( x vec ) (4.19)
and expand S prime in powers of chi ( x vec ) and keep terms up to chi^{2}( x vec ). The integrals in the fluctuating field chi ( x vec ) are Gaussian and they can be carried out explicitly. (b) There may be nontranslational invariant solutions to the saddle point equation with nontrivial topological structure. It is not known, however, how to calculate the entropy of all possible classical nonconstant configurations.
In the rest of this paper, however, we would like to focus on our results obtained by means of a nonperturbative method such as the Monte Carlo simulation. We will only use the results of the SPA for comparison and as a guide.
First, let us keep the physical temperatuare constant. Using Eq. (2.4) we obtain
a = { hbar c beta } over N_{beta . }(5.1)
Increasing N_{beta_}we approach the continuum limit (a -> 0 ) at constant temperature. To keep the correlation length xi constant in physical units for any a, we should find the value of g which gives the same value of xi. This is achieved through Eq. (2.8) which defines the function g (a). In Fig. 7 we plot xi_{roman}latt ( g , N_{beta})_ for lattice size 50^{2}times N_{beta_}where N_{beta_=2,}4, 8. The combination of Eq. (2.8) and Eq. (5.1) gives
xi = { xi_{roman}latt ( g , N_{beta}) } over N_{beta}{hbar c } over { K_{B}T } . (5.2)
In order to keep xi constant at a fixed temperature we should keep the ratio
b = { xi_{roman}latt ( g , N_{beta}) } over N_{beta }(5.3)
constant. b is the physical value of the correlation length at temperature T in units of a_{T}== ( hbar c / K_{B}T )_. In Fig. 8 we give b as a function of g for several values of N_{beta_.}We notice the lines for various N_{beta_}pass through the same point (g_{c}, b^{star})_=(1.45+-0.01, 0.80+-0.05). Let us say that we would like to define the theory's coupling constant at the value b = b_{0_}shown in Fig. 8. The line b = b_{0_}intersects the various curves for different N_{beta_'s}(i.e., in this case with constant temperature for different a's), and the value of g's at the intersections define g ( a_{T}/ N_{beta})_. We note that
lim from { N_{beta}-> inf } g ( a_{T}/ N_{beta}) = g_{c . }(5.4)
This value corresponds to the curve with infinite slope at g = g_{c_.}Note that if we choose to define the theory at b = b^{star}then g ( a_{T}/ N_{beta})_= g_{c_}for large N_{beta_,}i.e., for small a. Because
b = { xi_{roman}latt ( g , N_{beta})} over N_{beta}= b^{star
}at g_{c_,}we obtain
xi^{star}= b^{star}{hbar c } over { K_{B}T } , (5.5)
where b^{star}= 0 +- 0.05. Notice that at T=0 this point turns into a critical point (this is the critical point of the 3D classical Heisenberg model). We have also performed calculations for lattices with sizes 100^{2}times N_{beta_}with various values of g. Table I lists the values of the correlation lengths obtained for 50^{2}times N_{beta_}and 100^{2}times N_{beta_}sized lattices with N_{beta_=2,}4, 6, 8 as a function of g.
In Fig. 9 we give b calculated in the SPA for N_{beta_=2,}4, 6, 8 and N large enough so that the data shown are free of finite size effects. We notice the same behavior. In this approximation g_{c_=1.325+-0.010,}smaller than the MC value, and b^{star=0.98+-0}which is bigger than the MC value.
The renormalization group (RG) beta function,
beta_{roman}RG == - a dg(a) over da , (5.6)
can be calculated from the results of the SPA or MC calculation. The curves of Figs. 8 and 9 for N_{beta_=2,}4, 6, correspond to lattice spacings a_{T_/2,}a_{T_/4,}and a_{T_/6,}respectively. At a fixed value of xi we can find the intersections g ( a_{T}/ N_{beta})_ and take the derivative (5.6). We can repeat this for various xi's. The results for beta_{roman}RG_ obtained using the correlation lengths in the SPA behave as shown in Fig. 10. At g = g_{c_,}beta_{roman}RG_ changes sign. At T =0, xi_{c}= inf_, and for g < g_{c_}the system enters to a phase with spontaneous symmetry breaking, where the staggered magnetization is nonzero. We see that close to the critical point beta_{roman}RG (g)_ is clearly linear and
beta_{roman}RG (g) = - beta_{1}( g - g_{c}) + ... . (5.7)
In the case of SPA, we find g_{c_=1.33+-0}and beta_{1}= 1 +- 0.02_.
In Fig. 11 we present the beta_{roman}RG (g)_ obtained from the MC calculation using correlation lengths up to b=2.5. We see a similar linear behavior giving g_{c_=1.450+-0.003}and beta_{1}= 1 +- 0.05_.
We can integrate Eq. (5.6) to obtain a (g)
a = a_{sigma}exp ( - int^{g}dg over { beta_{roman}RG (g) } right ) , (5.8)
where a_{sigma_}is a constant of integration. The above equation defines the function g(a) which characterizes the continuum theory. a_{sigma_}is a characteristic parameter of the theory and the cutoff should be removed in such a way that the combination
a_{sigma}= a ^ exp ( int^{g}dg over { beta_{roman}RG (g) } right ) (5.9)
remains finite. In field theory, the phenomenon in which a vanishing length scale (a -> 0) and a dimensionless parameter (g) produce a dimensional quantity ( a_{sigma})_ with units of length is called dimensional transmutation.^{13}
Using the linear approximation [Eq. (5.7)] close to the critical point we find
a (g) = a_{sigma}| g - g_{c}|^{{}1 / beta_{1}} . (5.10)
The constant a_{sigma_}is determined by the physical value we assign to the correlation length at given values of the other parameters. Namely, its value will be determined from the phenomena which this theory is assumed to describe. This operation is the goal of the next section of this paper.
Combining Eqs. (2.4) and (5.10) we obtain
N_{beta}= | g - g_{c}|^{{}- 1 / beta_{1}} T_{sigma}over T , (5.11a)
where
K_{B}T_{sigma}= { hbar c } over a_{sigma . }(5.11b)
Substituting a (g) and N_{beta_}from Eqs. (5.10) and (5.11a) in to Eq. (2.8) we obtain
xi / a_{0}= f ( T over T_{sigma}right ) , (5.12a)
where we have defined f as follows:
f (T/ T_{sigma}) == xi_{roman}latt ( g , | g - g_{c}|^{{}- 1 / beta_{1}} T_{sigma}over T right ) | g - g_{c}|^{{}1 / beta_{1}} .
Since the constants a_{sigma_}and T_{sigma_}are independent of g and xi is also independent of g in the process of removing the cutoff, the function in Eq. (5.12b) is only a function of the ratio t == T / T_{sigma_.}In Fig. 12 we show the function f (t). The data points in the figure are those of Table I with xi_{roman}latt < 25_ and they correspond to various g < g_{c_}and N_{beta_}values. We see that all scale to a universal curve. Again, we emphasize the occurrence of dimensional transmutation where, although the lattice spacing is removed together with g we obtain correlation lengths in units of a finite constant a_{sigma_}as a function of temperature t in units of T_{sigma_.}
The curve f(t) can be approximated by an exponential (5.13) f (t) = A_{sigma}exp ( B_{sigma}/ t ) . (5.12b)
To demonstrate this, in Fig. 13 we plot the function t ^ ln [ f (t) ] and we see a straight line which intersects the y axis at B_{sigma_=4}and has a slope ln( A_{sigma}) = -_2.53. In Fig. 12 the solid line corresponds to the exponential (5.13) with the above parameters. We would like to remind the reader that in SPA we also found an exponential form [Eq. (4.18)].
In the next section we shall attempt to make contact between this model, the spin cf12 AF Heisenberg model and the experiment. To close the discussion about the nonlinear sigma model, going back to Fig. 7, we may notice that xi_{roman}latt_( g , ^ N_{beta})_= f (g) for g >> g_{c_,}i.e., a function independent of N_{beta_.}Hence, Eq. (2.8) yields that xi is only a function of g,
xi ( g >> g_{c}) = f (g) a (g) . (5.14)
The function xi (g) is independent of T and therefore can be determined by performing the calculation at T=0. Finally, for g < g_{c_,}i.e., in the region which is characterized with order at T =0 (where xi = inf ), xi is growing faster than 1/T with decreasing T. The SPA solution and our numerical data suggest an exponential increase with beta =1/T [Eqs. (4.18) and (5.13)].
These results confirm the crossover phase diagram given by CHN.^{10}In their more recent paper, they also use the exponential form for xi with a constant prefactor. They also obtain the relation (5.5) and find b^{star}=1.1 a value close to our SPA result but somewhat higher than our MC result.
We would like to discuss the possibility of making contact between the spin cf12 AF Heisenberg model and the nonlinear sigma model. In Refs. 5 and 6 we simulated the former and we found it to grow much more rapidly than 1/T. More precisely, in Ref. 5, we fit the correlation lengths to two different forms:
xi (T) = C / T e^{b/T , }(6.1)
and
xi (T) = C e^{{}b / | T - T_{c}|^{1/2}} , (6.2)
and we found that the latter form fits better and concluded that our simulation indicated that topological excitations may play an important role in the dynamics of the spin cf12 Heisenberg antiferromagnet. Following our findings for the sigma model we attempt to fit our numerical results for the Heisenberg model to
xi / a_{H}= A_{H}exp ( B_{H}J / T ) . (6.3)
This form, i.e., without the 1/T prefactor, fits our data equally well as the form (6.2), as shown in Fig. 14. The fit gives (see dashed line labeled l =0 in Fig. 7 of Ref. 6) A_{H_0.25+-0}and B_{H_=1.4+-0}(see Table II of Ref. 6 and B_{H}= 2 pi b_ in the notation of Ref. 6). In Fig. 14 we plot the function T / J ^ ln ( xi / a_{H})_ using the results of our calculation.^{5,6}The intersection with the T =0 axis gives B_{H_}and the slope ln( A_{H})_. The straight line fit gives A_{H_=0}and B_{H_=1.43.}In the last section we saw that the nonlinear sigma model has three different phases: (1) a phase with g > g_{c_,}where the correlation length is a constant independent of T, (2) a critical point at g = g_{c_}where the correlation length is proportional to 1/T, and (3) a phase with spontaneous staggered magnetization at T =0 for g < g_{c_,}where for finite T the correlation length grows approximately exponentially with 1/T as the temperature is lowered. Among the three forms the exponential fits better to the xi (T) of the spin cf12 AF Heisenberg model. Furthermore, assuming that the two models are equivalent at low T we conclude that the spin cf12 AF Heisenberg model should order at T =0 and the results of our simulation^{5,6}may also be consistent with spin wave theory and existence of an ordered state at T =0. Equivalence between the two models requires
A_{H}a_{H}= A_{sigma}a_{sigma , }(6.4)
B_{H}J = B_{sigma}T_{sigma . }(6.5)
From Eq. (6.4) we obtain a_{sigma}= 3 a_{H_,}from Eqs. (6.5) T_{sigma}= 0.325 J_, from (5.11b) we obtain hbar c apeq 1 J a_{H_.}In Oguchi's calculation^{14}for a spin-S antiferromagnet, the renormalized spin wave velocity is
hbar c_{r}= 2 sqrt 2 s ( 1 + 0.158 / 2 s ) J a_{H .
}For a spin cf12 antiferromagnet, its value hbar c_{r}= 1 J a_{H_,}is lower than our value for the bare spin wave velocity which enters in the nonlinear sigma model. More recently Gomez Santos, Joannopoulos, and Negele (GJN) (Ref. 15) have performed similar simulations of the spin cf12 AF Heisenberg model. They find overall agreement with our results at higher temperatures reported in Ref. 5, but they find smaller correlation lengths at lower temperatures. GJN argue that the origin of the discrepancy may be that their new algorithm searches the phase space more efficiently. We, however, believe that the discrepancy may also be due to finite size effects: their correlation lengths at low temperatures increase with the lattice size (see Fig. 6 of Ref. 15) whereas in our calculation finite size effects appear at larger correlation lengths (somewhat lower temperatures). This difference could be due to the different way of calculating the correlation function in the two calculations. Hence, it is possible that our results represented by the dashed line in Fig. 6 of Ref. 15, approximate better the infinite lattice. Using their values for A_{H_=0}and B_{H}apeq 1_, we obtain hbar c apeq 0 J a_{H_}which is somewhat lower than ours. If, on the other hand, we use the most recent form of Chakravarty, Halperin, and Nelson^{16}who found A_{H_=0.467}and B_{H_=0.94,}we find hbar c apeq 1 J a_{H_.}
In Fig. 15 we plot the inverse correlation length versus T as observed by neutron scattering experiments.^{3}The solid line corresponds to our Eq. (6.1) which fits both the nonlinear sigma model and the spin cf12 AF Heisenberg model using J = 1270 K, a value close to that reported by Raman scattering experiments.
We have studied the quantum mechanical nonlinear sigma model in 2+1 dimensions on a lattice. We have determined what should be the dependence of the coupling constant (spin stiffness) of the theory on the lattice spacing so that the results for the correlation length at any fixed physical temperature are independent of the cutoff in the continuum limit. At T = 0 we found the critical point g_{c_}that separates the disordered from the ordered phase of the sigma model. As T -> 0 and at g = g_{c_,}xi apeq0.8( hbar c / k_{B}T )_. By calculating the model's renormalization group beta function we rescale the calculated correlation lengths at various couplings and temperatures and find that they all collapse on the same curve xi / a_{sigma}= f ( T / T_{sigma})_. Even though the lattice spacing vanishes, dimensional transmutation occurs giving rise to a finite unit of length a_{sigma_.}Both parameters a_{sigma_}and the temperature scale T_{sigma}= hbar c / a_{sigma_}(c is the unrenormalized spin wave velocity which enters as a parameter in the partition function of sigma model) of the theory cannot be determined within the sigma model. We need to make contact with either the parameters of a microscopic model or with the experiment whose physics the sigma model is assumed to describe.
The parameters a_{sigma_}and T_{sigma_}of the sigma model can be related to the lattice spacing a_{H_}and the AF coupling J of the spin cf12 AF Heisenberg model. The numerical relationship between a_{sigma_,}T_{sigma_,}and a_{H_,}J is obtained by fitting the xi (T) = f (T/T_{sigma}) a_{sigma_}to the correlation length xi (T)\p = xi_{H}( T / J ) a_{H_}obtained from the simulations of the spin cf12 AF Heisenberg model.^{5,6,15}As a consequence of the assumption that the two models are equivalent at low T we find that (a) the spin cf12 AF Heisenberg model must order at T =0 and (b) the unrenormalized spin wave velocity c, a parameter of the sigma model, is obtained as hbar c = T_{sigma_a}sub sigma apeq 1 J a_{H_.}The value of hbar c obtained this way is not far from the value of the renormalized spin wave velocity obtained from spin wave theory of the spin cf12 quantum Heisenberg antiferromagnet.
Having obtained a common curve which fits the spin cf12 AF Heisenberg model and the nonlinear sigma model we find that we need J = 1270 K to fit the neutron scattering data for the correlation length in the insulator La_{2_CuO}sub 4. This value of J is close to that estimated by Raman scattering experiments.^{17}Smaller values of J will bring our results closer to the data in that region but further away at higher T.
We would like to thank S. Chakravarty, B. Halperin, and P. Weisz for useful discussions. This work was supported in part by the Center for Materials Research and Technology of The Florida State University (FSU), the U.S. Defense Advanced Research Projects Agency (DARPA) sponsored Florida Initiative for Microelectronics and Materials, under Contract No. MDA972 88-J-1006, and in part by the Supercomputer Computations Research Institute of FSU, which is partially funded by U.S. Department of Energy under Contract No. DE FC05 85ER 250000.
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