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We study the continuum limit of the nonlinear sigma model in 2+1 dimensions and at finite temperature T, using Monte Carlo simulation on large lattices. Even though the lattice spacing vanishes, dimensional transmutation occurs which makes the correlation lengths finite. With the assumption that the sigma model and the spin-(1/2 antiferromagnetic Heisenberg model are equivalent at low T, we make contact between the two models and find that the latter model must order at T=0. Our results are consistent with neutron-scattering experiments done on La2CuO4. .AE
©1989 The American Physical Society.
PACS: 75.10.Jm 74.20.-z
The discovery of the copper oxide superconductors, as well as the suggestion that the superconductivity mechanism in these new materials is related to the strong correlations among purely electronic degrees of freedom,^{1}has intensified the interest for understanding one of the simplest models to account for such correlations: the Hubbard model. The strong two dimensional (2D) spin correlations observed in neutron scattering experiments^{2}have given credit to the spin cf12 antiferromagnetic (AF) Heisenberg model defined as
H hat = J sum from {langle i,j rangle} ^ S vec_{i}cdot S vec_{j} , (1)
where S vec_{i_}is the spin cf12 operator of the conduction band electrons localized in the Wannier states around the ith unit cell of the copper oxide plane. J is the AF coupling and the sum is over the nearest neighbors. This model can be obtained from the Hubbard model at half filling by taking the strong coupling limit.^{3}In such a formulation, the Heisenberg model describes interactions that originate from virtual electron hopping processes.
Recently we simulated^{4,5}the spin cf12 2D AF Heisenberg model using Handscomb's quantum Monte Carlo (MC) method. We calculated the correlation length and found that it increases very rapidly with decreasing temperature. The results of Refs. 4 and 5 are consistent with neutron scattering experiments. It is, however, difficult to find an efficient quantum MC algorithm to study large systems and approach low temperatures.
The 2D quantum Heisenberg model is believed to be equivalent^{6}to the nonlinear sigma model in the two space plus one Elucidean time dimensions and at low temperatures. More recently, however, there was a suggestion^{7}that in the derivation of the nonlinear sigma model from the Heisenberg model one encounters a topological term. This term distinguishes the integer from the half integer spin case. The role which such topological terms might play in the development of the theory of superconductivity in the copper oxides was part of the reason for the excitement about this idea. Later, however, the necessity of such topological terms became less clear and in fact, today, the two models are believed to be equivalent without any additional terms.^{8}
The nonlinear sigma model in 2+1 dimensions has been recently studied by Chakavarty, Halperin, and Nelson (CHN).^{9}Using a one loop perturbative renormaliza\%tion group approach, CHN relate the nonlinear sigma model to the spin cf12 Heisenberg AF model at low temperatures and give a good fit to the data obtained from neutron scattering experiments done on La_{2}CuO_{4_.}
In this paper we study the nonlinear sigma model in two space plus one Euclidean time dimensions and at finite physical temperature using the MC method. The simulation of this model is easier than that of the quantum AF Heisenberg model. Using efficient vectorized algorithms suitable for ETA supercomputers we study large size lattices ( 100 times 100 times 8 is our largest lattice). We calculate the model's renormalization group beta function around the 3D critical point which separates the quantum disordered phase from the phase with spontaneous symmetry breaking. Using the beta function, we rescale the calculated correlation lengths at various values of the coupling g and temperature T and find that all collapse on the same curve independent of g and the lattice spacing. This gives rise to \fIdimensional transmutation\fR, a phenomenon well known in field theory, which produces a finite length scale. Assuming that the spin cf12 AF Heisenberg model and the sigma model are equivalent at low temperatures, we make contact between the two models by comparing the behavior of the correlation lengths at low temperatures. We find that the two models can be made equivalent if the spin cf12 AF Heisenberg model orders at T=0. We obtain a reasonable fit to the neutron scattering data^{2}of the insulator La_{2}CuO_{4_}bytaking J = 1270 K, a value close to that reported by Raman scattering experiments.^{10}
The nonlinear sigma model in two space plus one Euclidean time dimensions is defined as^{6,9}
S_{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
Here OMEGA vec is a three component vector field living on a unit sphere, c is the spin wave velocity, and beta = 1/ k_{B}T_. We discretize the space time and put the model on the (2+1)-dimensional lattice: L (3) S_{eff}= - 1 over 2g sum from {x vec} ^ sum from {mu = 1} to 3 ^ OMEGA vec ( x vec ) cdot [ OMEGA vec ( x vec + e bhat_{mu}) + OMEGA vec ( x vec - e bhat_{mu}) ] , (2)
where g = hbar c / rho_{0}a_, x vec covers the (2+1)-dimensional lattice of lattice spacing a, size N^{2}N_{beta_,}and
beta hbar c = N_{beta}^ a . (4)
We have to impose periodic boundary conditions (BC) in the Euclidean time direction. In this model the average of the field OMEGA vec is proportional to the average staggered magnetization and could describe the dynamics of the spins within one isolated CuO_{2_}layer.
From the two point function we can calculate the correlation length in lattice units xi_{roman}latt_ as a function of g, N_{beta_,}and N. For continuum limit behavior and for eliminating finite size effects, xi_{roman}latt_ must satisfy 1 << xi_{roman}latt << N _. We need to take the limit N -> inf and keep the time dimension finite so that Eq. (4) is satisfied. If, therefore, N is large enough so that xi_{roman}latt << N_, the correlation length is only a function of N_{beta_}and g. In physical units xi is given by
xi = xi_{roman}latt ( g, N_{beta}) a . (5)
In our simulation we used periodic BC in the space boundaries also. We used the heat bath algorithm and typically 5000 MC steps over the entire lattice for thermalization and about 10 ^ 000 for measurements. The correlation length is extracted from the correlation function G ( x_{1}- x_{1^{sprime})_}of the (Euclidean) time average of OMEGA vec ( x vec ) at two different points in space x_{1_}and x_{1^{sprime_.}We}fitted the long distance behavior of the correlation function with A ^ cosh [( x_{1}- x_{1^{sprime}-}N/_2) / xi_{roman}latt ]_. It is known that for g > g_{c_,}where g_{c_}is the 3D critical point (i.e., at T=0), the three modes of the theory have degenerate finite masses (inverse correlation lengths). For g < g_{c_,}however, there are two masses in the theory: Two modes correspond to the Goldstone mode excitations and become massless in the 3D theory ( beta -> inf ). They are related to the radial motion of the average field and give an exponentially small mass with the size of the finite beta. There is also a massive mode associated with fluctuations in the magnitude (radial component) of the average field. In this paper we study the mode having the smallest mass, which dominates the behavior of the correlation function at large distances.
Keeping the physical temperature constant we may approach the continuum limit a = hbar c beta / N_{beta}-> 0_ by increasing N_{beta_.}To keep the correlation length xi constant in physical units, for any a -> 0, we should find the value of g which gives the same value of xi. This is achieved through Eq. (6) which defines the function g(a). The combination of Eqs. (4) and (5) gives xi = b hbar c / k_{B}T_, where b = xi_{roman}latt (g,N_{beta}) / N_{beta_.}In order to keep xi constant at a fixed temperature we should keep the ratio b 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. 1 we give b as a function of g for several values of N_{beta_.}We notice that the lines for various N_{beta_}pass through the same point (g_{c}, b^{star}) = (1 +- 0 , 0 +- 0 )_. Let us say that we would like to define the theory's coupling constant at the value b = b_{0_}shown in Fig. 1. The line b = b_{0_}intersects the various curves for different N_{beta_'s}(i.e., in this case in which the temperature is constant, for different a's), and the values of g at the intersections define g(a_{T}/ N_{beta})_. We note that lim_{{}N_{beta}-> inf} g (a_{T}/ N_{beta}) = g_{c_.}Because b = b^{star}at g = g_{c_,}for large N_{beta_'s}(small a's) we obtain
xi^{star}= b^{star}hbar c / k_{B}T , (6)
where b^{star}= 0 +- 0.05. Notice that at T=0, g_{c_}turns into a critical point. These results confirm the crossover phase diagram given by CHN on general grounds (Ref. 9). Moreover, in their more recent work (Ref. 11), they obtain for the universal constant b^{star}= 1.1, a value somewhat higher than ours.
Using our results for the correlation length obtained on lattices of sizes 50^{2}times N_{beta_}and 100^{2}times N_{beta_}with N_{beta}= 2_, 4, 6, and 8, we calculate the renormalization group beta function beta_{roman}RG == - a ^ dg(a) / da_. Our results for beta_{roman}RG_ are shown as the inset in Fig. 1. To avoid finite size effects we used only those points for which b < 2.5. At g = g_{c_,}beta_{roman}RG_ changes sign. At T =0, xi^{star}= inf, and for g < g_{c_}the system enters 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 linear:
beta_{roman}RG (g) = - beta_{1}(g-g_{c}) + ^ . . . ^. (7)
We find g_{c}= 1.450 +- 0.003_ and beta_{1}= 1 +- 0.05_. Integrating both sides of the equation defining beta_{roman}RG_ one obtains a(g) = a_{sigma}^ exp [- tint^{g}d g / beta_{roman}RG (g)]_, where a_{sigma_}is a constant of integration having dimensions of length. 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 a way such that a_{sigma_}remains constant. In field theory, this limiting process where a vanishing length scale (a -> 0 ) and a dimensionless parameter (g) produce a dimensional quantity ( a_{sigma})_ is called \fIdimensional trans\%mutation\fR.^{12}
We have compared our numerical results with results obtained in the saddle point approximation. We find good agreement in the region g > g_{c_,}but poor agreement for g < g_{c_.}The saddle point approximation and details of the present calculation will be given elsewhere.^{13}
Using the linear approximation [Eq. (7)] close to the critical point we find
a (g) = a_{sigma}| g - g_{c}|^{back}30 {1/ beta_{1}} . (8)
Combining Eqs. (4) and (8) we obtain
N_{beta}= |g-g_{c}|^{back}30 {-1/ beta_{1}}T_{sigma}/ T , (9)
where k_{B}T_{sigma}= hbar c / a_{sigma}_. Substituting a(g) and N_{beta_}from Eqs. (8) and (9) into Eq. (5) we obtain
xi over {a_{sigma}}= f ( T over {T_{sigma}}right ) == xi_{roman}latt ( g,|g-g_{c}|^{back}30 {-1/ beta_{1}} {T_{sigma}}over T right ) |g-g_{c}|^{back}30 {1/ beta_{1}} . L
Since the constants a_{sigma_}and T_{sigma_}are independent of g and a and xi is also independent of g in the process of removing the cutoff, the function in Eq. (10) is only a function of t == T/T_{sigma_.}In Fig. 2 we show the function f(t). The data points in the figure 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 length scale a_{sigma_}as a function of temperature t in units of T_{sigma_.}
The curve f(t) can be approximated by an exponential, (11) f(t) = A_{sigma}^ exp ( B_{sigma}/ t ) , (10)
as the saddle point approximation (Ref. 13) and the most recent work of CHN (Ref. 11) suggest. The best fit gives A_{sigma}= 0.0795_ and B_{sigma}= 4.308_, and is shown as a solid line in Fig. 2.
For g >> g_{c_}the correlation length in the nonlinear sigma model is only a function of g and is independent of T. At the critical point g = g_{c_}we find that at low T, xi grows as 1/T as the temperature decreases.
It is possible to make contact between the spin- cf12 AF Heisenberg model and the nonlinear sigma model. In Refs. 4 and 5 we simulated the former and we found it to grow much more rapidly than 1/T. More precisely, in Ref. 4, we fitted the correlation lengths by xi (T) = C/ Te^{b/T,}as suggested by the spin wave theory, and by the Kos\%terlitz Thouless form xi (T) = C e^{{b/|T-T}sub c |^{1/2}} . 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 by
xi / a_{H}= A_{H}^ exp (B_{H}J/T ) . (12)
This form, i.e., without the 1/T prefactor, also fits our data well, giving A_{H}= 0.25_ and B_{H}= 1.4_ (Ref. 5, Table II). On this basis we may conclude that if data do not exist at very low temperatures, prefactors may play an important role. Hence the results of our simulation^{4,5}may also be consistent with spin wave theory and the existence of an ordered state at T=0.
Let us \fIassume\fR that the two models are equivalent at low T. In order to obtain the best fit between the correlation lengths calculated for the two models, we need to assume that the spin cf12 AF Heisenberg model corresponds to the broken phase (g < g_{c})_ of the sigma model in the continuum limit. Therefore the spin cf12 AF Heisenberg model should order at T=0 and A_{H}a_{H}= A_{sigma}a_{sigma_}and B_{H}J = B_{sigma}T_{sigma_.}We obtain a_{sigma}= 3 a_{H_}and hbar c apeq 1 J a_{H_.}In Oguchi's calculation,^{14}the value of the renormalized spin wave velocity for a spin cf12 antiferromagnet hbar c = 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)^^{15}have performed similar simulations of the spin cf12 AF Heisenberg model. They find overall agreement at higher temperatures with our results reported in Ref. 4, but they find some 20% smaller correlation lengths at lower temperatures. GJN argue that the origin of the discrepancy may be that their improved algorithm searches the phase space more efficiently. We believe that the discrepancy could also be due to finite size effects which may affect the two calculations differently because the correlation functions have been calculated in different ways. Notice that the correlation length (Fig. 6 of Ref. 15) at, for example, temperature T/J=0.5 systematically increases by increasing the size of their lattice. In our calculation finite size effects appear at larger correlation lengths (somewhat lower temperature). Hence, our results (dashed line in their Fig. 6) may approximate the infinite system better. Nevertheless, using the values for A_{H}= 0.32_ and B_{H}apeq 1_ reported by GJN, we obtain hbar c apeq 0 J a_{H_,}which is somewhat lower than our value. If, on the other hand, we use the most recent form of CHN,^{11}who find A_{H}= 0.467_ and B_{H}= 0.94_, we find hbar c apeq 1 J a_{H_,}which is closer to Oguchi's result.
In Fig. 3 we plot the inverse correlation length versus T as observed by neutron scattering experiments.^{2}The solid curve is the exponential given by Eq. (12) which fits both the nonlinear sigma model and the AF Heisenberg model. In the plot we used a_{H}= 3.8_ A ang, the Cu Cu distance, and J = 1270 K, which is close to the value reported by Raman scattering experiments.^{10}Our curve dis\%agrees with the data very close to the 3D Neaael critical temperature T_{N}app 200_ K. Smaller values of J will bring our results closer to the data in that region but further away from the data 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) 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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