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arXiv:cond-mat/9608124 · Physical Review Letters

Dimensional Crossover in Quantum Antiferromagnets

Sudip Chakravarty

Published Nov 18, 1996• 61 citations
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Abstract

The dimensional crossover in a spin-S nearest-neighbor Heisenberg antiferromagnet is discussed as it is tuned from a two-dimensional square lattice, of lattice spacing a, towards a spin chain by varying the width ${L}_{y}$ of a semi-infinite strip ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$. For integer spins and arbitrary ${L}_{y}$, and for half integer spins with ${L}_{y}/a$ an arbitrary even integer, explicit analytical expressions for the zero temperature correlation length and the spin gap are given. For half integer spins and ${L}_{y}/a$ an odd integer, it is argued that the $c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ behavior of the SU(2${)}_{1}$ Wess-Zumino-Witten fixed point is squeezed out as the width ${L}_{y}\ensuremath{\rightarrow}\ensuremath{\infty}$; here c is the conformal charge. The results specialized to $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ are applied to spin-ladder systems.

DOI
10.1103/physrevlett.77.4446

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The dimensional crossover in a spin-S nearest-neighbor Heisenberg antiferromagnet is discussed as it is tuned from a two-dimensional square lattice, of lattice spacing a, towards a spin chain by varying the width Ly{L}_{y}Ly​ of a semi-infinite strip Lx\ifmmode×\else\texttimes\fiLy{L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}Lx​\ifmmode×\else\texttimes\fiLy​. For integer spins and arbitrary Ly{L}_{y}Ly​, and for half integer spins with Ly/a{L}_{y}/aLy​/a an arbitrary even integer, explicit analytical expressions for the zero temperature correlation length and the spin gap are given. For half integer spins and Ly/a{L}_{y}/aLy​/a an odd integer, it is argued that the c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 behavior of the SU(2)1{)}_{1})1​ Wess-Zumino-Witten fixed point is squeezed out as the width Ly\ensuremath→\ensuremath∞{L}_{y}\ensuremath{\rightarrow}\ensuremath{\infty}Ly​\ensuremath→\ensuremath∞; here c is the conformal charge. The results specialized to S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ are applied to spin-ladder systems.
We study antiferromagnetic spin-\textonehalf{} Heisenberg ladders, comprised of nc{n}_{c}nc​ chains ($2\ensuremath{\le}{n}_{c}\ensuremath{\le}6$) with ratio $\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}$ of inter-chain to intra-chain couplings. The correlation length $\ensuremath{\xi}(T)$ is deduced from measurements of the correlation function. For even nc{n}_{c}nc​, the static structure factor exhibits a peak at a temperature below the corresponding spin gap. Results for isotropically coupled ladders ($\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}=1$) are compared to those for the single chain and the square lattice. For $\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}\ensuremath{\le}0.5$, the correlation function of the two-chain ladder is in excellent agreement with analytic results from conformal field theory, and $\ensuremath{\xi}(T)$ exhibits simple scaling behavior.
We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below Ec,{E}_{c},Ec​, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band (N+1)/2(N+1)/2(N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale Ec,{E}_{c},Ec​, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above Ec{E}_{c}Ec​ of the same type as in N/2N/2N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
We study scaling behavior in two-dimensional, square lattice, S=1/2 and S=1 Heisenberg antiferromagnets using the data on full q dependences of the equal-time structure factor and the static susceptibility, calculated through high-temperature series expansions. We also carry out comparisons with a model of two coupled S=1/2 Heisenberg planes with the interlayer exchange coupling tuned to the T=0 critical point (two-plane model hereafter). For both S=1/2 and S=1 models, we separately determine the spin-wave velocity c and mass m=c/\ensuremath{\xi}, in addition to the correlation length, \ensuremath{\xi}, and find that c is temperature dependent; only for temperatures below T\ensuremath{\lesssim}JS, where J is the exchange coupling, c approaches its known T=0 values, c0{\mathit{c}}_{0}c0​. This nonuniversal lattice effect is caused by the quantum nature of spin, and is therefore not captured by the quantum nonlinear \ensuremath{\sigma} model.Despite this temperature dependence of the spin-wave velocity, full q and \ensuremath{\omega} dependences of the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) agree with those of the universal scaling function, computable for the \ensuremath{\sigma} model, for temperatures up to T0{\mathit{T}}_{0}T0​\ensuremath{\sim}0.6c0{\mathit{c}}_{0}c0​/a, and their further analysis leads us to the inference that below T0{\mathit{T}}_{0}T0​ the S=1 model is in the renormalized classical (RC) regime, the two-plane model is in the quantum critical (QC) regime in agreement with earlier work, and the S=1/2 model exhibits a RC-QC crossover, centered around crossover and for the two-plane model at all T=0.55J. In particular, for the S=1/2 model above the RC-QC temperatures where calculated, the obtained spin-wave mass m=c/\ensuremath{\xi} is in excellent agreement with the known universal QC prediction, m\ensuremath{\simeq}1.04T. In contrast, for the S=1/2 model below the RC-QC crossover, and for the S=1 model at all temperatures, the behavior agrees with the exact RC expression. for all three models, nonuniversal behavior occurs above T\ensuremath{\sim}0.6c0{\mathit{c}}_{0}c0​/a. Our results strongly support the conjecture by Chubukov and Sachdev that the S=1/2 model is close enough to the T=0 critical point to exhibit QC behavior at intermediate temperatures.
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