arXiv:cond-mat/0608395 · Physical Review B

Scaling exponents in spin- 1 2 Heisenberg chains with dimerization and frustration studied with the density-matrix renormalization group

Manoranjan Kumar, S. Ramasesha, Diptiman Sen, Z. G. Soos

Published Feb 8, 2007• 27 citations

Abstract

In conformal field theory, key properties of spin-$\frac{1}{2}$ chains, such as the ground state energy per site and the excitation gap, scale with dimerization $\ensuremath{\delta}$ as ${\ensuremath{\delta}}^{\ensuremath{\alpha}}$ with known exponents $\ensuremath{\alpha}$ and logarithmic corrections. The logarithmic corrections vanish in a spin chain with nearest $(J=1)$ and next-nearest-neighbor interactions $({J}_{2})$, for ${J}_{2\mathrm{c}}=0.2411$. Density-matrix renormalization group analysis of a frustrated spin chain with no logarithmic corrections yields the field theoretic values of $\ensuremath{\alpha}$, and the scaling relation is valid up to the physically realized range, $\ensuremath{\delta}\ensuremath{\sim}0.1$. However, chains with logarithmic corrections $({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})$ are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately $3∕4$ to $2∕3$ for the spin gap and from approximately $3∕2$ to $4∕3$ for the energy per site, and error bars in the exponent also decrease as ${J}_{2}$ approaches to ${J}_{2\mathrm{c}}$.

DOI: 10.1103/physrevb.75.052404

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arXiv:cond-mat/0608395 · Physical Review B

Scaling exponents in spin- 1 2 Heisenberg chains with dimerization and frustration studied with the density-matrix renormalization group

Manoranjan Kumar, S. Ramasesha, Diptiman Sen, Z. G. Soos

Published Feb 8, 2007• 27 citations

arXiv HTML PDF

Abstract

In conformal field theory, key properties of spin-

\frac{1}{2}

chains, such as the ground state energy per site and the excitation gap, scale with dimerization

\ensuremath{\delta}

{\ensuremath{\delta}}^{\ensuremath{\alpha}}

with known exponents

\ensuremath{\alpha}

and logarithmic corrections. The logarithmic corrections vanish in a spin chain with nearest

(J=1)

and next-nearest-neighbor interactions

({J}_{2})

, for

{J}_{2\mathrm{c}}=0.2411

. Density-matrix renormalization group analysis of a frustrated spin chain with no logarithmic corrections yields the field theoretic values of

\ensuremath{\alpha}

, and the scaling relation is valid up to the physically realized range,

\ensuremath{\delta}\ensuremath{\sim}0.1

. However, chains with logarithmic corrections

({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})

are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately

3∕4

2∕3

for the spin gap and from approximately

3∕2

4∕3

for the energy per site, and error bars in the exponent also decrease as

{J}_{2}

approaches to

{J}_{2\mathrm{c}}

DOI: 10.1103/physrevb.75.052404

Related papers

Density-matrix renormalization-group studies of the spin-1/2 Heisenberg system with dimerization and frustration

R. Chitra, Swapan Kumar Pati, H. R. Krishnamurthy +2 more

Using the density-matrix renormalization-group technique, we study the ground-state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-1/2 chain with both dimerization and frustration, i.e., an alternation \ensuremath{\delta} of the nearest-neighbor exchanges and a next-nearest-neighbor exchange ${\mathit{J}}_{2}$ . For \ensuremath{\delta}=0, the system is gapless for ${\mathit{J}}_{2}$ ${\mathit{J}}_{2\mathit{c}}$ and has a gap for ${\mathit{J}}_{2}$ > ${\mathit{J}}_{2\mathit{c}}$ where ${\mathit{J}}_{2\mathit{c}}$ is about 0.241. For ${\mathit{J}}_{2}$ = ${\mathit{J}}_{2\mathit{c}}$ , the gap above the ground state grows as \ensuremath{\delta} to the power 0.667\ifmmode\pm\else\textpm\fi{}0.001. In the ${\mathit{J}}_{2}$ -\ensuremath{\delta} plane, there is a disorder line 2 ${\mathit{J}}_{2}$ +\ensuremath{\delta}=1. To the left of this line, the peak in the static structure factor S(q) is at ${\mathit{q}}_{\mathrm{max}}$ =\ensuremath{\pi} (N\'eel phase), while to the right of the line, ${\mathit{q}}_{\mathrm{max}}$ decreases from \ensuremath{\pi} to \ensuremath{\pi}/2 as ${\mathit{J}}_{2}$ is increased to large values (spiral phase). For \ensuremath{\delta}=1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.Using the density-matrix renormalization-group technique, we study the ground-state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-1/2 chain with both dimerization and frustration, i.e., an alternation \ensuremath{\delta} of the nearest-neighbor exchanges and a next-nearest-neighbor exchange ${\mathit{J}}_{2}$. For \ensuremath{\delta}=0, the system is gapless for ${\mathit{J}}_{2}$${\mathit{J}}_{2\mathit{c}}$ and has a gap for ${\mathit{J}}_{2}$>${\mathit{J}}_{2\mathit{c}}$ where ${\mathit{J}}_{2\mathit{c}}$ is about 0.241. For ${\mathit{J}}_{2}$=${\mathit{J}}_{2\mathit{c}}$, the gap above the ground state grows as \ensuremath{\delta} to the power 0.667\ifmmode\pm\else\textpm\fi{}0.001. In the ${\mathit{J}}_{2}$-\ensuremath{\delta} plane, there is a disorder line 2${\mathit{J}}_{2}$+\ensuremath{\delta}=1. To the left of this line, the peak in the static structure factor S(q) is at ${\mathit{q}}_{\mathrm{max}}$=\ensuremath{\pi} (N\'eel phase), while to the right of the line, ${\mathit{q}}_{\mathrm{max}}$ decreases from \ensuremath{\pi} to \ensuremath{\pi}/2 as ${\mathit{J}}_{2}$ is increased to large values (spiral phase). For \ensuremath{\delta}=1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.

arXiv:cond-mat/9412016Physical review. B, Condensed matterSep 1, 1995232 citations

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Davide Controzzi, C. Degli Esposti Boschi, F. Ortolani +1 more

We study the finite-size behavior of the low-lying excitations of spin- $\frac{1}{2}$ Heisenberg chains with dimerization and next-to-nearest neighbors interaction ${J}_{2}$ . The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where $L$ is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At ${J}_{2}={J}_{2c}$ , where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.We study the finite-size behavior of the low-lying excitations of spin-$\frac{1}{2}$ Heisenberg chains with dimerization and next-to-nearest neighbors interaction ${J}_{2}$. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where $L$ is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At ${J}_{2}={J}_{2c}$, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.

arXiv:cond-mat/0503539Physical Review BNov 17, 20054 citations

A dimerized and frustrated spin model with alternating ferromagnetic and antiferromagnetic exchanges: investigating quantum phases of Heisenberg spin chains with s = 1/2 and 1

Shaon Sahoo, V. M. L. Durga Prasad Goli, Diptiman Sen +1 more

We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic ( $J_1^F$ ) and antiferromagnetic ( $J_1^A$ ) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction ( $J_2^F$ ). In this model frustration is induced by the non-zero $J_2^F$ and the effect of dimerization is realized by changing the strength of $J_1^A$ for a fixed value of $J_1^F$ . The model with site spin $s$ behaves like a Haldane spin chain with site spin 2 $s$ in the limit of vanishing $J_2^F$ and large $J_1^F/J_1^A$ . We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of $J_1^A-J_2^F$ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic ($J_1^F$) and antiferromagnetic ($J_1^A$) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction ($J_2^F$). In this model frustration is induced by the non-zero $J_2^F$ and the effect of dimerization is realized by changing the strength of $J_1^A$ for a fixed value of $J_1^F$. The model with site spin $s$ behaves like a Haldane spin chain with site spin 2$s$ in the limit of vanishing $J_2^F$ and large $J_1^F/J_1^A$. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of $J_1^A-J_2^F$ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.

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Density-matrix renormalization-group analysis of quantum critical points: Quantum spin chains

Shan-Wen Tsai, John Bradley Marston

We present a simple method, combining the density-matrix renormalization-group algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin- $1/2$ Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.We present a simple method, combining the density-matrix renormalization-group algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin-$1/2$ Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.

arXiv:cond-mat/0001275Physical review. B, Condensed matterSep 1, 200027 citations

Spontaneous dimerization, critical lines, and short-range correlations in a frustrated spin-1 chain

Natalia Chepiga, Ian K. Affleck, Frédéric Mila

We report on a detailed investigation of the spin-$1 {J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Heisenberg model, a frustrated model with nearest-neighbor coupling ${J}_{1}$ , next-nearest neighbor coupling ${J}_{2}$ , and a three-site interaction ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$ previously studied in [Phys. Rev. B 93, 241108(R) (2016)]. Using density matrix renormalization group (DMRG) and exact diagonalizations, we show that the phase boundaries between the Haldane phase, the next-nearest neighbor Haldane phase, and the dimerized phase can be very accurately determined by combining the information deduced from the dimerization, the ground-state energy, the entanglement spectrum and the Berry phase. By a careful investigation of the finite-size spectrum, we also show that the transition between the next-nearest neighbor Haldane phase and the dimerized phase is in the Ising universality class all along the critical line. Furthermore, we justify the conformal embedding of the $\mathrm{SU}{(2)}_{2}$ Wess-Zumino-Witten conformal field theory in terms of a boson and an Ising field, and we explicitly derive a number of consequences of this embedding for the spectrum along the $\mathrm{SU}{(2)}_{2}$ transition line between the Haldane phase and the dimerized phase. We also show that the solitons along the first-order transition line between the Haldane phase and the dimerized phase carry a spin-1/2, while the domain walls between different dimerization domains inside the dimerized phase carry a spin 1. Finally, we show that short-range correlations change character in the Haldane and dimerized phases through disorder and Lifshitz lines, as well as through the development of short-range dimer correlations in the Haldane phase, leading to a remarkably rich phase diagram.We report on a detailed investigation of the spin-$1 {J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Heisenberg model, a frustrated model with nearest-neighbor coupling ${J}_{1}$, next-nearest neighbor coupling ${J}_{2}$, and a three-site interaction ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$ previously studied in [Phys. Rev. B 93, 241108(R) (2016)]. Using density matrix renormalization group (DMRG) and exact diagonalizations, we show that the phase boundaries between the Haldane phase, the next-nearest neighbor Haldane phase, and the dimerized phase can be very accurately determined by combining the information deduced from the dimerization, the ground-state energy, the entanglement spectrum and the Berry phase. By a careful investigation of the finite-size spectrum, we also show that the transition between the next-nearest neighbor Haldane phase and the dimerized phase is in the Ising universality class all along the critical line. Furthermore, we justify the conformal embedding of the $\mathrm{SU}{(2)}_{2}$ Wess-Zumino-Witten conformal field theory in terms of a boson and an Ising field, and we explicitly derive a number of consequences of this embedding for the spectrum along the $\mathrm{SU}{(2)}_{2}$ transition line between the Haldane phase and the dimerized phase. We also show that the solitons along the first-order transition line between the Haldane phase and the dimerized phase carry a spin-1/2, while the domain walls between different dimerization domains inside the dimerized phase carry a spin 1. Finally, we show that short-range correlations change character in the Haldane and dimerized phases through disorder and Lifshitz lines, as well as through the development of short-range dimer correlations in the Haldane phase, leading to a remarkably rich phase diagram.

arXiv:1608.08109Physical review. B./Physical review. BNov 8, 201631 citations

Density matrix renormalization group study of critical behavior of the s p i n − 1 2 alternating Heisenberg chain

T. Papenbrock, Ted Barnes, David J. Dean +2 more

We investigate the critical behavior of the $S=1/2$ alternating Heisenberg chain using the density matrix renormalization group. The ground-state energy per spin, ${e}_{0},$ and singlet-triplet energy gap $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}$ are determined for a range of alternations $\ensuremath{\delta}.$ Our results for the approach of ${e}_{0}$ to the uniform chain limit are well described by $c{\ensuremath{\delta}}^{p},$ with $p\ensuremath{\approx}1.45.$ The singlet-triplet gap is also well described by a power law, with $p\ensuremath{\approx}0.73,$ half of the ${e}_{0}$ power. The renormalization group predictions of power laws with logarithmic corrections can also accurately describe our data provided that a surprisingly large-scale parameter ${\ensuremath{\delta}}_{0}$ is present in the logarithms.We investigate the critical behavior of the $S=1/2$ alternating Heisenberg chain using the density matrix renormalization group. The ground-state energy per spin, ${e}_{0},$ and singlet-triplet energy gap $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}$ are determined for a range of alternations $\ensuremath{\delta}.$ Our results for the approach of ${e}_{0}$ to the uniform chain limit are well described by $c{\ensuremath{\delta}}^{p},$ with $p\ensuremath{\approx}1.45.$ The singlet-triplet gap is also well described by a power law, with $p\ensuremath{\approx}0.73,$ half of the ${e}_{0}$ power. The renormalization group predictions of power laws with logarithmic corrections can also accurately describe our data provided that a surprisingly large-scale parameter ${\ensuremath{\delta}}_{0}$ is present in the logarithms.

arXiv:cond-mat/0212254Physical review. B, Condensed matterJul 28, 200337 citations

In conformal field theory, key properties of spin-

\frac{1}{2}

chains, such as the ground state energy per site and the excitation gap, scale with dimerization

\ensuremath{\delta}

{\ensuremath{\delta}}^{\ensuremath{\alpha}}

with known exponents

\ensuremath{\alpha}

and logarithmic corrections. The logarithmic corrections vanish in a spin chain with nearest

(J=1)

and next-nearest-neighbor interactions

({J}_{2})

, for

{J}_{2\mathrm{c}}=0.2411

. Density-matrix renormalization group analysis of a frustrated spin chain with no logarithmic corrections yields the field theoretic values of

\ensuremath{\alpha}

, and the scaling relation is valid up to the physically realized range,

\ensuremath{\delta}\ensuremath{\sim}0.1

. However, chains with logarithmic corrections

({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})

are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately

3∕4

2∕3

for the spin gap and from approximately

3∕2

4∕3

for the energy per site, and error bars in the exponent also decrease as

{J}_{2}

approaches to

{J}_{2\mathrm{c}}

{\mathit{J}}_{2}

. For \ensuremath{\delta}=0, the system is gapless for

{\mathit{J}}_{2}

{\mathit{J}}_{2\mathit{c}}

and has a gap for

{\mathit{J}}_{2}

{\mathit{J}}_{2\mathit{c}}

where

{\mathit{J}}_{2\mathit{c}}

is about 0.241. For

{\mathit{J}}_{2}

{\mathit{J}}_{2\mathit{c}}

, the gap above the ground state grows as \ensuremath{\delta} to the power 0.667\ifmmode\pm\else\textpm\fi{}0.001. In the

{\mathit{J}}_{2}

-\ensuremath{\delta} plane, there is a disorder line 2

{\mathit{J}}_{2}

+\ensuremath{\delta}=1. To the left of this line, the peak in the static structure factor S(q) is at

{\mathit{q}}_{\mathrm{max}}

=\ensuremath{\pi} (N\'eel phase), while to the right of the line,

{\mathit{q}}_{\mathrm{max}}

decreases from \ensuremath{\pi} to \ensuremath{\pi}/2 as

{\mathit{J}}_{2}

is increased to large values (spiral phase). For \ensuremath{\delta}=1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.

We study the finite-size behavior of the low-lying excitations of spin-

\frac{1}{2}

Heisenberg chains with dimerization and next-to-nearest neighbors interaction

{J}_{2}

. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where

L

is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At

{J}_{2}={J}_{2c}

, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.

We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (

J_1^F

) and antiferromagnetic (

J_1^A

) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (

J_2^F

). In this model frustration is induced by the non-zero

J_2^F

and the effect of dimerization is realized by changing the strength of

J_1^A

for a fixed value of

J_1^F

. The model with site spin

s

behaves like a Haldane spin chain with site spin 2

s

in the limit of vanishing

J_2^F

and large

J_1^F/J_1^A

. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of

J_1^A-J_2^F

is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.

1/2

Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.

We report on a detailed investigation of the spin-$1 {J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Heisenberg model, a frustrated model with nearest-neighbor coupling

{J}_{1}

, next-nearest neighbor coupling

{J}_{2}

, and a three-site interaction ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$ previously studied in [Phys. Rev. B 93, 241108(R) (2016)]. Using density matrix renormalization group (DMRG) and exact diagonalizations, we show that the phase boundaries between the Haldane phase, the next-nearest neighbor Haldane phase, and the dimerized phase can be very accurately determined by combining the information deduced from the dimerization, the ground-state energy, the entanglement spectrum and the Berry phase. By a careful investigation of the finite-size spectrum, we also show that the transition between the next-nearest neighbor Haldane phase and the dimerized phase is in the Ising universality class all along the critical line. Furthermore, we justify the conformal embedding of the

\mathrm{SU}{(2)}_{2}

Wess-Zumino-Witten conformal field theory in terms of a boson and an Ising field, and we explicitly derive a number of consequences of this embedding for the spectrum along the

\mathrm{SU}{(2)}_{2}

transition line between the Haldane phase and the dimerized phase. We also show that the solitons along the first-order transition line between the Haldane phase and the dimerized phase carry a spin-1/2, while the domain walls between different dimerization domains inside the dimerized phase carry a spin 1. Finally, we show that short-range correlations change character in the Haldane and dimerized phases through disorder and Lifshitz lines, as well as through the development of short-range dimer correlations in the Haldane phase, leading to a remarkably rich phase diagram.

We investigate the critical behavior of the

S=1/2

alternating Heisenberg chain using the density matrix renormalization group. The ground-state energy per spin,

{e}_{0},

and singlet-triplet energy gap $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}$ are determined for a range of alternations $\ensuremath{\delta}.$ Our results for the approach of

{e}_{0}

to the uniform chain limit are well described by $c{\ensuremath{\delta}}^{p},$ with $p\ensuremath{\approx}1.45.$ The singlet-triplet gap is also well described by a power law, with $p\ensuremath{\approx}0.73,$ half of the

{e}_{0}

power. The renormalization group predictions of power laws with logarithmic corrections can also accurately describe our data provided that a surprisingly large-scale parameter ${\ensuremath{\delta}}_{0}$ is present in the logarithms.