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

arXiv:cond-mat/0608395Physical Review BFeb 8, 200727 citations

Scaling and Renormalization

Gérard G. Emch, Chuang Liu

Springer eBooksJan 1, 2002274 citations

Logarithmic corrections to scaling in the XY2—model

Ralph Kenna B, A.C. Irving

arXiv:hep-lat/9411027Nuclear Physics B - Proceedings SupplementsApr 1, 19951 citation

Entanglement of Coupled Harmonic Oscillators

H. Snijder

The entropy of a black hole scales with its area. This is a puzzling feature in the context of statistical mechanics, where entropy scales with volume. It is argued that entanglement entropy underlays such an area law. This quantity is hard to calculate because a precise understanding of the underlying quantum state is needed. Therefore it is interesting to study a Schwarzschild analogue in a photon condensate which is an exactly known quantum system in the ground-state mimicking the behaviour of an astronomic black hole. A harmonic chain of size $N$ with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out $N-l$ oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for $N\\rightarrow\\infty$ , which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large $N$ , the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by $S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421$ in the discrete case and $S=\\frac{0.3744}{3} \\ln{l} + 0.6421$ in the infinite case.The entropy of a black hole scales with its area. This is a puzzling feature in the context of statistical mechanics, where entropy scales with volume. It is argued that entanglement entropy underlays such an area law. This quantity is hard to calculate because a precise understanding of the underlying quantum state is needed. Therefore it is interesting to study a Schwarzschild analogue in a photon condensate which is an exactly known quantum system in the ground-state mimicking the behaviour of an astronomic black hole. A harmonic chain of size $N$ with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out $N-l$ oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for $N\\rightarrow\\infty$, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large $N$, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by $S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421$ in the discrete case and $S=\\frac{0.3744}{3} \\ln{l} + 0.6421$ in the infinite case.

Jan 1, 20190 citations

Scaling behavior of the order parameter and its conjugated field in an absorbing phase transition around the upper critical dimension

S. Lübeck

We analyze numerically the critical behavior of an absorbing phase transition in a conserved lattice gas in an external field. The external field is realized as a spontaneous creation of active particles that drives the system away from criticality. Nevertheless, the order parameter obeys certain scaling laws for sufficiently small external fields. These scaling laws are investigated and the corresponding exponents are determined in various dimensions (D=2,3,4,5). At the so-called upper critical dimension D(c)=4 one has to modify the usual scaling laws by logarithmic corrections.We analyze numerically the critical behavior of an absorbing phase transition in a conserved lattice gas in an external field. The external field is realized as a spontaneous creation of active particles that drives the system away from criticality. Nevertheless, the order parameter obeys certain scaling laws for sufficiently small external fields. These scaling laws are investigated and the corresponding exponents are determined in various dimensions (D=2,3,4,5). At the so-called upper critical dimension D(c)=4 one has to modify the usual scaling laws by logarithmic corrections.

arXiv:cond-mat/0202014Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsApr 12, 200232 citations

From Nonlinear Scaling Fields to Critical Amplitudes

Yannick Meurice, S. Niermann

arXiv:cond-mat/0105380Journal of Statistical PhysicsJul 1, 20027 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 $\ensuremath{\delta}$ 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

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}}

N

with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out

N-l

oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for

N\\rightarrow\\infty

, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large

N

, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by

S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421

in the discrete case and

S=\\frac{0.3744}{3} \\ln{l} + 0.6421

in the infinite case.