We discuss variationally optimized matrix-product states for the transverse-field Ising chain using $D\ifmmode\times\else\texttimes\fi{}D$ matrices with small $D∊{2--10}$ . For finite system size $N$ there are energy minimums for symmetric as well as symmetry-broken states, which cross each other at a field value ${h}_{c}(N,D)$ ; thus the transition is first order. A continuous transition develops as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The asymptotic critical behavior is then always of mean-field type (the magnetization exponent $\ensuremath{\beta}=1/2$) but a window of field strengths where true Ising scaling holds $(\ensuremath{\beta}=1/8)$ emerges with increasing $D$ . We also demonstrate asymptotic mean-field behavior for infinite-size two-dimensional tensor-product (iPEPS) states with small tensors. The behaviors should be generic at symmetry-breaking transitions.We discuss variationally optimized matrix-product states for the transverse-field Ising chain using $D\ifmmode\times\else\texttimes\fi{}D$ matrices with small $D∊{2--10}$. For finite system size $N$ there are energy minimums for symmetric as well as symmetry-broken states, which cross each other at a field value ${h}_{c}(N,D)$; thus the transition is first order. A continuous transition develops as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The asymptotic critical behavior is then always of mean-field type (the magnetization exponent $\ensuremath{\beta}=1/2$) but a window of field strengths where true Ising scaling holds $(\ensuremath{\beta}=1/8)$ emerges with increasing $D$. We also demonstrate asymptotic mean-field behavior for infinite-size two-dimensional tensor-product (iPEPS) states with small tensors. The behaviors should be generic at symmetry-breaking transitions.

arXiv:1002.1657Physical Review BAug 26, 201032 citations

Zero-temperature renormalization-group method for quantum systems. III. Ising model in a transverse field in two dimensions

K. A. Penson, Rémi Jullien, Pierre M. Pfeuty

A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.

Physical review. B, Condensed matterMay 1, 197961 citations

Crossover critical behavior in three-dimensional Ising-like systems

M. A. Anisimov, Jan V. Sengers, Andrei F. Kostko

APSMar 1, 20031 citation

Interdimensional Scaling Laws

Y. Imry, G. Deutscher, David J. Bergman +1 more

We consider in detail the critical behavior of a $d$ -dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: $d$ -dimensional mean field, $d$ -dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.We consider in detail the critical behavior of a $d$-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: $d$-dimensional mean field, $d$-dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.

Physical Review AFeb 1, 197323 citations

We discuss variationally optimized matrix-product states for the transverse-field Ising chain using $D\ifmmode\times\else\texttimes\fi{}D$ matrices with small

D∊{2--10}

. For finite system size

N

there are energy minimums for symmetric as well as symmetry-broken states, which cross each other at a field value

{h}_{c}(N,D)

; thus the transition is first order. A continuous transition develops as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The asymptotic critical behavior is then always of mean-field type (the magnetization exponent $\ensuremath{\beta}=1/2$) but a window of field strengths where true Ising scaling holds $(\ensuremath{\beta}=1/8)$ emerges with increasing

D

. We also demonstrate asymptotic mean-field behavior for infinite-size two-dimensional tensor-product (iPEPS) states with small tensors. The behaviors should be generic at symmetry-breaking transitions.

We consider in detail the critical behavior of a

d

-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior:

d

-dimensional mean field,

d

-dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in

d

and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.