The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. It is known that this model exhibits a phase transition at the critical inverse temperature <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\beta _\textrm{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>β</mi> <mtext>c</mtext> </msub> </math></EquationSource> </InlineEquation>, which is determined by the spin-spin couplings and the transverse field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Björnberg Commun. Math. Phys. 323, 329–366 (2013) investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing the spin-spin coupling <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$J\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q$</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation> in a proper manner, with fixed temperature. In this paper, we fix J and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$q$</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation> and show that the susceptibility diverges as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$({\beta _\textrm{c}}-\beta )^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mtext>c</mtext> </msub> <mo>-</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta \uparrow {\beta _\textrm{c}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>β</mi> <mo stretchy="false">↑</mo> </mrow> <msub> <mi>β</mi> <mtext>c</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$d>4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>></mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> assuming an infrared bound on the space-time two-point function. One of the key elements is a stochastic-geometric representation in Björnberg & Grimmett J. Stat. Phys. 136, 231–273 (2009) and Crawford & Ioffe Commun. Math. Phys. 296, 447–474 (2010). As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$q=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>).

Mean-field behavior of the quantum Ising susceptibility and a new lace expansion for the classical Ising model

期刊论文
发表日期：

2025年9月15日

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Yoshinori Kamijima,
Akira Sakai

摘要

The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. It is known that this model exhibits a phase transition at the critical inverse temperature $\beta _\textrm{c}$ $β_{c}$ , which is determined by the spin-spin couplings and the transverse field $q\ge 0$ $q \geq 0$ . Björnberg Commun. Math. Phys. 323, 329–366 (2013) investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing the spin-spin coupling $J\ge 0$ $J \geq 0$ and $q$ $q$ in a proper manner, with fixed temperature. In this paper, we fix J and $q$ $q$ and show that the susceptibility diverges as $({\beta _\textrm{c}}-\beta )^{-1}$ ${(β_{c} - β)}^{- 1}$ as $\beta \uparrow {\beta _\textrm{c}}$ $β ↑ β_{c}$ for $d>4$ $d > 4$ assuming an infrared bound on the space-time two-point function. One of the key elements is a stochastic-geometric representation in Björnberg & Grimmett J. Stat. Phys. 136, 231–273 (2009) and Crawford & Ioffe Commun. Math. Phys. 296, 447–474 (2010). As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., $q=0$ $q = 0$ ).