Mean-field behavior of the quantum Ising susceptibility and a new lace expansion for the classical Ising model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908509074882560 |
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| author | Kamijima, Yoshinori Sakai, Akira |
| author_facet | Kamijima, Yoshinori Sakai, Akira |
| contents | 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 $β_{\mathrm{c}}$, which is determined by the spin-spin couplings and the transverse field $q \geq 0$. Björnberg [Commun. Math. Phys., 232 (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 $J \geq 0$ and $q$ in a proper manner, with fixed temperature. In this paper, we fix $J$ and $q$ and show that the susceptibility diverges as $(β_{\mathrm{c}} - β)^{-1}$ as $β\uparrowβ_{\mathrm{c}}$ for $d>4$ 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 (2009)] and Crawford & Ioffe [Commun. Math. Phys., 296 (2010)]. As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., $q=0$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06592 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean-field behavior of the quantum Ising susceptibility and a new lace expansion for the classical Ising model Kamijima, Yoshinori Sakai, Akira Mathematical Physics Probability 82B20, 82B26, 82B27, 82B10 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 $β_{\mathrm{c}}$, which is determined by the spin-spin couplings and the transverse field $q \geq 0$. Björnberg [Commun. Math. Phys., 232 (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 $J \geq 0$ and $q$ in a proper manner, with fixed temperature. In this paper, we fix $J$ and $q$ and show that the susceptibility diverges as $(β_{\mathrm{c}} - β)^{-1}$ as $β\uparrowβ_{\mathrm{c}}$ for $d>4$ 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 (2009)] and Crawford & Ioffe [Commun. Math. Phys., 296 (2010)]. As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., $q=0$). |
| title | Mean-field behavior of the quantum Ising susceptibility and a new lace expansion for the classical Ising model |
| topic | Mathematical Physics Probability 82B20, 82B26, 82B27, 82B10 |
| url | https://arxiv.org/abs/2501.06592 |