Abstract
For the Hubbard model on the two-dimensional copper-oxide lattice, equal-time four-point correlation functions at positive temperature are proved to decay exponentially in the thermodynamic limit if the magnitude of the on-site interactions is smaller than some power of temperature. This result especially implies that the equal-time correlation functions for singlet Cooper pairs of various symmetries decay exponentially in the distance between the Cooper pairs in high temperatures or in low-temperature weak-coupling regimes. The proof is based on a multi-scale integration over the Matsubara frequency.
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Abbreviations
- Notation :
-
Description
- L :
-
Size of lattice of the position variable
- t :
-
Hopping amplitude
- U c :
-
Coupling constant on the Cu sites
- U o :
-
Coupling constant on the O sites
- \({\epsilon _{c}^{\sigma}, \epsilon _o^{\sigma} (\sigma \in \{ \uparrow, \downarrow \}) }\) :
-
Spin-dependent on-site energies
- \({\beta}\) :
-
Proportional to the inverse of temperature
- E max :
-
\({{\rm max}_{\sigma \in \{ \uparrow, \downarrow \}} { \{1, |t|, |\epsilon _c^ {\sigma}|, |\epsilon _o^{\sigma}| \}}}\)
- \({\hat{\mathcal{X}} _j, \hat{\mathcal{Y}} _j}\) :
-
Same as \({(\hat{\rho}_j, \hat{\rm x}_j, \hat{\sigma}_j), (\hat{\eta} _j, \hat{\rm y} _j, \hat{\tau}_j) (j = 1, 2)}\),
- (j = 1, 2):
-
Fixed sites to define the correlation function
- h :
-
Step size of the discretization of \({[0, \beta), [-\beta, \beta)}\)
- N L, h :
-
\({6L^2 \beta h}\), Cardinality of I L, h
- \({\lambda _1, \lambda _{-1}}\) :
-
Used to modify the interaction
- c :
-
Generic constant depending only on a fixed smooth function
- M :
-
Parameter to control the size of the support of the cut-off function
- N h :
-
\({\lfloor \log(2h) / \log (M) \rfloor}\)
- \({N_{\beta}}\) :
-
\({{\rm max} \{ \lfloor \log (1 /\beta) / \log (M) \rfloor + 1, 1\}}\)
- c 0 :
-
Constant depending on M and \({\beta}\)
- \({\alpha}\) :
-
Additional parameter used in the multi-scale integration
- Notation :
-
Description
- \({\Gamma}\) :
-
\({( \mathbb{Z} /L\mathbb{Z})^2}\)
- \({[0, \beta) _h}\) :
-
\({\{0, 1/h, \ldots, \beta - 1/h \}}\)
- \({[-\beta, \beta)_h}\) :
-
\({\{-\beta,-\beta + 1/h, \ldots, -1/h\} \cup [0, \beta) _h}\)
- \({\Gamma ^*}\) :
-
\({(\frac{2 \pi}{L} \mathbb{Z} / (2 \pi \mathbb{Z}))^2}\)
- \({\mathcal{M}_h}\) :
-
\({\{\omega \in \pi (2 \mathbb{Z} +1)/\beta\ |\ | \omega | < \pi h \}}\)
- I L, h :
-
\({\{1,2,3 \} \times \Gamma \times \{ \uparrow, \downarrow \} \times [0, \beta)_h}\)
- \({\tilde{I} _{L, h}}\) :
-
\({I_{L, h} \times \{1,-1 \}}\)
- \({(I_{L, h})_o^m}\) :
-
Subset of \({I_{L, h}^m}\)
- D small :
-
Subset of \({ \mathbb{C} ^4}\)
- D R :
-
Subset of \({ \mathbb{C}}\)
- Notation :
-
Description
- \({\mathcal{F} _{t, \beta}(\cdot)}\) :
-
Used to specify the domain of analyticity of the covariance
- \({\hat{s}(\cdot)}\) :
-
Fixed function of spin
- \({\mathcal{C} (\cdot,\cdot)}\) :
-
Covariance of full scale
- \({\mathcal{X}_l(\cdot)}\) :
-
Cut-off function of \({l}\)-th scale
- \({\mathcal{C}_l(\cdot,\cdot)}\) :
-
Covariance of l-th scale
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Communicated by Vieri Mastropietro.
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Kashima, Y. Exponential Decay of Equal-Time Four-Point Correlation Functions in the Hubbard Model on the Copper-Oxide Lattice. Ann. Henri Poincaré 15, 1453–1522 (2014). https://doi.org/10.1007/s00023-013-0278-0
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DOI: https://doi.org/10.1007/s00023-013-0278-0