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d4091693-a614-453b-a58c-a64c9e63a378 | To the first order, \(G(h)\psi = |\nabla |\psi \) (see Theorem 2.1.1 in [1]}), so the linearization of (REF ) is
\({\left\lbrace \begin{array}{ll}h_t = |\nabla |\psi ,\\\psi _t = -h,\end{array}\right.}\)
| [1] | [
[
73,
76
]
] | https://openalex.org/W1584473947 |
cd0c35ad-91fd-48d6-9d97-be6b45e30bd1 | This reveals the dispersive nature of the equation and suggests that it is amenable to \(L^2\) energy estimates and \(L^\infty \) decay estimates. Indeed,
energy estimates has been worked out in [1]} and we only need to quote their results in Proposition REF below. It roughly says
\(\frac{d}{dt}\Vert u\Vert _{H^s}^2 \lesssim \Vert u\Vert _{H^s}^2\Vert u\Vert _{C^r}^2.\)
| [1] | [
[
197,
200
]
] | https://openalex.org/W1584473947 |
280e3415-584f-4f67-9f0c-5b48cb6c4b06 | While the whole structure of the proof resembles that in [1]}, here we aim not only to improve on known results on lifespans of two dimensional water waves, but also to show that the framework established in [1]} is easily adaptable, and specifically, that the wealth of estimates already present in the literature, for example [3]}, [4]}, can be readily assembled to yield a short proof of previously inaccessible results.
| [3] | [
[
328,
331
]
] | https://openalex.org/W1584473947 |
77c06042-080e-46e3-8ad2-f33f2430079d | While the whole structure of the proof resembles that in [1]}, here we aim not only to improve on known results on lifespans of two dimensional water waves, but also to show that the framework established in [1]} is easily adaptable, and specifically, that the wealth of estimates already present in the literature, for example [3]}, [4]}, can be readily assembled to yield a short proof of previously inaccessible results.
| [4] | [
[
334,
337
]
] | https://openalex.org/W2592223562 |
e4643065-529e-4d0e-ab78-23b8fe765bc9 | It should be added however, that in the energy estimates, one can have three factors of \(L^\infty \) norms on the right-hand side of (REF ),
using additional integrability in two dimensions [1]}, [2]},
but this extra saving does not easily carry over to dispersive estimates,
because the trivial four wave resonances (\(\sqrt{|\xi _1|} + \sqrt{|\xi _2|} - \sqrt{|\xi _1|} - \sqrt{|\xi _2|} = 0\) ) lead to modified scattering [3]}. Controlling this effect seems to require more regularity in the frequency space, i.e., weights in the physical space, which is out of the scope of this paper.
| [1] | [
[
192,
195
]
] | https://openalex.org/W2898175050 |
7b842127-da55-487f-957c-680f8fefd983 | It should be added however, that in the energy estimates, one can have three factors of \(L^\infty \) norms on the right-hand side of (REF ),
using additional integrability in two dimensions [1]}, [2]},
but this extra saving does not easily carry over to dispersive estimates,
because the trivial four wave resonances (\(\sqrt{|\xi _1|} + \sqrt{|\xi _2|} - \sqrt{|\xi _1|} - \sqrt{|\xi _2|} = 0\) ) lead to modified scattering [3]}. Controlling this effect seems to require more regularity in the frequency space, i.e., weights in the physical space, which is out of the scope of this paper.
| [2] | [
[
198,
201
]
] | https://openalex.org/W2916835505 |
f5c1fa46-6649-47a4-b31e-28c3c9b38e43 | It should be added however, that in the energy estimates, one can have three factors of \(L^\infty \) norms on the right-hand side of (REF ),
using additional integrability in two dimensions [1]}, [2]},
but this extra saving does not easily carry over to dispersive estimates,
because the trivial four wave resonances (\(\sqrt{|\xi _1|} + \sqrt{|\xi _2|} - \sqrt{|\xi _1|} - \sqrt{|\xi _2|} = 0\) ) lead to modified scattering [3]}. Controlling this effect seems to require more regularity in the frequency space, i.e., weights in the physical space, which is out of the scope of this paper.
| [3] | [
[
428,
431
]
] | https://openalex.org/W2141636416 |
e7e6b63b-db2e-4e39-9bb8-39d4cc281955 | Definition 3 (See (4.35)–(4.37) in [1]})
\(\begin{aligned}G(h)\psi &= |\nabla |\psi + \int _0^1 \partial _sG(sh)\psi ds,\\\partial _sG(sh)\psi &= -G(sh)[hB(sh)\psi ] - (hV(sh)\psi )^{\prime },\\B = B(h)\psi &= \frac{G(h)\psi + h^{\prime }\psi ^{\prime }}{1 + h^{\prime 2}},\\V = V(h)\psi &= \psi ^{\prime } - h^{\prime }B(h)\psi .\end{aligned}\)
| [1] | [
[
35,
38
]
] | https://openalex.org/W3133514565 |
fe8b5628-2045-4ec2-bf6c-02016d1c07e7 | Lemma 1 (Lemma 2.0.5 in [1]})
Let \(\gamma > 3\) be such that \(2\gamma \notin {\mathbb {Z}}\) . Then for all \((h, |\nabla |^{1/2}\psi ) \in C_*^\gamma \times C_*^{\gamma -1/2}\) such that \(\Vert h^{\prime }\Vert _{C_*^{\gamma -1}} + \Vert h^{\prime }\Vert _{C_*^{-1}}^{1/2}\Vert h^{\prime }\Vert _{H^{-1}}^{1/2} \le c_r\) , we have \(\Vert G(h)\psi \Vert _{C_*^{\gamma -1}} + \Vert B\Vert _{C_*^{\gamma -1}} + \Vert V\Vert _{C_*^{\gamma -1}} \lesssim _r \Vert |\nabla |^{1/2}\psi \Vert _{C_*^{\gamma -1/2}}\) .
| [1] | [
[
24,
27
]
] | https://openalex.org/W1584473947 |
d42ecf77-9900-4d83-aa8c-e5b397f7fb6e | By Remark REF we can assume that \(\psi \) is Schwartz.
By Remark REF we have the necessary smallness condition to apply Theorem 2.1.1 in [1]} to get
\(\begin{aligned}\Vert (G(h)\psi , B, V)\Vert _{H^{s-1}}&\lesssim _{\Vert h\Vert _{C_*^\gamma }} \Vert |\nabla |^{1/2}\psi \Vert _{C_*^{\gamma -1/2}}\Vert h\Vert _{H^s} + \Vert |\nabla |^{1/2}\psi \Vert _{H^{s-1/2}}\\&\lesssim \Vert |\nabla |^{1/2}\psi \Vert _{H^{s-1/2}}\end{aligned}\)
| [1] | [
[
141,
144
]
] | https://openalex.org/W1584473947 |
60295d07-6e1f-45d8-8822-01baf39489f2 | By Remark REF we can assume that \(\psi \) is Schwartz.
By Remark REF we have the necessary smallness condition to apply (2.5.1) in [1]} to get the bound for \(G(h)\psi \) . To get the other bound, we also need the expression of \(B = B(h)\psi \) in (REF ), the Sobolev multiplication theorem and the smallness of \(\Vert h\Vert _{H^s}\) .
| [1] | [
[
135,
138
]
] | https://openalex.org/W1584473947 |
4df80c15-afa4-4476-963c-6bdd082d0c84 | Here we collect the assumptions on which Chapters 1–3 of [1]} are based.
| [1] | [
[
57,
60
]
] | https://openalex.org/W1584473947 |
dc6a2efb-1f2b-46fc-bb3c-9e2f673f1cd6 | Definition 4 (Definition A.1.2 of [1]})
Define \(w = \psi - T_Bh\) , where
\(\widehat{T_fg}(\xi ) = \int _{\xi _1+\xi _2=\xi } \varphi (\xi _1, \xi _2)\hat{f}(\xi _1)\hat{g}(\xi _2)d\xi \)
| [1] | [
[
34,
37
]
] | https://openalex.org/W1584473947 |
c549a5d9-d324-4183-9e00-18a7c920123d | Assumption 1 (Assumption 3.1.1 (i) in [1]})
\((h, |\nabla |^{1/2}\psi ) \in C([0, T], H^s \times H^{s-1/2}\) and \(|\nabla |^{1/2}w \in C([0, T], H^s)\) .
| [1] | [
[
38,
41
]
] | https://openalex.org/W1584473947 |
e79d1a25-b8f0-4841-9ebf-b69f8391fd67 | Assumption 2 (Assumption 3.1.1 (ii) in [1]})
\(\sup _{t\in [0,T]} (\Vert h^{\prime }\Vert _{C_*^{\rho -1}} + \Vert h^{\prime }\Vert _{C_*^{-1}}^{1/2}\Vert h^{\prime }\Vert _{H^{-1}}^{1/2}) \le c_{s,\rho }\text{ is small enough}\)
| [1] | [
[
39,
42
]
] | https://openalex.org/W1584473947 |
a0d65d71-e032-4c07-90a1-6467bd02fb10 | Remark 4
By the remark after Assumption 3.1.1 in [1]}, Assumption REF is guaranteed if \(\sup _{t\in [0,T]} \Vert h\Vert _{C_*^\rho }\) , \(\Vert h(0)\Vert _{L^2}\) and \(\Vert |\nabla |^{1/2}\psi (0)\Vert _{L^2}\) are small enough.
| [1] | [
[
50,
53
]
] | https://openalex.org/W1584473947 |
262cbd80-8f82-49ab-af7a-9450d2e1e886 | Assumption 3 (Assumption 3.1.5 in [1]})
\(\sup _{t\in [0,T]} (\Vert h(t)\Vert _{C_*^\rho } + \Vert |\nabla |^{1/2}\psi (t)\Vert _{C_*^\rho }) \le c_{s,\rho }\text{ is small enough}\)
| [1] | [
[
34,
37
]
] | https://openalex.org/W1584473947 |
140efdd2-dd1b-4a03-84cd-c782f1c54036 | Then the arguments up to Chapter 3 of [1]} applies. In more detail,
there is a change of variable \((T_ah, |\nabla |^{1/2}w) \mapsto \Phi \) satisfying \(E_s/3 \le \Vert \Phi \Vert _{H^s} \le 3E_s\) by (3.7.2) and (3.7.3) in [1]} and the quartic energy estimate
\(\Vert \Phi (t)\Vert _{H^s}^2 \le \Vert \Phi (0)\Vert _{H^s}^2 + C\int _0^t (\Vert h(\tau )\Vert _{C_*^\rho }^2 + \Vert |\nabla |^{1/2}\psi (\tau )\Vert _{C_*^\rho }^2)\Vert \Phi (\tau )\Vert _{H^s}^2d\tau \)
| [1] | [
[
38,
41
],
[
227,
230
]
] | https://openalex.org/W1584473947 |
7577e678-da52-48d3-8975-1c0b83e1803c | by (3.7.7) in [1]}, from which the result now follows.
| [1] | [
[
14,
17
]
] | https://openalex.org/W1584473947 |
afca9c27-d3b9-47da-8aab-5fd5a3e423dc | Let \(u = h + i|\nabla |^{1/2}\psi \) , whose evolution equation is \(u_t + i\Lambda u = N\) , where the dispersion relation \(\Lambda = |\nabla |^{1/2}\) and (see (6.1) and (4.43) in [1]})
\(\begin{aligned}N&=(G(h)-|\nabla |)\psi +\frac{i}{2}|\nabla |^{1/2}((1+h^{\prime 2})B^2-\psi ^{\prime 2})=N_2+N_3,\\N_2&=-|\nabla |(h|\nabla |\psi )-(h\psi ^{\prime })^{\prime }+\frac{i}{2}|\nabla |^{1/2}((|\nabla |\psi )^2-\psi ^{\prime 2}),\\N_3&=B_3+h^{\prime 2}B+\frac{i}{2}|\nabla |^{1/2}(B^2-(|\nabla |\psi )^2+h^{\prime 2}B^2),\\B_3&=B - |\nabla |\psi + |\nabla |(h|\nabla |\psi ) + h\psi ^{\prime \prime }.\end{aligned}\)
| [1] | [
[
185,
188
]
] | https://openalex.org/W3133514565 |
1504e98a-c1c7-44ff-94df-bfb643b55860 | Using integration by parts in time we get (see (6.4) to (6.6) in [1]})
\(\begin{aligned}u_2(t) &= \sum _{\mu ,\nu =\pm } \left( Q_{\mu \nu }(t) - e^{-it\Lambda }Q_{\mu \nu }(0) - \int _0^t e^{-i(t-\tau )\Lambda }C_{\mu \nu }(\tau )d\tau \right),\\\hat{Q}_{\mu \nu }(\xi , t) &= C\int _{\xi _1+\xi _2=\xi } \frac{m_{\mu \nu }(\xi _1, \xi _2)}{i\Phi _{\mu \nu }(\xi _1, \xi _2)}\hat{u}_\mu (\xi _1, t)\hat{u}_\nu (\xi _2, t)d\xi _1,\\\hat{C}_{\mu \nu }(\xi , t) &= C\int _{\xi _1+\xi _2=\xi } \frac{m_{\mu \nu }(\xi _1, \xi _2)}{i\Phi _{\mu \nu }(\xi _1, \xi _2)}(\hat{u}_\mu (\xi _1, t)\hat{N}_\nu (\xi _2, t) + \hat{N}_\mu (\xi _1, t)u_\nu (\xi _2, t))d\xi _1\end{aligned}\)
| [1] | [
[
65,
68
]
] | https://openalex.org/W3133514565 |
6e5f8755-407c-4c21-92ce-04864e850a2e | It suffices to show that \(\Vert P_ke^{-it\Lambda }u\Vert _{L^4_tL^\infty _x} \lesssim 2^{3k/8}\Vert u\Vert _{L^2}\) for \(k \in {\mathbb {Z}}\) , where \(P_k\) denotes the Littlewood–Paley decomposition. Since \(\Lambda \) is homogeneous of degree \(1/2\) , the scaling \((x, t) \mapsto (2^kx, 2^{k/2}t)\) is a symmetry, so we can assume \(k = 0\) . Then the result from the standard \(t^{-1/2}\) dispersion estimates and the Hardy–Littlewood–Sobolev inequality, see Theorem 2.3 in [1]} for details.
| [1] | [
[
488,
491
]
] | https://openalex.org/W1547431580 |
b3349843-e9b7-4351-8388-5ec7ae0979f3 | Definition 5 (Definition C.1–C.2 in [1]}) A multiplier \(m(\xi _1, \xi _2)\) is of class \(\mathcal {B}_s\) if:
| [1] | [
[
36,
39
]
] | https://openalex.org/W2592223562 |
e6775664-7b0e-4ea9-a805-2e08ee2a592c | By Section 3 of [1]}, \(m_{\mu \nu } \in \mathcal {B}_{3/2}\)
and \(\Phi _{\mu \nu } \in \mathcal {B}_{1/2}\) . Hence \(m_{\mu \nu }/\Phi _{\mu \nu } \in \mathcal {B}_1\) , and can be decomposed as \(m_1 + m_2\) where \(m_j\) captures the contribution where the frequency of the \(j\) -th slot is bounded below by a (small) constant times the frequency of the other slot. Thus, for example, \(m_1 \in \mathcal {\tilde{B}}_1\) .
| [1] | [
[
16,
19
]
] | https://openalex.org/W2592223562 |
be15e791-84c2-45d7-b1c6-6ea241a3ab11 | Let \(Q_{\mu \nu ,j}\) be the corresponding bilinear product. Then the multiplier of \(\sqrt{1-\Delta }^\gamma Q_{\mu \nu ,1}(\sqrt{1-\Delta }^{-\gamma -1}\cdot , \cdot )\) is of class \(\mathcal {B}_0\) and by Theorem C.1 (i) of [1]} satisfies
\(\Vert \sqrt{1-\Delta }^\gamma Q_{\mu \nu ,1}(\sqrt{1-\Delta }^{-\gamma -1}f, g)\Vert _{L^2} \lesssim _{p,q} \Vert f\Vert _{L^o}\Vert g\Vert _{L^q}\)
| [1] | [
[
233,
236
]
] | https://openalex.org/W2592223562 |
c1a2e0a8-3aab-424f-a099-a52c9f2c8e96 | Note that in (REF ), \(B_3 = B - B_{\le 2}\) as defined in (2.6.1) in [1]}. Thus we have:
| [1] | [
[
71,
74
]
] | https://openalex.org/W1584473947 |
316434e3-2fb6-4f4b-a622-cc6c89a05bb1 | Lemma 7 (Proposition 2.6.1 in [1]})
Let \(s, \gamma , \mu \) be such that \(s - 1/2 > \gamma > 14\) , \(s \ge \mu \ge 5\) and \(2\gamma \notin {\mathbb {Z}}\) . Then for all \((h, |\nabla |^{1/2}\psi ) \in H^{s+1/2} \times (C_*^{\gamma -1/2} \cap H^\mu )\) such that \(\Vert h\Vert _{C_*^\gamma } \le c_{s,\gamma ,\mu }\) is small enough,
\(\Vert B_3\Vert _{H^{\mu -1}} \lesssim _{s,\gamma ,\mu } \Vert h\Vert _{C_*^\gamma }(\Vert |\nabla |^{1/2}\psi \Vert _{C_*^{\gamma -1/2}}\Vert h\Vert _{H^s} + \Vert h\Vert _{C_*^\gamma }\Vert |\nabla |^{1/2}\psi \Vert _{H^\mu }).\)
| [1] | [
[
30,
33
]
] | https://openalex.org/W1584473947 |
d6e4b42e-b6cd-462e-9575-1e725223198e | [Proof of Theorem REF ]
Since the water wave equation is locally wellposed (see [1]} for example), we only need to show a priori estimates that can be closed.
| [1] | [
[
80,
83
]
] | https://openalex.org/W2916835505 |
d26f2767-356e-4e3b-8e09-b4688a4a17b8 | If \(1 \le R \le \epsilon ^{-2}\) then we resort to [1]}, noting that that result carries over to the periodic case.
| [1] | [
[
53,
56
]
] | https://openalex.org/W2916835505 |
60ef1e18-5e76-4bbc-a53c-6b81b3e7a4b5 | The nature of dark matter (DM) is still shrouded in mystery. While a weakly interacting massive particle (WIMP) remains a simple and appealing possibility, the sheer lack of any experimental evidence for such states provides ever stronger motivation for also exploring alternative avenues. One promising direction is to consider hidden sector (HS) scenarios, which feature a DM candidate that strongly interacts with itself and potentially other dark-sector states, while only being – at best – feebly coupled to the Standard Model (SM), or more generally to the visible sector (VS).Such scenarios are further motivated as a potential solution for some of the issues that collisionless DM faces on galactic and sub-galactic scales [1]}, [2]}. Such a HS may be in thermal equilibrium with itself, but not with the VS, meaning that its temperature \(T^{\prime }\) might differ substantially from the temperature \(T\) of the VS [3]}, [4]}.
| [1] | [
[
731,
734
]
] | https://openalex.org/W2083034637 |
018fc30a-a116-4604-848f-641b91e79fea | The nature of dark matter (DM) is still shrouded in mystery. While a weakly interacting massive particle (WIMP) remains a simple and appealing possibility, the sheer lack of any experimental evidence for such states provides ever stronger motivation for also exploring alternative avenues. One promising direction is to consider hidden sector (HS) scenarios, which feature a DM candidate that strongly interacts with itself and potentially other dark-sector states, while only being – at best – feebly coupled to the Standard Model (SM), or more generally to the visible sector (VS).Such scenarios are further motivated as a potential solution for some of the issues that collisionless DM faces on galactic and sub-galactic scales [1]}, [2]}. Such a HS may be in thermal equilibrium with itself, but not with the VS, meaning that its temperature \(T^{\prime }\) might differ substantially from the temperature \(T\) of the VS [3]}, [4]}.
| [2] | [
[
737,
740
]
] | https://openalex.org/W2612559938 |
df1192d3-86c8-4814-8ed2-5cf56b4ff2db | The nature of dark matter (DM) is still shrouded in mystery. While a weakly interacting massive particle (WIMP) remains a simple and appealing possibility, the sheer lack of any experimental evidence for such states provides ever stronger motivation for also exploring alternative avenues. One promising direction is to consider hidden sector (HS) scenarios, which feature a DM candidate that strongly interacts with itself and potentially other dark-sector states, while only being – at best – feebly coupled to the Standard Model (SM), or more generally to the visible sector (VS).Such scenarios are further motivated as a potential solution for some of the issues that collisionless DM faces on galactic and sub-galactic scales [1]}, [2]}. Such a HS may be in thermal equilibrium with itself, but not with the VS, meaning that its temperature \(T^{\prime }\) might differ substantially from the temperature \(T\) of the VS [3]}, [4]}.
| [3] | [
[
928,
931
]
] | https://openalex.org/W2161075311 |
ebd87658-fc0b-4e44-bffd-bacf3113e6eb | The nature of dark matter (DM) is still shrouded in mystery. While a weakly interacting massive particle (WIMP) remains a simple and appealing possibility, the sheer lack of any experimental evidence for such states provides ever stronger motivation for also exploring alternative avenues. One promising direction is to consider hidden sector (HS) scenarios, which feature a DM candidate that strongly interacts with itself and potentially other dark-sector states, while only being – at best – feebly coupled to the Standard Model (SM), or more generally to the visible sector (VS).Such scenarios are further motivated as a potential solution for some of the issues that collisionless DM faces on galactic and sub-galactic scales [1]}, [2]}. Such a HS may be in thermal equilibrium with itself, but not with the VS, meaning that its temperature \(T^{\prime }\) might differ substantially from the temperature \(T\) of the VS [3]}, [4]}.
| [4] | [
[
934,
937
]
] | https://openalex.org/W3098837140 |
0f21b0e0-c43f-4226-a13e-1032beba31fc | Exponential frequency power spectra of fluctuation time series appears to be an intrinsic property of deterministic chaos in continuous time systems.[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]} This has been observed in numerous experiments and model simulations of fluids and magnetized plasmas.[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]} Recently, the exponential spectrum has been attributed to the presence of Lorentzian pulses in the temporal dynamics.[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]} Weakly non-linear systems are often characterized by quasi-periodic oscillations, resulting in a frequency power spectral density resembling a Dirac comb.[29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]} Far from the linear instability threshold the spectral peaks broaden and in many cases an exponential spectrum results.[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]}
| [4] | [
[
167,
170
],
[
329,
332
],
[
1036,
1039
]
] | https://openalex.org/W1975964493 |
e26eb417-7bce-46c6-b449-2cde51148cd5 | Here, we briefly present the well-known Poisson summation formula, which is treated in a number of textbooks [1]}, [2]}, [3]}, [4]}. For our purposes, the formulation used in Corollary VII.2.6 in [2]} is the most useful. The statement in the book is for functions on general Euclidian spaces, but we repeat it here only for our special case (the real line):
| [1] | [
[
109,
112
]
] | https://openalex.org/W2319026302 |
448e6340-9797-4ed1-b994-06a83a8eb93f | Here, we briefly present the well-known Poisson summation formula, which is treated in a number of textbooks [1]}, [2]}, [3]}, [4]}. For our purposes, the formulation used in Corollary VII.2.6 in [2]} is the most useful. The statement in the book is for functions on general Euclidian spaces, but we repeat it here only for our special case (the real line):
| [3] | [
[
121,
124
]
] | https://openalex.org/W1531371725 |
f30fbe79-b765-4764-8ac5-d1a9de194674 | In contrast to coil-coil block copolymers, rod-coil diblock copolymers
have rich phase behaviors. Rod-coil
block copolymers, as the simplest rod-coil block copolymers,
self-assembled into ordered structures, e.g., zigzag
lamellae [1]}, stripes [2]},
honeycombs [3]}, and hollow spherical and cylindrical
micelles [4]}. Compared with AB
diblock copolymers, ABC triblock copolymers have more independent
parameters controlling their phase behavior. The phase behavior of
the two-component rod-coil block copolymers is controlled mainly by
three parameters: the volume fraction of the rod block \(f_{\text{A}}\) , the
Flory-Huggins interaction between different blocks \(\chi _{\text{AB}}\) , and
the total degree of polymerization of the copolymer \(N\) . For ABC
triblock copolymers, the number of parameters increases to six,
including three interaction parameters \(\chi _{\text{AB}} \) , \(\chi _{\text{BC}} \) ,
and \(\chi _{\text{AC}} \) ; two independent volume fractions \(f_{\text{A}}\) and \(f_{\text{B}}\)
and the total degree of polymerization of the copolymer \(N\) .
This increased number of molecular variables will impose varieties
and complexities on the self-assembly of the rod-coil block
copolymers, meanwhile, leading to a great model system for
engineering of a large number of intriguing nanostructures.
| [4] | [
[
313,
316
]
] | https://openalex.org/W4240817298 |
5621148a-7083-44e0-a80f-f80a364ed73f | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [4] | [
[
183,
186
]
] | https://openalex.org/W3159222123 |
23590172-5ae2-41d0-ab0d-a6fac3e986af | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [5] | [
[
330,
333
]
] | https://openalex.org/W2006613135 |
165ecc8b-b450-4c7f-9338-7405d9ece9d5 | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [6] | [
[
336,
339
]
] | https://openalex.org/W2092660500 |
1a304aaa-41a9-47d5-b95d-156ecb1bf60f | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [7] | [
[
342,
345
]
] | https://openalex.org/W2080509755 |
d46a2639-01bc-4a74-829d-cca9989256c7 | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [8] | [
[
348,
351
]
] | https://openalex.org/W2169067738 |
45f798db-0a31-4a36-bdd7-4d6a453be38f | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [9] | [
[
354,
357
]
] | https://openalex.org/W3104988532 |
f876d4e4-8cfa-46cb-97f3-17f968a8d437 | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [13] | [
[
575,
579
]
] | https://openalex.org/W2974635060 |
441db285-9be3-4c24-84d4-6f10fec9ee0d | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [14] | [
[
582,
586
]
] | https://openalex.org/W3099516921 |
6ec8a38d-209f-4c8b-a4bb-5fa8c6add7a3 | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [16] | [
[
596,
600
]
] | https://openalex.org/W3132260257 |
71694726-d2a2-4e45-9987-ae4225126390 | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [17] | [
[
603,
607
]
] | https://openalex.org/W3133629120 |
c05ec679-6d97-465b-a6a2-f22096bd193f | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [19] | [
[
617,
621
]
] | https://openalex.org/W3132395781 |
b85b661a-5213-494b-9878-a9f50f348b11 | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [22] | [
[
875,
879
]
] | https://openalex.org/W3153145871 |
4ca0ee79-2ef6-4ace-8917-36daec4c5393 | Rich quantum phase transitions
in strongly correlated metals with
multiple degrees of freedom and geometrical frustration
were discovered one after another recently
[1]}, [2]}, [3]}, [4]}.
To understand such rich phase transitions,
a significant ingredient is various
quantum interference processes between different fluctuations
[5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
The recent discovery of unconventional density-wave (DW) order
and exotic superconductivity in kagome lattice metal AV\(_3\) Sb\(_5\) (A=K, Rb, Cs)
have triggered enormous amount of researches
[13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
The prominent interplay between density-wave and superconductivity
in addition to geometrical frustration have
attracted considerable attention
in kagome metals with strong correlation.
This discovery sparked lots of theoretical studies
[21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}.
| [23] | [
[
882,
886
]
] | https://openalex.org/W3133870740 |
21aa70eb-52ef-4f8a-a9a1-89a399d912b2 | At ambient pressure,
AV\(_3\) Sb\(_5\) exhibits charge-channel DW order
at \(T_{\rm DW}=78\) , 94 and 102 K for A=K, Cs and Rb, respectively
[1]}, [2]}, [3]}, [4]}.
Below \(T_{\rm DW}\) , \(2\times 2\) (inverse) star of David pattern
is observed by STM studies
[5]}, [6]}.
The absence of acoustic phonon anomaly at \(T_{\rm DW}\)
[7]}
would exclude strong electron-phonon coupling driven DW state.
As possible electron-correlation-driven DW orders,
charge/bond orders and loop current orders
[8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}
have been proposed theoretically,
mainly based on the extended Hubbard model
with the on-site (\(U\) ) and the nearest-site (\(V\) ) Coulomb interactions.
However, when \(V\ll U\) due to Thomas-Fermi screening,
previous studies predicted the strong spin-density-wave (SDW) instability,
in contrast to the tiny SDW instability in AV\(_3\) Sb\(_5\) at \(T_{\rm DW}\)
[4]}, [3]}, [18]}.
| [1] | [
[
142,
145
]
] | https://openalex.org/W2974635060 |
e6c4e900-abfd-4988-8600-6d378ef334e4 | At ambient pressure,
AV\(_3\) Sb\(_5\) exhibits charge-channel DW order
at \(T_{\rm DW}=78\) , 94 and 102 K for A=K, Cs and Rb, respectively
[1]}, [2]}, [3]}, [4]}.
Below \(T_{\rm DW}\) , \(2\times 2\) (inverse) star of David pattern
is observed by STM studies
[5]}, [6]}.
The absence of acoustic phonon anomaly at \(T_{\rm DW}\)
[7]}
would exclude strong electron-phonon coupling driven DW state.
As possible electron-correlation-driven DW orders,
charge/bond orders and loop current orders
[8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}
have been proposed theoretically,
mainly based on the extended Hubbard model
with the on-site (\(U\) ) and the nearest-site (\(V\) ) Coulomb interactions.
However, when \(V\ll U\) due to Thomas-Fermi screening,
previous studies predicted the strong spin-density-wave (SDW) instability,
in contrast to the tiny SDW instability in AV\(_3\) Sb\(_5\) at \(T_{\rm DW}\)
[4]}, [3]}, [18]}.
| [2] | [
[
148,
151
]
] | https://openalex.org/W3099516921 |
aecf4cf8-953e-4312-a090-861d0dc765c7 | At ambient pressure,
AV\(_3\) Sb\(_5\) exhibits charge-channel DW order
at \(T_{\rm DW}=78\) , 94 and 102 K for A=K, Cs and Rb, respectively
[1]}, [2]}, [3]}, [4]}.
Below \(T_{\rm DW}\) , \(2\times 2\) (inverse) star of David pattern
is observed by STM studies
[5]}, [6]}.
The absence of acoustic phonon anomaly at \(T_{\rm DW}\)
[7]}
would exclude strong electron-phonon coupling driven DW state.
As possible electron-correlation-driven DW orders,
charge/bond orders and loop current orders
[8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}
have been proposed theoretically,
mainly based on the extended Hubbard model
with the on-site (\(U\) ) and the nearest-site (\(V\) ) Coulomb interactions.
However, when \(V\ll U\) due to Thomas-Fermi screening,
previous studies predicted the strong spin-density-wave (SDW) instability,
in contrast to the tiny SDW instability in AV\(_3\) Sb\(_5\) at \(T_{\rm DW}\)
[4]}, [3]}, [18]}.
| [8] | [
[
495,
498
]
] | https://openalex.org/W3153145871 |
ae62f5c8-fb87-4d2c-8f11-4037d74dc2b0 | At ambient pressure,
AV\(_3\) Sb\(_5\) exhibits charge-channel DW order
at \(T_{\rm DW}=78\) , 94 and 102 K for A=K, Cs and Rb, respectively
[1]}, [2]}, [3]}, [4]}.
Below \(T_{\rm DW}\) , \(2\times 2\) (inverse) star of David pattern
is observed by STM studies
[5]}, [6]}.
The absence of acoustic phonon anomaly at \(T_{\rm DW}\)
[7]}
would exclude strong electron-phonon coupling driven DW state.
As possible electron-correlation-driven DW orders,
charge/bond orders and loop current orders
[8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}
have been proposed theoretically,
mainly based on the extended Hubbard model
with the on-site (\(U\) ) and the nearest-site (\(V\) ) Coulomb interactions.
However, when \(V\ll U\) due to Thomas-Fermi screening,
previous studies predicted the strong spin-density-wave (SDW) instability,
in contrast to the tiny SDW instability in AV\(_3\) Sb\(_5\) at \(T_{\rm DW}\)
[4]}, [3]}, [18]}.
| [13] | [
[
528,
532
]
] | https://openalex.org/W2076451752 |
36f6c159-275c-4ddc-aa8c-026a58e942dd | At ambient pressure,
AV\(_3\) Sb\(_5\) exhibits charge-channel DW order
at \(T_{\rm DW}=78\) , 94 and 102 K for A=K, Cs and Rb, respectively
[1]}, [2]}, [3]}, [4]}.
Below \(T_{\rm DW}\) , \(2\times 2\) (inverse) star of David pattern
is observed by STM studies
[5]}, [6]}.
The absence of acoustic phonon anomaly at \(T_{\rm DW}\)
[7]}
would exclude strong electron-phonon coupling driven DW state.
As possible electron-correlation-driven DW orders,
charge/bond orders and loop current orders
[8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}
have been proposed theoretically,
mainly based on the extended Hubbard model
with the on-site (\(U\) ) and the nearest-site (\(V\) ) Coulomb interactions.
However, when \(V\ll U\) due to Thomas-Fermi screening,
previous studies predicted the strong spin-density-wave (SDW) instability,
in contrast to the tiny SDW instability in AV\(_3\) Sb\(_5\) at \(T_{\rm DW}\)
[4]}, [3]}, [18]}.
| [14] | [
[
535,
539
]
] | https://openalex.org/W2041469486 |
dc311a0e-cf8b-4c41-b603-2326d3ed8d47 | At ambient pressure,
AV\(_3\) Sb\(_5\) exhibits charge-channel DW order
at \(T_{\rm DW}=78\) , 94 and 102 K for A=K, Cs and Rb, respectively
[1]}, [2]}, [3]}, [4]}.
Below \(T_{\rm DW}\) , \(2\times 2\) (inverse) star of David pattern
is observed by STM studies
[5]}, [6]}.
The absence of acoustic phonon anomaly at \(T_{\rm DW}\)
[7]}
would exclude strong electron-phonon coupling driven DW state.
As possible electron-correlation-driven DW orders,
charge/bond orders and loop current orders
[8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}
have been proposed theoretically,
mainly based on the extended Hubbard model
with the on-site (\(U\) ) and the nearest-site (\(V\) ) Coulomb interactions.
However, when \(V\ll U\) due to Thomas-Fermi screening,
previous studies predicted the strong spin-density-wave (SDW) instability,
in contrast to the tiny SDW instability in AV\(_3\) Sb\(_5\) at \(T_{\rm DW}\)
[4]}, [3]}, [18]}.
| [15] | [
[
542,
546
]
] | https://openalex.org/W2060884327 |
0a3bee44-5129-464d-a26f-7bcdf6434626 | Below \(T_{\rm DW}\) , exotic nodal superconductivity
occurs at \(T_{\rm c}=1 \sim 3\) K at ambient pressure.
[1]}, [2]}.
The Hebel-Slichter peak in \(1/T_1T\) [3]}
and small impurity effect on \(T_{\rm c}\)
[4]}
indicate the singlet \(s\) -wave superconducting (SC) state.
On the other hand,
possibilities of triplet pairing state
[5]}
and nematic SC state
[6]}, [7]}
have been reported, which indicates non-\(s\) wave pairing states.
In addition, possibility of topological states
has been discussed intensively
[8]}.
Under pressure, \(T_{\rm DW}\) decreases and
vanishes at the DW quantum critical point (DW-QCP)
at \(P\sim 2\) GPa.
For A=Cs, \(T_{\rm c}\) exhibits nontrivial
double SC dome in the DW phase,
and the highest \(T_{\rm c}\ (\lesssim 10 {\rm K})\)
is realized at the DW-QCP
[9]}.
In addition, theoretical phonon-mediated \(s\) -wave \(T_{\rm c}\)
is too low to explain experiments
[10]}.
Thus, unconventional SC state due to DW fluctuations
[11]}, [12]}
is naturally expected in AV\(_3\) Sb\(_5\) .
| [2] | [
[
117,
120
]
] | https://openalex.org/W3132395781 |
b3973ad8-76cf-4c20-a77a-4c1e748a2bd7 | Below \(T_{\rm DW}\) , exotic nodal superconductivity
occurs at \(T_{\rm c}=1 \sim 3\) K at ambient pressure.
[1]}, [2]}.
The Hebel-Slichter peak in \(1/T_1T\) [3]}
and small impurity effect on \(T_{\rm c}\)
[4]}
indicate the singlet \(s\) -wave superconducting (SC) state.
On the other hand,
possibilities of triplet pairing state
[5]}
and nematic SC state
[6]}, [7]}
have been reported, which indicates non-\(s\) wave pairing states.
In addition, possibility of topological states
has been discussed intensively
[8]}.
Under pressure, \(T_{\rm DW}\) decreases and
vanishes at the DW quantum critical point (DW-QCP)
at \(P\sim 2\) GPa.
For A=Cs, \(T_{\rm c}\) exhibits nontrivial
double SC dome in the DW phase,
and the highest \(T_{\rm c}\ (\lesssim 10 {\rm K})\)
is realized at the DW-QCP
[9]}.
In addition, theoretical phonon-mediated \(s\) -wave \(T_{\rm c}\)
is too low to explain experiments
[10]}.
Thus, unconventional SC state due to DW fluctuations
[11]}, [12]}
is naturally expected in AV\(_3\) Sb\(_5\) .
| [5] | [
[
335,
338
]
] | https://openalex.org/W3113330169 |
31b646a8-7e00-4bf0-a449-cf3df11eca6e | Below \(T_{\rm DW}\) , exotic nodal superconductivity
occurs at \(T_{\rm c}=1 \sim 3\) K at ambient pressure.
[1]}, [2]}.
The Hebel-Slichter peak in \(1/T_1T\) [3]}
and small impurity effect on \(T_{\rm c}\)
[4]}
indicate the singlet \(s\) -wave superconducting (SC) state.
On the other hand,
possibilities of triplet pairing state
[5]}
and nematic SC state
[6]}, [7]}
have been reported, which indicates non-\(s\) wave pairing states.
In addition, possibility of topological states
has been discussed intensively
[8]}.
Under pressure, \(T_{\rm DW}\) decreases and
vanishes at the DW quantum critical point (DW-QCP)
at \(P\sim 2\) GPa.
For A=Cs, \(T_{\rm c}\) exhibits nontrivial
double SC dome in the DW phase,
and the highest \(T_{\rm c}\ (\lesssim 10 {\rm K})\)
is realized at the DW-QCP
[9]}.
In addition, theoretical phonon-mediated \(s\) -wave \(T_{\rm c}\)
is too low to explain experiments
[10]}.
Thus, unconventional SC state due to DW fluctuations
[11]}, [12]}
is naturally expected in AV\(_3\) Sb\(_5\) .
| [10] | [
[
906,
910
]
] | https://openalex.org/W3133870740 |
1ff83307-1a7f-411c-a13b-fb8266c1c32e | The current central issues would be summarized as:
(i) Origin of the DW state and its driving mechanism,
(ii) SC state and its mechanism, and
(iii) Interplay between DW and superconductivity.
To attack these issues,
it is useful to focus on the similarity to Fe-based superconductors,
in which \(s\) -wave state appears next to
the nematic and smectic orbital/bond orders.
These orders are naturally explained
in terms of the quantum interference mechanism:
The interference among optical phonons
[1]},
and/or spin fluctuations
[2]}, [3]}, [4]}, [5]}
(at wavevectors \({ {\mathbf {q}} }\) and \({ {\mathbf {q}} }^{\prime }\) )
give rise to unconventional DW at \({ {\mathbf {q}} }+{ {\mathbf {q}} }^{\prime }\) ,
which is shown in Fig. REF (a).
This mechanism is also applicable to
various nematic/smectic orders in transition metal compounds
[6]}, [7]}, [8]}, [9]}
and \(f\) -electron systems
[10]}.
It is meaningful to investigate the interplay
between interference mechanism,
van Hove singularity (vHS) points
and sublattice degrees of freedom in kagome metal.
Its lattice and band structure and Fermi surface (FS)
are shown in Figs. REF (b), (c) and (d), respectively.
| [1] | [
[
497,
500
]
] | https://openalex.org/W2006613135 |
1725f40f-0a62-4656-ab3c-8e8d0554c625 | The current central issues would be summarized as:
(i) Origin of the DW state and its driving mechanism,
(ii) SC state and its mechanism, and
(iii) Interplay between DW and superconductivity.
To attack these issues,
it is useful to focus on the similarity to Fe-based superconductors,
in which \(s\) -wave state appears next to
the nematic and smectic orbital/bond orders.
These orders are naturally explained
in terms of the quantum interference mechanism:
The interference among optical phonons
[1]},
and/or spin fluctuations
[2]}, [3]}, [4]}, [5]}
(at wavevectors \({ {\mathbf {q}} }\) and \({ {\mathbf {q}} }^{\prime }\) )
give rise to unconventional DW at \({ {\mathbf {q}} }+{ {\mathbf {q}} }^{\prime }\) ,
which is shown in Fig. REF (a).
This mechanism is also applicable to
various nematic/smectic orders in transition metal compounds
[6]}, [7]}, [8]}, [9]}
and \(f\) -electron systems
[10]}.
It is meaningful to investigate the interplay
between interference mechanism,
van Hove singularity (vHS) points
and sublattice degrees of freedom in kagome metal.
Its lattice and band structure and Fermi surface (FS)
are shown in Figs. REF (b), (c) and (d), respectively.
| [2] | [
[
528,
531
]
] | https://openalex.org/W2092660500 |
2024f0e2-3b2c-48e7-bbbd-42a20bf65091 | The current central issues would be summarized as:
(i) Origin of the DW state and its driving mechanism,
(ii) SC state and its mechanism, and
(iii) Interplay between DW and superconductivity.
To attack these issues,
it is useful to focus on the similarity to Fe-based superconductors,
in which \(s\) -wave state appears next to
the nematic and smectic orbital/bond orders.
These orders are naturally explained
in terms of the quantum interference mechanism:
The interference among optical phonons
[1]},
and/or spin fluctuations
[2]}, [3]}, [4]}, [5]}
(at wavevectors \({ {\mathbf {q}} }\) and \({ {\mathbf {q}} }^{\prime }\) )
give rise to unconventional DW at \({ {\mathbf {q}} }+{ {\mathbf {q}} }^{\prime }\) ,
which is shown in Fig. REF (a).
This mechanism is also applicable to
various nematic/smectic orders in transition metal compounds
[6]}, [7]}, [8]}, [9]}
and \(f\) -electron systems
[10]}.
It is meaningful to investigate the interplay
between interference mechanism,
van Hove singularity (vHS) points
and sublattice degrees of freedom in kagome metal.
Its lattice and band structure and Fermi surface (FS)
are shown in Figs. REF (b), (c) and (d), respectively.
| [4] | [
[
540,
543
]
] | https://openalex.org/W3104988532 |
77fa6370-86fb-45f2-beaa-f1b432fbaa6e | The current central issues would be summarized as:
(i) Origin of the DW state and its driving mechanism,
(ii) SC state and its mechanism, and
(iii) Interplay between DW and superconductivity.
To attack these issues,
it is useful to focus on the similarity to Fe-based superconductors,
in which \(s\) -wave state appears next to
the nematic and smectic orbital/bond orders.
These orders are naturally explained
in terms of the quantum interference mechanism:
The interference among optical phonons
[1]},
and/or spin fluctuations
[2]}, [3]}, [4]}, [5]}
(at wavevectors \({ {\mathbf {q}} }\) and \({ {\mathbf {q}} }^{\prime }\) )
give rise to unconventional DW at \({ {\mathbf {q}} }+{ {\mathbf {q}} }^{\prime }\) ,
which is shown in Fig. REF (a).
This mechanism is also applicable to
various nematic/smectic orders in transition metal compounds
[6]}, [7]}, [8]}, [9]}
and \(f\) -electron systems
[10]}.
It is meaningful to investigate the interplay
between interference mechanism,
van Hove singularity (vHS) points
and sublattice degrees of freedom in kagome metal.
Its lattice and band structure and Fermi surface (FS)
are shown in Figs. REF (b), (c) and (d), respectively.
| [6] | [
[
845,
848
]
] | https://openalex.org/W2616446735 |
9ed611e9-4520-4267-a2eb-2400f8599d25 | In this paper,
we study the unconventional DW order in AV\(_3\) Sb\(_5\)
due to beyond-mean-field effects,
and its interplay with exotic SC states.
By optimizing the form factor that represents the
nonlocal (=\({ {\mathbf {k}} }\) -dependent) particle-hole (p-h) condensation,
we derive the smectic bond-order at wavevector \({ {\mathbf {q}} }_n\) (\(n=1,2,3\) )
even when spin fluctuations are tiny.
Its driving force is the paramagnon interference, which
provides large “nonlocal” backward and umklapp scattering
among different vHS points.
In addition, the smectic DW fluctuations
induce sizable “beyond-Migdal” pairing interaction.
For this reason, both nodal \(s\) -wave pairing
and \(p\) -wave pairing states are expected to emerge.
The coexistence of both states
would explain exotic SC states in \(T\) -\(P\) phase diagram
[1]}, [2]}.
<FIGURE> | [2] | [
[
840,
843
]
] | https://openalex.org/W3132260257 |
afb23d41-361c-4eec-b919-4b85b7825b82 | where \(l,m=A,B,C,A^{\prime },B^{\prime },C^{\prime }\) .
Hereafter the unit of energy is eV.
The nearest-neighbor hopping integrals are
\(t_{b{3g}}=0.5\) , \(t_{b{2g}}=1\) and \(t_{b{3g},b{2g}}=0.002\) ,
and the on-site energies are \(E_{b{3g}}=-0.055\) and \(E_{b{2g}}=2.17\)
[1]}.
In the numerical study, it is convenient to
analyze the six-orbital triangular lattice model
in Fig. REF in the Supplemental Materials (SM) A [2]},
which is completely equivalent to
the kagome metal in Fig. REF (b).
In the \(b_{3g}\) -orbital band shown in Fig. REF (d),
each vHS point (A, B and C) is
composed of pure orbital (\(A\) , \(B\) and \(C\) ),
while the point \({ {\mathbf {k}} }_{\rm AB}=({ {\mathbf {k}} }_{\rm A}+{ {\mathbf {k}} }_{\rm B})/2\)
is composed of orbitals \(A\) and \(B\) .
The present \(b_{3g}\) -orbital FS in the vicinity of three vHS points,
on which the pseudogap opens below \(T_{\rm DW}\)
[3]}, [4]}, [5]}, [6]},
well captures the observed FS
[7]}, [8]}, [9]}.
| [5] | [
[
928,
931
]
] | https://openalex.org/W3160466619 |
a0537f9a-da25-46d3-baa3-ba91b6f14f39 | Nonmagnetic DW orders cannot be explained in the RPA
unless large nearest-neighbor Coulomb interaction \(V\) (\(V>0.5U\) ) exists.
However, beyond-RPA nonlocal correlations,
called the vertex corrections (VCs),
can induce various DW orders even for \(V=0\)
[1]}, [2]}, [3]}, [4]}, [5]}.
To consider the VCs due to the paramagnon interference
in Fig. REF (a),
which causes the nematicity in Fe-based and cuprate superconductors,
we employ the linearized DW equation
[3]}, [7]}:
\(\lambda _{{ {\mathbf {q}} }}f_{ {\mathbf {q}} }^{L}(k)&=& -\frac{T}{N}\sum _{p,M_1,M_2}I_{ {\mathbf {q}} }^{L,M_1}(k,p)\nonumber \\& &\times \lbrace G(p)G(p+{ {\mathbf {q}} }) \rbrace ^{M_1,M_2} f_{ {\mathbf {q}} }^{M_2}(p) ,\)
| [1] | [
[
259,
262
]
] | https://openalex.org/W2006613135 |
f5f0a010-3681-48af-9256-59514c44318d | Nonmagnetic DW orders cannot be explained in the RPA
unless large nearest-neighbor Coulomb interaction \(V\) (\(V>0.5U\) ) exists.
However, beyond-RPA nonlocal correlations,
called the vertex corrections (VCs),
can induce various DW orders even for \(V=0\)
[1]}, [2]}, [3]}, [4]}, [5]}.
To consider the VCs due to the paramagnon interference
in Fig. REF (a),
which causes the nematicity in Fe-based and cuprate superconductors,
we employ the linearized DW equation
[3]}, [7]}:
\(\lambda _{{ {\mathbf {q}} }}f_{ {\mathbf {q}} }^{L}(k)&=& -\frac{T}{N}\sum _{p,M_1,M_2}I_{ {\mathbf {q}} }^{L,M_1}(k,p)\nonumber \\& &\times \lbrace G(p)G(p+{ {\mathbf {q}} }) \rbrace ^{M_1,M_2} f_{ {\mathbf {q}} }^{M_2}(p) ,\)
| [2] | [
[
265,
268
]
] | https://openalex.org/W2092660500 |
0de499ad-ffed-40ee-92f1-0476b634e1f3 | Nonmagnetic DW orders cannot be explained in the RPA
unless large nearest-neighbor Coulomb interaction \(V\) (\(V>0.5U\) ) exists.
However, beyond-RPA nonlocal correlations,
called the vertex corrections (VCs),
can induce various DW orders even for \(V=0\)
[1]}, [2]}, [3]}, [4]}, [5]}.
To consider the VCs due to the paramagnon interference
in Fig. REF (a),
which causes the nematicity in Fe-based and cuprate superconductors,
we employ the linearized DW equation
[3]}, [7]}:
\(\lambda _{{ {\mathbf {q}} }}f_{ {\mathbf {q}} }^{L}(k)&=& -\frac{T}{N}\sum _{p,M_1,M_2}I_{ {\mathbf {q}} }^{L,M_1}(k,p)\nonumber \\& &\times \lbrace G(p)G(p+{ {\mathbf {q}} }) \rbrace ^{M_1,M_2} f_{ {\mathbf {q}} }^{M_2}(p) ,\)
| [4] | [
[
277,
280
]
] | https://openalex.org/W3104988532 |
04d2a0bd-5f12-48ca-9c58-32dec1d31cec | Here, we apply the one-loop approximation for \(\Phi _{\rm LW}\)
[1]}, [2]}.
Then, \(I_{ {\mathbf {q}} }^{L,M}\) is
composed of one single-magnon exchange Maki-Thompson (MT) term
and two double-magnon interference AL terms.
Their diagrammatic expression (Fig. REF )
and analytic one are explained in the SM B [3]}.
Due to the AL terms,
nonmagnetic nematic order in FeSe is naturally reproduced
even if spin fluctuations are very weak
[1]}.
The importance of AL terms was
verified by the functional-renormalization-group (fRG) study
with constrained-RPA,
in which higher-order parquet VCs are produced
in an unbiased way,
for several Hubbard models
[5]}, [6]}, [7]}.
Later, we see that the AL diagrams induce the
backward and umklapp scattering shown in Fig. REF (f),
and they mediate the p-h condensation
at the inter-vHS nesting vector \({ {\mathbf {q}} }_1={ {\mathbf {k}} }_{\rm B}-{ {\mathbf {k}} }_{\rm A}\) .
<FIGURE> | [1] | [
[
66,
69
],
[
436,
439
]
] | https://openalex.org/W2092660500 |
fa7cdf96-6dd7-4ca8-85f2-be25290c828d | Here, we apply the one-loop approximation for \(\Phi _{\rm LW}\)
[1]}, [2]}.
Then, \(I_{ {\mathbf {q}} }^{L,M}\) is
composed of one single-magnon exchange Maki-Thompson (MT) term
and two double-magnon interference AL terms.
Their diagrammatic expression (Fig. REF )
and analytic one are explained in the SM B [3]}.
Due to the AL terms,
nonmagnetic nematic order in FeSe is naturally reproduced
even if spin fluctuations are very weak
[1]}.
The importance of AL terms was
verified by the functional-renormalization-group (fRG) study
with constrained-RPA,
in which higher-order parquet VCs are produced
in an unbiased way,
for several Hubbard models
[5]}, [6]}, [7]}.
Later, we see that the AL diagrams induce the
backward and umklapp scattering shown in Fig. REF (f),
and they mediate the p-h condensation
at the inter-vHS nesting vector \({ {\mathbf {q}} }_1={ {\mathbf {k}} }_{\rm B}-{ {\mathbf {k}} }_{\rm A}\) .
<FIGURE> | [5] | [
[
650,
653
]
] | https://openalex.org/W3159222123 |
40288cd8-5894-497f-b70f-5fc64a72dfb4 | Here, we apply the one-loop approximation for \(\Phi _{\rm LW}\)
[1]}, [2]}.
Then, \(I_{ {\mathbf {q}} }^{L,M}\) is
composed of one single-magnon exchange Maki-Thompson (MT) term
and two double-magnon interference AL terms.
Their diagrammatic expression (Fig. REF )
and analytic one are explained in the SM B [3]}.
Due to the AL terms,
nonmagnetic nematic order in FeSe is naturally reproduced
even if spin fluctuations are very weak
[1]}.
The importance of AL terms was
verified by the functional-renormalization-group (fRG) study
with constrained-RPA,
in which higher-order parquet VCs are produced
in an unbiased way,
for several Hubbard models
[5]}, [6]}, [7]}.
Later, we see that the AL diagrams induce the
backward and umklapp scattering shown in Fig. REF (f),
and they mediate the p-h condensation
at the inter-vHS nesting vector \({ {\mathbf {q}} }_1={ {\mathbf {k}} }_{\rm B}-{ {\mathbf {k}} }_{\rm A}\) .
<FIGURE> | [6] | [
[
656,
659
]
] | https://openalex.org/W2080509755 |
20203f0f-87ae-40ce-bdd0-c2c36c83714f | Here, we apply the one-loop approximation for \(\Phi _{\rm LW}\)
[1]}, [2]}.
Then, \(I_{ {\mathbf {q}} }^{L,M}\) is
composed of one single-magnon exchange Maki-Thompson (MT) term
and two double-magnon interference AL terms.
Their diagrammatic expression (Fig. REF )
and analytic one are explained in the SM B [3]}.
Due to the AL terms,
nonmagnetic nematic order in FeSe is naturally reproduced
even if spin fluctuations are very weak
[1]}.
The importance of AL terms was
verified by the functional-renormalization-group (fRG) study
with constrained-RPA,
in which higher-order parquet VCs are produced
in an unbiased way,
for several Hubbard models
[5]}, [6]}, [7]}.
Later, we see that the AL diagrams induce the
backward and umklapp scattering shown in Fig. REF (f),
and they mediate the p-h condensation
at the inter-vHS nesting vector \({ {\mathbf {q}} }_1={ {\mathbf {k}} }_{\rm B}-{ {\mathbf {k}} }_{\rm A}\) .
<FIGURE> | [7] | [
[
662,
665
]
] | https://openalex.org/W2616446735 |
64e2923f-ce5c-4815-9f3c-82b87b2db0ca | In summary, we derived the smectic bond-order in AV\(_3\) Sb\(_5\)
due to the paramagnon interference mechanism,
irrespective of tiny magnetic criticality in kagome metal
because of the prominent vHS and geometrical frustration.
In addition, we predicted that the emergence of
nodal \(s\) -wave and \(p\) -wave SC states
owing to the cooperation of bond-order and SDW fluctuations.
This mechanism may explain high-\(T_{\rm c}\) state
in CsV\(_3\) Sb\(_5\) under pressure.
The present study would give crucial hints to understand
recently discovered “smectic order and adjacent high-\(T_{\rm c}\) state”
in FeSe/SrTiO\(_3\) [1]}.
It is noteworthy that the paramagnon interference
mechanism also responsible for loop current orders,
which are the condensations of “odd-parity” p-h pairs
[2]}, [3]}.
It is important to study the current order mechanism in kagome metals
in future because the emergence of current orders
has been hotly discussed experimentally
[4]}, [5]}, [6]}.
| [2] | [
[
790,
793
]
] | https://openalex.org/W3132123652 |
f54e3195-914a-4b50-b889-8647f911a938 | In summary, we derived the smectic bond-order in AV\(_3\) Sb\(_5\)
due to the paramagnon interference mechanism,
irrespective of tiny magnetic criticality in kagome metal
because of the prominent vHS and geometrical frustration.
In addition, we predicted that the emergence of
nodal \(s\) -wave and \(p\) -wave SC states
owing to the cooperation of bond-order and SDW fluctuations.
This mechanism may explain high-\(T_{\rm c}\) state
in CsV\(_3\) Sb\(_5\) under pressure.
The present study would give crucial hints to understand
recently discovered “smectic order and adjacent high-\(T_{\rm c}\) state”
in FeSe/SrTiO\(_3\) [1]}.
It is noteworthy that the paramagnon interference
mechanism also responsible for loop current orders,
which are the condensations of “odd-parity” p-h pairs
[2]}, [3]}.
It is important to study the current order mechanism in kagome metals
in future because the emergence of current orders
has been hotly discussed experimentally
[4]}, [5]}, [6]}.
| [6] | [
[
974,
977
]
] | https://openalex.org/W3131186123 |
94893ddc-a687-443e-a348-734288f237bb | Next, we explain the multiorbital Coulomb interaction.
The matrix expression of the
spin-channel Coulomb interaction is
[1]}, [2]}, [3]}
\(U_{l_{1}l_{2},l_{3}l_{4}}^s = {\left\lbrace \begin{array}{ll}U, & l_1=l_2=l_3=l_4 \\U^{\prime } , & l_1=l_3 \ne l_2=l_4 \\J, & l_1=l_2 \ne l_3=l_4 \\J^{\prime } , & l_1=l_4 \ne l_2=l_3\end{array}\right.}\)
| [3] | [
[
132,
135
]
] | https://openalex.org/W2092660500 |
471814f8-478f-43cb-be74-c50a4b33a84e | in the case that
\(l_1 \sim l_4\) are orbitals (\(X\) , \(X^{\prime }\) ) at site X (=A,B,C).
In other cases, \(U_{l_{1}l_{2},l_{3}l_{4}}^s =0\) .
Also, the matrix expression of the
charge-channel Coulomb interaction is
[1]}, [2]}, [3]}
\(U_{l_{1}l_{2},l_{3}l_{4}}^c = {\left\lbrace \begin{array}{ll}-U, & l_1=l_2=l_3=l_4 \\U^{\prime }-2J , & l_1=l_3 \ne l_2=l_4 \\-2U^{\prime } + J, & l_1=l_2 \ne l_3=l_4 \\-J^{\prime } , & l_1=l_4 \ne l_2=l_3\end{array}\right.}\)
| [3] | [
[
233,
236
]
] | https://openalex.org/W2092660500 |
17f88bac-c669-4f79-935e-871717699704 | The spin (charge) susceptibility in the RPA,
\(\chi ^{s(c)}_{ll^{\prime },mm^{\prime }}(q)\) , is given as
[1]}, [2]}, [3]}
\(\hat{\chi }^{s(c)}(q)= \hat{\chi }^0(q)(\hat{1}-\hat{U}^{s(c)}\hat{\chi }^0(q))^{-1} ,\)
| [3] | [
[
119,
122
]
] | https://openalex.org/W2092660500 |
7242f394-0e55-48c1-ae02-c36241a566f2 | Here, we derive the kernel function in the
DW equation, \(I^{l l^{\prime }, m m^{\prime }}_{{ {\mathbf {q}} }}(k,k^{\prime })\) ,
studied in the main text.
It is given as
\(\delta ^2 \Phi _{\rm LW}/\delta G_{l^{\prime }l}(k)\delta G_{mm^{\prime }}(p)\)
at \({ {\mathbf {q}} }={\mathbf {0}}\) in the conserving approximation scheme
[1]}, [2]}, [3]},
where \(\Phi _{\rm LW}\) is the Luttinger-Ward function.
Here, we apply the one-loop approximation for \(\Phi _{\rm LW}\)
[4]}, [3]}.
Then, \(I_{ {\mathbf {q}} }^{L,M}\) in this kagome model is given as
\(&& I^{l l^{\prime }, m m^{\prime }}_{{ {\mathbf {q}} }} (k,k^{\prime })= \sum _{b = s, c} \frac{a^b}{2}\Bigl [ -V^{b}_{l m, l^{\prime } m^{\prime }} (k-k^{\prime })\nonumber \\&&+\frac{T}{N} \sum _{p} \sum _{l_1 l_2, m_1 m_2}V^{b}_{l l_1, m m_1} \left( p+{ {\mathbf {q}} }\right)V^{b}_{m^{\prime } m_2, l^{\prime } l_2} \left( p \right)\nonumber \\&& \qquad \qquad \qquad \quad \times G_{l_1 l_2} (k-p) G_{m_2 m_1} (k^{\prime }-p)\nonumber \\&&+\frac{T}{N} \sum _{p} \sum _{l_1 l_2, m_1 m_2}V^{b}_{l l_1, m_2 m^{\prime }} \left( p+{ {\mathbf {q}} }\right)V^{b}_{m_1 m, l^{\prime } l_2} \left( p \right)\nonumber \\&& \qquad \qquad \qquad \times G_{l_1 l_2} (k-p) G_{m_2 m_1} (k^{\prime }+p+{ {\mathbf {q}} }) \Bigr ] ,\)
| [4] | [
[
475,
478
]
] | https://openalex.org/W2092660500 |
d68e315c-6e7c-4e89-9f8b-be6f7d301985 | which is depicted in Fig. REF (b).
Here \(\lambda _{ {\mathbf {q}} }\) is the eigenvalue
that reaches unity at the transition temperature.
\(\hat{f}_{ {\mathbf {q}} }\) is the form factor of the DW order,
which corresponds to the “symmetry-breaking in the self-energy”.
By solving Eq. (REF ), we can obtain the optimized
momentum and orbital dependences of \(\hat{f}\) .
This mechanism has been successfully applied to
explain the electronic nematic orders in Fe-based
[1]}, [2]}, [3]}
and cuprate superconductors [4]},
and multipole orders in \(f\) -electron systems
[5]}.
| [1] | [
[
472,
475
]
] | https://openalex.org/W2092660500 |
9e07e8f0-56d5-4bf8-b4cb-682e26c84aef | which is depicted in Fig. REF (b).
Here \(\lambda _{ {\mathbf {q}} }\) is the eigenvalue
that reaches unity at the transition temperature.
\(\hat{f}_{ {\mathbf {q}} }\) is the form factor of the DW order,
which corresponds to the “symmetry-breaking in the self-energy”.
By solving Eq. (REF ), we can obtain the optimized
momentum and orbital dependences of \(\hat{f}\) .
This mechanism has been successfully applied to
explain the electronic nematic orders in Fe-based
[1]}, [2]}, [3]}
and cuprate superconductors [4]},
and multipole orders in \(f\) -electron systems
[5]}.
| [2] | [
[
478,
481
]
] | https://openalex.org/W3104988532 |
f91a90fd-de51-4452-9658-6ee259ef9f17 | which is depicted in Fig. REF (b).
Here \(\lambda _{ {\mathbf {q}} }\) is the eigenvalue
that reaches unity at the transition temperature.
\(\hat{f}_{ {\mathbf {q}} }\) is the form factor of the DW order,
which corresponds to the “symmetry-breaking in the self-energy”.
By solving Eq. (REF ), we can obtain the optimized
momentum and orbital dependences of \(\hat{f}\) .
This mechanism has been successfully applied to
explain the electronic nematic orders in Fe-based
[1]}, [2]}, [3]}
and cuprate superconductors [4]},
and multipole orders in \(f\) -electron systems
[5]}.
| [4] | [
[
517,
520
]
] | https://openalex.org/W3159222123 |
83b0d243-33fd-438b-8dfa-af5ec9394f04 | It is noteworthy that
both the DW equation and the functional-renormalization group (fRG) method
explain the nematic and smectic bond-order in single-orbital
square lattice Hubbard models [1]}, [2]}
and anisotropic triangular lattice ones
[3]}.
This fact means that higher-order diagrams other than MT or AL terms,
that are included in the fRG method, are not essential
in explaining the bond-order.
Note that the contributions away from the conduction bands
are included into \(N\) -patch fRG
by applying the RG+cRPA method
[1]}, [5]}, [3]}.
| [1] | [
[
188,
191
],
[
525,
528
]
] | https://openalex.org/W2616446735 |
fde0b944-a9e5-4b23-a337-189503145c90 | It is noteworthy that
both the DW equation and the functional-renormalization group (fRG) method
explain the nematic and smectic bond-order in single-orbital
square lattice Hubbard models [1]}, [2]}
and anisotropic triangular lattice ones
[3]}.
This fact means that higher-order diagrams other than MT or AL terms,
that are included in the fRG method, are not essential
in explaining the bond-order.
Note that the contributions away from the conduction bands
are included into \(N\) -patch fRG
by applying the RG+cRPA method
[1]}, [5]}, [3]}.
| [5] | [
[
531,
534
]
] | https://openalex.org/W3159222123 |
22892fa7-3a0f-469a-acc8-8416b2b23fc7 | Here, we discuss the reason
why bond-order fluctuations mediate the pairing interaction.
In Ref. [1]}, the authors studied
the orbital fluctuation mediated \(s\) -wave superconductivity
in Fe-based superconductors.
In that study, the electron-boson coupling (=form factor)
is an orbital-dependent but \({ {\mathbf {k}} }\) -independent charge quadrupole operator:
\(\hat{f}^{{ {\mathbf {q}} }}({ {\mathbf {k}} })=\hat{O}_\Gamma \) \((\Gamma =xz,yz,xy)\) .
In the main text,
we obtain the development of bond-order fluctuations with
the \({ {\mathbf {k}} }\) -dependent form factor in AV\(_3\) Sb\(_5\) , which is given by the nonlocal
vertex corrections (VCs) that are dropped in the RPA.
We reveal that bond-order fluctuations
mediate significant “beyond-Migdal” pairing interaction
thanks to the \({ {\mathbf {k}} }\) -dependent form factor [2]},
and therefore \(s\) -wave and \(p\) -wave SC states emerge in AV\(_3\) Sb\(_5\) .
<FIGURE> | [1] | [
[
97,
100
]
] | https://openalex.org/W2006613135 |
04841a27-6b02-4a85-a72e-5c728a40ba45 | In feature attribution, we seek to allocate an individual model
prediction \(\hat{y}\) to the features: \(\hat{y} = \mathrm {constant} +\phi _1 + \cdots + \phi _p\) . Recently, an attribution method based on
Shapley values has gained popularity because of its nice theoretical
properties and the availability of an efficient software
implementation for tree-based models ([1]}).
The contributions can be positive or negative, and for a
feature \(x_i\) , its total absolute Shapley contribution \(\sum |\phi _i|\)
over the training set may be used as another definition of the
importance of \(x_i\) .
| [1] | [
[
373,
376
]
] | https://openalex.org/W2787070805 |
80fb3e25-14ef-4e6e-b313-5e16c5656d60 | Another body of research targets ICU readmissions. ICU readmissions
may be a more difficult target for machine learning than the first ICU
transfer since the former involves patients who have been discharged
from the ICU by a human expert (the physician) presumably after
extensive tests and monitoring. Desautels et al. [1]} learned a model
to predict death and 48-hour ICU readmission when a patient is first
discharged from the ICU. Their AdaBoost model containing 1,000
decision trees was trained on the MIMIC III dataset
[2]} and achieved an accuracy of 70%, a sensitivity
of 59%, and a specificity of 66%.
| [2] | [
[
526,
529
]
] | https://openalex.org/W2396881363 |
b9e407f7-9e9d-4392-aca6-68a611110205 | A number of other works studying transcription and splicing dynamics
(e.g. [1]}, [2]}, [3]}) forgo detailed
dynamical modelling, which limits their ability to properly account
for varying mRNA half-lives.
Our statistical model incorporates a linear ordinary differential
equation of transcription dynamics, including mRNA degradation.
Similar linear differential equation models have been proposed as models of
mRNA dynamics previously [4]}, [5]}, [6]},
but assuming a specific parametric form for the transcriptional activity.
In contrast, we apply a non-parametric Gaussian process framework
that can accommodate a quite general shape of transcriptional activity. As
demonstrated previously [7]}, [8]}, [9]}, the
linearity of the differential equation allows efficient exact Bayesian
inference of the transcriptional activity function. Before presenting our
results we outline our modelling approach.
| [7] | [
[
693,
696
]
] | https://openalex.org/W2124584833 |
afcd6e83-2c0a-4890-b4e5-b48235017672 | A number of other works studying transcription and splicing dynamics
(e.g. [1]}, [2]}, [3]}) forgo detailed
dynamical modelling, which limits their ability to properly account
for varying mRNA half-lives.
Our statistical model incorporates a linear ordinary differential
equation of transcription dynamics, including mRNA degradation.
Similar linear differential equation models have been proposed as models of
mRNA dynamics previously [4]}, [5]}, [6]},
but assuming a specific parametric form for the transcriptional activity.
In contrast, we apply a non-parametric Gaussian process framework
that can accommodate a quite general shape of transcriptional activity. As
demonstrated previously [7]}, [8]}, [9]}, the
linearity of the differential equation allows efficient exact Bayesian
inference of the transcriptional activity function. Before presenting our
results we outline our modelling approach.
| [8] | [
[
699,
702
]
] | https://openalex.org/W2125373697 |
6d3d5187-b49c-4568-8d73-3bd1a5aa628e | We measure the transcriptional activity \(p(t)\) using RNA polymerase (pol-II)
ChIP-Seq time course data collected close to the 3' end of the gene
(reads lying in the last 20% of the transcribed region). Our main assumption is that
pol-II abundance at the 3' end of the gene is proportional to
the production rate of mature mRNA after a possible delay \(\Delta \) due to
disengaging from the polymerase and processing. The mRNA abundance is measured using RNA-Seq reads mapping
to annotated transcripts, taking all annotated transcripts into
account and resolving mapping ambiguities using a probabilistic
method [1]} (see Methods Section for details). As we limit
our analysis to pol-II data collected from the 3'-end of the
transcribed region, we do not expect a significant contribution to
\(\Delta \) from transcriptional delays when fitting the model. Such transcriptional delays
have recently been studied by modelling transcript elongation dynamics
using pol-II ChIP-Seq time course
data [2]} and nascent mRNA (GRO-Seq) data [3]} in the
same system. Here we instead focus on production delays that can occur
after elongation is essentially complete.
| [1] | [
[
615,
618
]
] | https://openalex.org/W2097860373 |
ab84e4f9-60cb-43e4-a55f-a223d5fda99d | We have previously shown how to perform inference over
differential equations driven by functions modelled using Gaussian
processes [1]}, [2]}, [3]}. The main methodological novelty in
the current work is the inclusion of the delay term in
equation (REF ) and the development of a Bayesian inference
scheme for this and other model parameters. In brief, we cast the
problem as Bayesian inference with a Gaussian process prior
distribution over \(p(t)\) that can be integrated out to obtain the data
likelihood under the model in Eqn. (REF ) assuming
Gaussian observation noise. This likelihood function and its gradient are used for inference
with a Hamiltonian MCMC algorithm [4]} to obtain a
posterior distribution over all model parameters and the full pol-II
and mRNA functions \(p(t)\) and \(m(t)\) .
| [1] | [
[
132,
135
]
] | https://openalex.org/W2124584833 |
efeda6bb-fcec-43d3-a6e2-77a56fa4d67c | We have previously shown how to perform inference over
differential equations driven by functions modelled using Gaussian
processes [1]}, [2]}, [3]}. The main methodological novelty in
the current work is the inclusion of the delay term in
equation (REF ) and the development of a Bayesian inference
scheme for this and other model parameters. In brief, we cast the
problem as Bayesian inference with a Gaussian process prior
distribution over \(p(t)\) that can be integrated out to obtain the data
likelihood under the model in Eqn. (REF ) assuming
Gaussian observation noise. This likelihood function and its gradient are used for inference
with a Hamiltonian MCMC algorithm [4]} to obtain a
posterior distribution over all model parameters and the full pol-II
and mRNA functions \(p(t)\) and \(m(t)\) .
| [2] | [
[
138,
141
]
] | https://openalex.org/W2125373697 |
ece7a335-8b46-47a4-842e-22128fe2d66e | We have previously shown how to perform inference over
differential equations driven by functions modelled using Gaussian
processes [1]}, [2]}, [3]}. The main methodological novelty in
the current work is the inclusion of the delay term in
equation (REF ) and the development of a Bayesian inference
scheme for this and other model parameters. In brief, we cast the
problem as Bayesian inference with a Gaussian process prior
distribution over \(p(t)\) that can be integrated out to obtain the data
likelihood under the model in Eqn. (REF ) assuming
Gaussian observation noise. This likelihood function and its gradient are used for inference
with a Hamiltonian MCMC algorithm [4]} to obtain a
posterior distribution over all model parameters and the full pol-II
and mRNA functions \(p(t)\) and \(m(t)\) .
| [4] | [
[
678,
681
]
] | https://openalex.org/W2059448777 |
72488c18-3a53-4d80-8ba4-5cb5a0b98e3c | mRNA concentration was estimated from RNA-seq read data using BitSeq [1]}. BitSeq is a probabilistic method to infer transcript expression from
RNA-seq data after mapping to an annotated transcriptome. We
estimated expression levels to all entries in the transcriptome,
including the pre-mRNA transcripts, and
used the sum of the mRNA transcript expressions in FPKM
units to estimate the mRNA expression level of a gene.
Different time points of the RNA-seq time series were normalised using
the method of [2]}.
| [1] | [
[
69,
72
]
] | https://openalex.org/W2097860373 |
9f69fd18-8ac1-45d8-86e8-4386aeddef85 | mRNA concentration was estimated from RNA-seq read data using BitSeq [1]}. BitSeq is a probabilistic method to infer transcript expression from
RNA-seq data after mapping to an annotated transcriptome. We
estimated expression levels to all entries in the transcriptome,
including the pre-mRNA transcripts, and
used the sum of the mRNA transcript expressions in FPKM
units to estimate the mRNA expression level of a gene.
Different time points of the RNA-seq time series were normalised using
the method of [2]}.
| [2] | [
[
506,
509
]
] | https://openalex.org/W2152239989 |
d6d9c240-f611-4e7a-9357-7a52e596bc4b | Given the differential equation parameters, GP inference yields a full
posterior distribution over the shape of the Pol-II and mRNA
functions \(p(t)\) and \(m(t)\) . We infer the differential equation
parameters from the data using MCMC sampling which
allows us to assign a level of
uncertainty to our parameter estimates. To infer a full
posterior over the differential equation parameters \(\beta _0\) ,
\(\beta \) , \(\alpha \) , \(\Delta \) , \(m_0\) , \(E[p_0]=\mu _p\) ,
the observation model parameters \(\sigma _{p}^2\) , \(\sigma _{m}^2\) ,
and a magnitude parameter \(C_p\) and width parameter \(l\) of the GP prior,
we set near-flat priors for them over reasonable value ranges, except for the
delay \(\Delta \) whose prior is biased toward 0 (exact ranges and full
details are presented in Supplementary Material).
We combine these priors with the likelihood obtained from the GP model
after marginalising out \(p(t)\) and \(m(t)\) , which can be performed
analytically. We infer the
posterior over the parameters by Hamiltonian MCMC sampling.
This full MCMC approach utilises gradients of the distributions for
efficient sampling and rigorously takes uncertainty over differential
equation parameters into account. Thus the final posterior accounts
for both the uncertainty about differential equation parameters, and
uncertainty over the underlying functions for each differential
equation.
We ran 4 parallel chains starting from different random initial states
for convergence checking using the potential scale reduction
factor of [1]}. We obtained 500 samples from each of the
4 chains after discarding the first half of the samples as burn-in and
thinning by a factor of 10.
Posterior distributions over the functions \(p(t)\) and \(m(t)\) are
obtained by sampling 500 realisations of \(p(t)\) and \(m(t)\) for each
parameter sample from the exact Gaussian conditional posterior given
the parameters in the sample. The resulting posteriors for \(p(t)\) and
\(m(t)\) are non-Gaussian, and are summarised by posterior mean and
posterior quantiles.
Full details of the MCMC procedure are in Supplementary Material.
| [1] | [
[
1552,
1555
]
] | https://openalex.org/W2148534890 |
5548062d-8dc6-4c22-ba3f-d13cb682cc9d | RNA-seq data were analysed at each time point separately using
BitSeq [1]}. The reads were first mapped to human
reference transcriptome (Ensembl v68) using Bowtie version 0.12.7 [2]}.
In order to separate pre-mRNA activity as well, we augmented the
reference transcriptome with pre-mRNA transcripts for each gene that
consisted of the genomic sequence from the beginning of the first
exon to the end of the last exon of the gene.
| [1] | [
[
70,
73
]
] | https://openalex.org/W2097860373 |
a0674dcf-34f3-41f8-b8c9-689ca937d900 | RNA-seq data were analysed at each time point separately using
BitSeq [1]}. The reads were first mapped to human
reference transcriptome (Ensembl v68) using Bowtie version 0.12.7 [2]}.
In order to separate pre-mRNA activity as well, we augmented the
reference transcriptome with pre-mRNA transcripts for each gene that
consisted of the genomic sequence from the beginning of the first
exon to the end of the last exon of the gene.
| [2] | [
[
179,
182
]
] | https://openalex.org/W2124985265 |
8349e1ec-0188-4b8c-9300-cb7c39d10cc2 | BitSeq uses a probabilistic model to probabilistically assign
multimapping reads to transcript isoforms [1]}, in our
case also including the pre-mRNA transcripts. We obtained gene
expression estimates by adding the corresponding mRNA transcript
expression levels. In addition to the mean expression levels, BitSeq
provides variances of the transcript isoform expression levels. We
further used the biological variance estimation procedure from BitSeq
differential expression analysis on the estimated gene expression
levels by treating the first three time points (0, 5, 10 min) as
biological replicates. Genes with similar mean expression levels
(log-RPKM) were grouped together such that each group contained 500
genes except for the last group with 571 genes with the highest
expression. Then, the biological variances were estimated for each
group of genes by using the Metropolis–Hastings algorithm used in
BitSeq stage 2 [1]}. Biological variances for the single
measurements were determined according to the gene expression levels
at each time point, where each gene was considered to belong to the
closest gene group according to its expression level. The observation
noise variance for each observation was defined as the sum of the
technical (BitSeq stage 1) and biological (BitSeq stage 2) variances,
and transformed from log-expression to raw expression using
\(\sigma ^2_{\text{raw}} = \sigma ^2_{\text{log}} \exp (\mu _{\text{log}})^2.\)
| [1] | [
[
104,
107
],
[
927,
930
]
] | https://openalex.org/W2097860373 |
2e4b92f1-282b-49c8-b491-0e750e36813b | Different time points of the RNA-seq time series were normalised using
the method of [1]} as implemented in the edgeR
R/Bioconductor package [2]}.
| [1] | [
[
85,
88
]
] | https://openalex.org/W2152239989 |
96319471-737d-4d05-9fc6-17757aa0a334 | Different time points of the RNA-seq time series were normalised using
the method of [1]} as implemented in the edgeR
R/Bioconductor package [2]}.
| [2] | [
[
141,
144
]
] | https://openalex.org/W2114104545 |