Manuscript Hawk04-A@1 · Hawk04-A.tex · section · 526092ba

Frozen source and reservoir

A regulated interacting microscopic application › Frozen source and reservoir · lines 994–1075

Hawk04-A@1#eq:HS, Hawk04-A@1#eq:H

Let $N = 2 n \geq 4$ , with Majoranas

χ_{i}^{†} = χ_{i}, {χ_{i}, χ_{j}} = 2 δ_{ij} I,

acting on $H_{S} = F_{f} (C^{n})$ . Freeze real quartic coefficients and set

H_{bare} B_{N} H_{S} = i < j < k < l \sum J_{ij k l} χ_{i} χ_{j} χ_{k} χ_{l}, = i < j < k < l \sum ∣ J_{ij k l} ∣, = H_{bare} + B_{N} I, M_{N} = 2 B_{N} .

Every quartic monomial is self-adjoint with norm one, so

0 \leq H_{S} \leq M_{N} I .

Write

S_{j} = i χ_{2 j - 1} χ_{2 j}, O_{s^{'}} = \frac{1}{n} j = 1 \sum^{n} s^{'}_{j} S_{j}, A = α O_{s^{'}} + ζ O_{s^{'}}^{2} .

The $S_{j}$ commute, and $∥ A ∥ \leq ∣ α ∣ + ∣ ζ ∣$ . For the estimates, one may use a fixed finite family of even self-adjoint $A_{a}$ , including the choice $A_{a} = A$ .

Fix

0 < μ < Λ < \infty, h = L^{2} ([μ, Λ], d ω; C^{r}), h_{B} f (ω) = ω f (ω) .

On $H_{bath} = F_{s} (h)$ , let

H_{B} = d Γ (h_{B}), f_{a} (ω) = \frac{e _{a}}{2 ω}, Φ (f) = a (f) + a^{†} (f),

with annihilation conjugate-linear in $f$ . The inner product is conjugate-linear in its first argument.

For frozen real $g_{a}$ , define

H = H_{S} + H_{B} + V_{int}, V_{int} = a \sum g_{a} A_{a} \otimes Φ (f_{a}) .

Set

G G_{ω} = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ h_{B}^{- 1/2} f_{a} .

Here

∥ f_{a} ∥^{2} = \frac{1}{2} lo g \frac{Λ}{μ}, h_{B}^{- 1/2} f_{a}^{2} = \frac{1}{2} (\frac{1}{μ} - \frac{1}{Λ}) .

This is a fixed-realization quartic SYK–scalar-reservoir specification. The normalization is printed rather than inferred from a different SYK convention. The bath is initially vacuum. No gravitational geometry is assigned to these objects.

Displayed formulas in this passage

Hawk04-A@1#eq:HS · line 1004
$H_{bare} B_{N} H_{S} = i < j < k < l \sum J_{ij k l} χ_{i} χ_{j} χ_{k} χ_{l}, = i < j < k < l \sum ∣ J_{ij k l} ∣, = H_{bare} + B_{N} I, M_{N} = 2 B_{N} .$
Hawk04-A@1#eq:H · line 1051
$H = H_{S} + H_{B} + V_{int}, V_{int} = a \sum g_{a} A_{a} \otimes Φ (f_{a}) .$
unlabeled · line 1058
$G G_{ω} = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ h_{B}^{- 1/2} f_{a} .$
unlabeled · line 997
$χ_{i}^{†} = χ_{i}, {χ_{i}, χ_{j}} = 2 δ_{ij} I,$
unlabeled · line 1014
$0 \leq H_{S} \leq M_{N} I .$
unlabeled · line 1019
$S_{j} = i χ_{2 j - 1} χ_{2 j}, O_{s^{'}} = \frac{1}{n} j = 1 \sum^{n} s^{'}_{j} S_{j}, A = α O_{s^{'}} + ζ O_{s^{'}}^{2} .$
unlabeled · line 1032
$0 < μ < Λ < \infty, h = L^{2} ([μ, Λ], d ω; C^{r}), h_{B} f (ω) = ω f (ω) .$
unlabeled · line 1040
$H_{B} = d Γ (h_{B}), f_{a} (ω) = \frac{e _{a}}{2 ω}, Φ (f) = a (f) + a^{†} (f),$
unlabeled · line 1063
$∥ f_{a} ∥^{2} = \frac{1}{2} lo g \frac{Λ}{μ}, h_{B}^{- 1/2} f_{a}^{2} = \frac{1}{2} (\frac{1}{μ} - \frac{1}{Λ}) .$