Quantum Sun (QSun)

The Quantum Sun model is a compact many-body model for avalanche-driven ergodicity restoration. In Polfed it is useful both physically and numerically: physically because it separates a small ergodic grain from a set of outer spins with decaying couplings, and numerically because it produces sparse disordered Hamiltonians that are natural inputs for polfed.

The main constructor is qsun_hamiltonian:

using Polfed.Models: qsun_hamiltonian

H = qsun_hamiltonian(L_loc, L_grain, g0, α; kwargs...)

For the full Julia constructor signature, argument types, keyword meanings, and return types, see Models, in particular qsun_hamiltonian.

Model Definition

QSun supports two closely related conventions:

the default full-Hilbert-space model, which does not enforce U(1) conservation,
the U(1)-symmetric model, which works in a fixed total-$S_z$ sector.
both constructions generalize directly to higher spins such as S = 1, S = 3/2, and so on.

Both are built through the same qsun_hamiltonian interface.

Conventional Quantum Sun Model

In the default construction the Hamiltonian has the form

\[\hat H = \hat H_{\mathrm{grain}} + g_0 \sum_{j=1}^{L_{\mathrm{loc}}} \alpha^{u_j}\, \hat S^x_{n(j)} \hat S^x_j + \sum_{j=1}^{L_{\mathrm{loc}}} h_j\, \hat S^z_j .\]

Here the first $L_{\mathrm{grain}}$ sites belong to the ergodic grain, while the remaining $L_{\mathrm{loc}}$ sites are the outer localized spins. In the implementation:

$h_j$ are drawn uniformly from $[h_z - w,\; h_z + w]$.
The neighbor map $n(j)$ picks which grain site couples to the outer spin $j$.
The coupling profile is

\[\alpha^{u_1} = 1, \qquad \alpha^{u_j}, \qquad u_j = (j-1) + \eta_j, \qquad \eta_j \in [-\zeta,\zeta] \quad (j \ge 2).\]

The parameter $\zeta$ therefore introduces randomness into the effective distance exponent, while $\alpha$ controls how fast the couplings decay away from the grain.

The grain term is constructed as a GOE random matrix:

\[\hat H_{\mathrm{grain}} = \frac{\gamma}{\sqrt{d_{\mathrm{grain}} + 1}}\, G,\]

where $G$ is a real symmetric GOE-like matrix and $d_{\mathrm{grain}} = (2S+1)^{L_{\mathrm{grain}}}$ is the grain Hilbert-space dimension.

The role of $\gamma$ is important:

$\gamma$ sets the overall energy scale of the grain block.
Larger $\gamma$ makes the internal grain dynamics stronger relative to the outer-spin disorder and the grain-to-ray couplings.
Smaller $\gamma$ weakens the grain and makes the competition with disorder more pronounced.

In other words, $\gamma$ does not change the structure of the model, but it does change how strongly the ergodic grain acts as a bath for the outer spins.

Parameters in the Conventional Model

L_loc: number of outer localized spins.
L_grain: number of spins inside the ergodic grain.
g0: overall prefactor multiplying all grain-to-ray couplings.
α: decay parameter for the long-range couplings. For 0 < α < 1, the couplings decay exponentially with effective distance; smaller α gives faster decay.
γ: overall scale of the random grain Hamiltonian.
w: half-width of the uniform disorder window for the local fields. The fields are sampled from [hz - w, hz + w].
hz: center of the disorder window for the local fields.
ζ: randomness in the effective distance exponent u_j. It broadens the coupling profile around the deterministic value j - 1.
S: on-site spin. It can be integer or half-integer, and the local Hilbert space dimension is 2S + 1.
rng: random-number generator controlling disorder, couplings, and the grain matrix.
use_sparse: if true, return a sparse matrix; if false, return a dense matrix.

The full Hilbert-space dimension is

\[(2S + 1)^{L_{\mathrm{grain}} + L_{\mathrm{loc}}}.\]

U(1) Symmetric Quantum Sun Model

The U(1)-symmetric constructor keeps only states in a fixed total-$S_z$ sector and replaces the grain-to-ray coupling by the U(1)-preserving combination

\[\hat S^x_{n(j)} \hat S^x_j + \hat S^y_{n(j)} \hat S^y_j = \frac{1}{2} \left( \hat S^+_{n(j)} \hat S^-_j + \hat S^-_{n(j)} \hat S^+_j \right).\]

The Hamiltonian then reads

\[\hat H_{U(1)} = \hat H_{\mathrm{grain}}^{(S_z)} + g_0 \sum_{j=1}^{L_{\mathrm{loc}}} \alpha^{u_j}\, \left( \hat S^x_{n(j)} \hat S^x_j + \hat S^y_{n(j)} \hat S^y_j \right) + \sum_{j=1}^{L_{\mathrm{loc}}} h_j\, \hat S^z_j .\]

In this construction, the grain matrix is first organized in a block-diagonal way with respect to magnetization sectors inside the grain subspace. The final reduced Hamiltonian is then obtained by keeping only matrix elements that conserve the total magnetization of the full system, not by fixing the grain magnetization independently of the outer spins.

The reduced grain block is therefore obtained by projecting the random grain matrix into the chosen total-$S_z$ sector and then rescaling it so that γ still sets its overall strength:

\[\hat H_{\mathrm{grain}}^{(S_z)} = \gamma\, \frac{R_{\mathrm{cons}}} {\sqrt{\mathrm{tr}(R_{\mathrm{cons}}^2)/d_{\mathrm{red}}}} .\]

This means that γ plays the same conceptual role in both conventions: it is the scale of the grain Hamiltonian. The difference is only that in the U(1)-symmetric case the grain matrix is first projected into the selected sector before being normalized.

Additional Parameters in the U(1) Symmetric Model

The parameters L_loc, L_grain, g0, α, γ, w, hz, ζ, S, rng, and use_sparse keep the same meaning as in the conventional model. The new ingredients are:

use_U1=true: tells qsun_hamiltonian to build the U(1)-symmetric model instead of the conventional one.
S_z: total magnetization sector kept in the reduced Hilbert space.

The chosen sector must satisfy

\[|S_z| \le (L_{\mathrm{grain}} + L_{\mathrm{loc}})S,\]

and

\[S_z + (L_{\mathrm{grain}} + L_{\mathrm{loc}})S \in \mathbb{Z}.\]

For spin-1/2 and an even number of sites, S_z = 0.0 is the central sector and is usually the most natural first choice.

Basic Example

The most direct way to work with QSun is to construct the Hamiltonian with qsun_hamiltonian and then pass it into polfed.

If you want the complete constructor reference while reading this page, jump to Models.

Full-Hilbert-Space Example

using Polfed
using Polfed.Models: qsun_hamiltonian
using LinearAlgebra
using Random

# random number generator
rng = MersenneTwister(1234)

# setting model parameters
L_loc = 8
L_grain = 2
g0 = 1.0
α = 0.55

# constructing the Hamiltonian
H = qsun_hamiltonian(
    L_loc,
    L_grain,
    g0,
    α;
    S=0.5,
    γ=1.0,
    w=0.5,
    hz=1.0,
    ζ=0.2,
    rng=rng,
    use_sparse=true,
)

# generating a random initial state
x0 = rand(rng, size(H, 1))
x0 ./= norm(x0)

# setting POLFED parameters:
# number of states and position in the spectrum
howmany = 40
target = :middle

vals, vecs = polfed(H, x0, howmany, target)

In this example:

L_grain = 2 builds a small ergodic grain.
L_loc = 8 adds eight outer spins coupled to that grain.
γ = 1.0 sets the strength of the random grain block.
w = 0.5 and hz = 1.0 define the disorder interval for the local $S^z$ fields.
ζ = 0.2 makes the effective coupling distances slightly random.
use_sparse=true returns a sparse matrix, which is typically the right choice for POLFED workflows.

U(1)-Symmetric Example

using Random

# random number generator
rng = MersenneTwister(1234)

# constructing the Hamiltonian in a fixed total-S_z sector
H_u1 = qsun_hamiltonian(
    L_loc,
    L_grain,
    g0,
    α;
    S=0.5,
    γ=1.0,
    w=0.5,
    hz=1.0,
    ζ=0.2,
    rng=rng,
    use_U1=true,
    S_z=0.0,
    use_sparse=true,
)

# checking the reduced Hilbert-space size
println(size(H_u1))

This call uses the same physical parameters but restricts the Hamiltonian to a fixed total-$S_z$ sector. For larger systems this can reduce the matrix size substantially, so it is often the preferred choice when the symmetry is relevant for the problem you want to study.

Why Quantum Sun?

The Quantum Sun model is a useful benchmark for several reasons:

It is a clean toy model of the avalanche picture of ergodicity restoration: a small ergodic grain is coupled to a set of otherwise localized spins.
The parameter α gives direct control over how quickly couplings decay away from the grain, which makes it a natural tuning parameter for studying ergodicity breaking.
The model is easy to vary across disorder realizations and parameter regimes, while still remaining rich enough to display nontrivial many-body behavior.
It is particularly well matched to polfed, because one usually wants interior or near-middle-spectrum eigenpairs of fairly large sparse Hamiltonians to study ergodicity-breaking transitions at infinite temperature.
The U(1)-symmetric version provides a second, smaller representation of the same physical idea, which is very helpful when exploring larger systems or specific symmetry sectors.

The Quantum Sun model also serves as a natural playground for universal features near a many-body ergodicity-breaking transition (EBT). In particular, the way eigenstate thermalization breaks down as the transition is approached provides a useful framework for detecting an upcoming EBT; see for example Fading ergodicity meets maximal chaos (PRB 111, 184203) and Fading ergodicity (PRB 110, 134206).

The non-ergodic regime is equally informative: it allows one to quantify localization properties in many-body Hilbert space and to track the emergence of area-law or sub-volume-law entanglement structures. Recent variants of the model also show that nonstandard combinations of spectral statistics and eigenstate structure can appear in generalized Quantum Sun settings; see Many-body mobility edge in quantum sun models (PRB 109, L180201) and Unconventional Thermalization of a Localized Chain Interacting with an Ergodic Bath (arXiv:2507.18286).

In short, QSun is a good model when you want a physically meaningful, disorder-driven many-body problem that is still straightforward to generate and vary across many realizations.

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