We study a stability criterion hypothesis for superfluids expressed in terms of the local density spectral function ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})$ that is applicable to both homogeneous and inhomogeneous systems. We evaluate the local density spectral function in the presence of a one-dimensional repulsive or attractive external potential within Bogoliubov theory, using solutions for the tunneling problem. We also evaluate the local density spectral function using an orthogonal basis, and calculate the autocorrelation function ${C}_{n}(\mathbf{r},t)$ . When superfluids in a $d$ -dimensional system flow below a threshold, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{d}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{d+1}$ holds in the long-time regime. However, when superfluids flow with the critical current, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{\beta}}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{\ensuremath{\beta}+1}$ holds in the long-time regime with $\ensuremath{\beta}<d$. These results support the stability criterion hypothesis recently proposed.We study a stability criterion hypothesis for superfluids expressed in terms of the local density spectral function ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})$ that is applicable to both homogeneous and inhomogeneous systems. We evaluate the local density spectral function in the presence of a one-dimensional repulsive or attractive external potential within Bogoliubov theory, using solutions for the tunneling problem. We also evaluate the local density spectral function using an orthogonal basis, and calculate the autocorrelation function ${C}_{n}(\mathbf{r},t)$. When superfluids in a $d$-dimensional system flow below a threshold, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{d}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{d+1}$ holds in the long-time regime. However, when superfluids flow with the critical current, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{\beta}}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{\ensuremath{\beta}+1}$ holds in the long-time regime with $\ensuremath{\beta}<d$. These results support the stability criterion hypothesis recently proposed.

arXiv:1305.6984Physical Review ADec 6, 20134 citations

Theoretical Aspects of Mainly Low Dimensional Magnetic Systems

Hans C. Fogedby

Springer eBooksJan 1, 198029 citations

Zero- and Low-Temperature Behavior of the Two-Dimensional ±J Ising Spin Glass

Creighton K. Thomas, David A. Huse, A. Alan Middleton

Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.

arXiv:1012.3444Physical Review LettersJul 20, 201154 citations

Magnetism in one-dimensional disordered systems

George Theodorou, Morrel H. Cohen

Springer eBooksApr 6, 20081 citation

Low-dimensional interacting electron systems in a magnetic field

Veikko Halonen

Jan 1, 19920 citations

Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codesGeneralized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes

Zijian Liang, Yu-An Chen

We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions, generalizing the Kitaev $\mathbb{Z}_p$ toric code by augmenting each stabilizer with two additional qudits. Using the Laurent-polynomial formalism, we adapt the Gröbner basis to compute the logical dimension $k$ efficiently, without explicitly constructing large parity-check matrices. We then perform a systematic search over various stabilizer realizations and lattice geometries for $p\in\{3,5,7,11\}$ , identifying qudit low-density parity-check codes with the optimal finite-size performance. Representative examples include $[[242,10,22]]_3$ and $[[120,6,20]]_{11}$ , both achieving $k d^{2}/n=20$ . Across the searched regime, the best observed $k d^{2}$ at fixed $n$ increases with $p$ , with an empirical relation $k d^{2} = 0.0541 \, n^{2}\ln p + 3.84 \, n$ , compatible with a Bravyi--Poulin--Terhal-type tradeoff when the interaction range grows with system size.We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions, generalizing the Kitaev $\mathbb{Z}_p$ toric code by augmenting each stabilizer with two additional qudits. Using the Laurent-polynomial formalism, we adapt the Gröbner basis to compute the logical dimension $k$ efficiently, without explicitly constructing large parity-check matrices. We then perform a systematic search over various stabilizer realizations and lattice geometries for $p\in\{3,5,7,11\}$, identifying qudit low-density parity-check codes with the optimal finite-size performance. Representative examples include $[[242,10,22]]_3$ and $[[120,6,20]]_{11}$, both achieving $k d^{2}/n=20$. Across the searched regime, the best observed $k d^{2}$ at fixed $n$ increases with $p$, with an empirical relation $k d^{2} = 0.0541 \, n^{2}\ln p + 3.84 \, n$, compatible with a Bravyi--Poulin--Terhal-type tradeoff when the interaction range grows with system size.

quant-pharXiv:2602.20158Feb 24, 20260 citations

{C}_{n}(\mathbf{r},t)

. When superfluids in a

d

-dimensional system flow below a threshold, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{d}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{d+1}$ holds in the long-time regime. However, when superfluids flow with the critical current, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{\beta}}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{\ensuremath{\beta}+1}$ holds in the long-time regime with $\ensuremath{\beta}<d$. These results support the stability criterion hypothesis recently proposed.

Generalized

\mathbb{Z}_p

toric codes as qudit low-density parity-check codes

We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions, generalizing the Kitaev

\mathbb{Z}_p

toric code by augmenting each stabilizer with two additional qudits. Using the Laurent-polynomial formalism, we adapt the Gröbner basis to compute the logical dimension

k

efficiently, without explicitly constructing large parity-check matrices. We then perform a systematic search over various stabilizer realizations and lattice geometries for

p\in\{3,5,7,11\}

, identifying qudit low-density parity-check codes with the optimal finite-size performance. Representative examples include

[[242,10,22]]_3

and

[[120,6,20]]_{11}

, both achieving

k d^{2}/n=20

. Across the searched regime, the best observed

k d^{2}

at fixed

n

increases with

p

, with an empirical relation

k d^{2} = 0.0541 \, n^{2}\ln p + 3.84 \, n

, compatible with a Bravyi--Poulin--Terhal-type tradeoff when the interaction range grows with system size.