arXiv subjects

Sinong Liu

Publications and source records attributed to Sinong Liu.

13 recordsLinked to original sources

arXiv2026

Fortuitous BMN Operators from Multiple Cores

We develop a systematic construction of fortuitous cohomology classes in the BMN sector of $\mathcal N=4$ super Yang--Mills theory from multiple lower-rank building blocks. For arbitrary partitions of colour space, we construct corrected effective letters that obey the supersymmetry transformations of the corresponding lower-rank theories. These allow us to build gauge-invariant, $Q$-closed polynomial operators. A restriction map recovers the constituent cohomology classes, establishing non-exactness and fortuity without brute-force computations. As an explicit example, we combine two fortuitous $SU(2)$ cores to construct a fortuitous class in $SU(4)$. Our results provide a systematic approach to the structure of fortuitous cohomology at finite $N$.

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arXiv2026

Fortuitous BMN Cohomology at Every $N$

Finite-$N$ trace relations can produce $Q$-cohomology classes with no stable large-$N$ ancestor. The brane-fused construction at $SU(3)$ suggests that such states can retain an $SU(2)$ nonabelian core accompanied by scalar hair. We develop this idea into a general construction for every $SU(N)$, $N\geq3$. Starting from the $SU(2)$ threshold operator $O_{24}$, we select pairs of eigenvalues of one scalar and construct corrected two-colour letters whose $Q$-transformations obey the $SU(2)$ rules exactly. Substitution into $O_{24}$ gives a closed rational operator for each pair. Multiplying by eigenvalue-separation factors removes the denominators, and summing over all pairs produces a gauge-invariant polynomial. An exact block-restriction identity recovers a nonzero multiple of $O_{24}$. The restriction maps every stable class to a stable $SU(2)$ class, proving that the resulting operator is non-exact and fortuitous, even when non-Coulombic stable classes are included. This proves the existence of fortuitous BMN cohomology at every $N\geq3$. The construction is an exact algebraic realization of a nonabelian core with scalar dressing.

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arXiv2026

Fate of "Space-like singularities" in $c=1$ Matrix Model

A class of time dependent backgrounds in two dimensional String Theory leads to superluminal Liouville walls on the worldsheet. In the dual double scaled $c=1$ matrix model these backgrounds involve eigenvalues leaking out to infinity, and the collective field fluctuations become strongly coupled along space-like regions, resembling singularities. We realize these backgrounds as results of quantum quenches in the matrix model, retaining non-linear terms in the matrix potential, thus departing from a double scaling limit. Working in the fermion picture in a Thomas-Fermi approximation, we show that while the early time behavior of the phase space density near the maximum of the potential agrees with that obtained in the double scaled theory, at times of the order $(\log N)$ the effect of the IR wall becomes significant. At later times, with a characteristic winding time of order $(\log N)^2$, folds on the fermi surface proliferate and eventually cover the allowed region in phase space densely. Using action-angle variables, we show that the phase space density oscillates around a time independent and angle independent value rapidly at late times. A coarse-grained density in the angle space relaxes to a time independent equilibrium value as a power law with an exponent largely independent of the details of the initial state. Thus, the appearance of a space-like singularity is an artifact of the strict double scaling limit. We comment on the interpretation of the final state in String Theory.

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arXiv2025

Superluminal Liouville walls in 2d String Theory and space-like singularities

An interesting class of time dependent backgrounds in $1+1$ dimensional string theory involves worldsheet Liouville walls which move in (target space) time. When a parameter in such a background exceeds a certain critical value, the speed of the Liouville wall exceeds the speed of light, and there is no usual S-Matrix. We examine such backgrounds in the dual $c=1$ matrix model from the point of view of fluctuations of the collective field, and determine the nature of the emergent space-time perceived by these fluctuations. We show that so long as the corresponding Liouville wall remains time-like, the emergent space time is conformal to full Minkowski space with a time-like wall. However, for the cases where the Liouville wall is superluminal, the emergent space-time has a {\em space-like boundary} where the collective field couplings diverge. This appears as a space-like singularity in perturbative collective field theory. We comment on the necessity of incorporating finite $N$, as well as finite (double-scaled) coupling, effects to understand the behavior of the exact theory near this boundary.

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arXiv2025

Complexity of PXP scars revisited

We revisit a quantum quench scenario in which either a scarring or thermalizing initial state evolves under the PXP Hamiltonian. Within this framework, we study the time evolution of spread complexity and related quantities in the Krylov basis. We find that the Lanczos coefficients $b_n$, as functions of the iteration number $n$, exhibit a characteristic arched growth and decay, followed by erratic oscillations which we refer to as buttress. The arched profile predominantly arises from contributions within the quantum many-body scar subspace, while the buttress is linked to thermalization dynamics. To explain this behavior, we utilize the representation theory of $\mathfrak{s}l_3(\mathbb{C})$, allowing us to decompose the PXP Hamiltonian into a linear component and a residual part. The linear term governs the formation and width of the arch, and we observe that that there exists a threshold of arch width which determines whether a given initial state exhibits scarring. Meanwhile, the residual term accounts qualitatively for the emergence of the buttress. We estimate an upper bound for the extent of the buttress using Lucas numbers. Finally, we demonstrate that spread complexity oscillates periodically over time for scarred initial states, whereas such oscillations are suppressed in thermalizing cases.

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arXiv2024

Krylov complexity of density matrix operators

Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ($C_K$) and Spread complexity ($C_S$) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between $C_K$ and $2C_S$. Furthermore, for maximally entangled pure states, we find that the moment-generating function of $C_K$ becomes the Spectral Form Factor and, at late-times, $C_K$ is simply related to $NC_S$ for $N\geq2$ within the $N$-dimensional Hilbert space. Notably, we confirm that $C_K = 2C_S$ holds across all times when $N=2$. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.

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arXiv2022

Quantum complexity and topological phases of matter

In this work, we find that the complexity of quantum many-body states, defined as a spread in the Krylov basis, may serve as a new probe that distinguishes topological phases of matter. We illustrate this analytically in one of the representative examples, the Su-Schrieffer-Heeger model, finding that spread complexity becomes constant in the topological phase. Moreover, in the same setup, we analyze exactly solvable quench protocols where the evolution of the spread complexity shows distinct dynamical features depending on the topological vs non-topological phase of the initial state as well as the quench Hamiltonian.

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arXiv2022

Spread Complexity and Topological Transitions in the Kitaev Chain

A number of recent works have argued that quantum complexity, a well-known concept in computer science that has re-emerged recently in the context of the physics of black holes, may be used as an efficient probe of novel phenomena such as quantum chaos and even quantum phase transitions. In this article, we provide further support for the latter, using a 1-dimensional p-wave superconductor - the Kitaev chain - as a prototype of a system displaying a topological phase transition. The Hamiltonian of the Kitaev chain manifests two gapped phases of matter with fermion parity symmetry; a trivial strongly-coupled phase and a topologically non-trivial, weakly-coupled phase with Majorana zero-modes. We show that Krylov-complexity (or, more precisely, the associated spread-complexity) is able to distinguish between the two and provides a diagnostic of the quantum critical point that separates them. We also comment on some possible ambiguity in the existing literature on the sensitivity of different measures of complexity to topological phase transitions.

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arXiv2022

Entanglement Entropy and Phase Space Density: Lowest Landau Levels and 1/2 BPS states

We consider the entanglement entropy of an arbitrary subregion in a system of $N$ non-relativistic fermions in $2+1$ dimensions in Lowest Landau Level (LLL) states. Using the connection of these states to those of an auxiliary $1+1$ dimensional fermionic system, we derive an expression for the leading large-$N$ contribution in terms of the expectation value of the phase space density operator in $1+1$ dimensions. For appropriate subregions the latter can replaced by its semiclassical Thomas-Fermi value, yielding expressions in terms of explicit integrals which can be evaluated analytically. We show that the leading term in the entanglement entropy is a perimeter law with a shape independent coefficient. Furthermore, we obtain analytic expressions for additional contributions from sharp corners on the entangling curve. Both the perimeter and the corner pieces are in good agreement with existing calculations for special subregions. Our results are relevant to the integer quantum Hall effect problem, and to the half-BPS sector of $\mathcal N=4$ Yang Mills theory on $S^3$. In this latter context, the entanglement we consider is an entanglement in target space. We comment on possible implications to gauge-gravity duality.

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arXiv2021

Gauge Invariant Target Space Entanglement in D-Brane Holography

It has been suggested in arXiv:2004.00613 that in Dp-brane holography, entanglement in the target space of the D-brane Yang-Mills theory provides a precise notion of bulk entanglement in the gravity dual. We expand on this discussion by providing a gauge invariant characterization of operator sub-algebras corresponding to such entanglement. This is achieved by finding a projection operator which imposes a constraint characterizing the target space region of interest. By considering probe branes in the Coulomb branch we provide motivation for why the operator sub-algebras we consider are appropriate for describing a class of measurements carried out with low-energy probes in the corresponding bulk region of interest. We derive expressions for the corresponding Renyi entropies in terms of path integrals which can be directly used in numerical calculations.

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arXiv2020

Quantum Quench in $c=1$ Matrix Model and Emergent Space-times

We consider quantum quench in large-N singlet sector quantum mechanics of a single hermitian matrix in the double scaling limit. The time dependent parameter is the self-coupling of the matrix. We find exact classical solutions of the collective field theory of eigenvalue density with abrupt smooth quench profiles which asymptote to constant couplings at early and late times, and the system is initially in its ground state. With adiabatic initial conditions we find that adiabaticity is always broken regardless of the quench speed. In a class of quench profiles the saddle point solution for the collective field diverges at a finite time, and a further time evolution becomes ambiguous. However the underlying matrix model expressed in terms of fermions predict a smooth time evolution across this point. By studying fluctuations around the saddle point solution we interpret the emergent space-times. They generically have spacelike boundaries where the couplings of the fluctuations diverge. Only for very finely tuned quench profiles, the space-time is normal.

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arXiv2020

Complexity and scaling in quantum quench in $1+1$ dimensional fermionic field theories

We consider the scaling behavior of circuit complexity under quantum quench in an a relativistic fermion field theory on a one dimensional spatial lattice. This is done by finding an exactly solvable quench protocol which asymptotes to massive phases at early and late times and crosses a critical point in between. We find a variety of scaling behavior as a function of the quench rate, starting with a saturation for quenches at the lattice scale, a "fast quench scaling" at intermediate rate and a Kibble Zurek scaling at slow rates.

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arXiv2019

Quantum Quench in Non-relativistic Fermionic Field Theory: Harmonic traps and 2d String Theory

We investigate a class of exactly solvable quantum quench protocols with a finite quench rate in systems of one dimensional non-relativistic fermions in external harmonic oscillator or inverted harmonic oscillator potentials, with time dependent masses and frequencies. These hamiltonians arise, respectively, in harmonic traps, and the $c=1$ Matrix Model description of two dimensional string theory with time dependent string coupling. We show how the dynamics is determined by a single function of time which satisfies a generalized Ermakov-Pinney equation. The quench protocols we consider asymptote to constant masses and frequencies at early times, and cross or approach a gapless potential. In a right side up harmonic oscillator potential we determine the scaling behavior of the one point function and the entanglement entropy of a subregion by obtaining analytic approximations to the exact answers. The results are consistent with Kibble-Zurek scaling for slow quenches and with perturbation calculations for fast quenches. For cis-critical quench protocols the entanglement entropy oscillates at late times around its initial value. For end-critical protocols the entanglement entropy monotonically goes to zero inversely with time, reflecting the spread of fermions over the entire line. For the inverted harmonic oscillator potential, the dual collective field description is a scalar field in a time dependent metric and dilaton background.

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