arXiv subjects

Tianci Zhou

Publications and source records attributed to Tianci Zhou.

At least 37 records · Page 2Linked to original sources

arXiv2021

Non-unitary Entanglement Dynamics in Continuous Variable Systems

We construct a random unitary Gaussian circuit for continuous-variable (CV) systems subject to Gaussian measurements. We show that when the measurement rate is nonzero, the steady state entanglement entropy saturates to an area-law scaling. This is different from a many-body qubit system, where a generic entanglement transition is widely expected. Due to the unbounded local Hilbert space, the time scale to destroy entanglement is always much shorter than the one to build it, while a balance could be achieved for a finite local Hilbert space. By the same reasoning, the absence of transition should also hold for other non-unitary Gaussian CV dynamics.

quant-ph↗

arXiv2021

Operator Growth from Global Out-of-time-order Correlators

In the context of chaotic quantum many-body systems, we show that operator growth, as diagnosed by out-of-time-order correlators of local operators, also leaves a sharp imprint in out-of-time-order correlators of global operators. In particular, the characteristic spacetime shape of growing local operators can be accessed using global measurements without any local control or readout. Building on an earlier conjectured phase diagram for operator growth in chaotic systems with power-law interactions, we show that existing nuclear spin data for out-of-time-order correlators of global operators are well fit by our theory. We also predict super-polynomial operator growth in dipolar systems in 3d and discuss the potential observation of this physics in future experiments with nuclear spins and ultra-cold polar molecules.

quant-ph↗

arXiv2020

Entanglement Negativity at Measurement-Induced Criticality

We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their "mutual negativity" and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a class of "hybrid" circuit models obtained by perturbing the measurement-only circuit with unitary gates of progressive levels of complexity. While other critical exponents vary appreciably for different choices of unitary gate ensembles at their respective critical points, the mutual negativity has scaling dimension 3 across remarkably many of the hybrid circuits, which is notably different from that in percolation. We contrast our results with limiting cases where a geometrical minimal-cut picture is available.

cond-mat.stat-mech↗

arXiv2020

The entanglement membrane in chaotic many-body systems

In certain analytically-tractable quantum chaotic systems, the calculation of out-of-time-order correlation functions, entanglement entropies after a quench, and other related dynamical observables, reduces to an effective theory of an ``entanglement membrane'' in spacetime. These tractable systems involve an average over random local unitaries defining the dynamical evolution. We show here how to make sense of this membrane in more realistic models, which do not involve an average over random unitaries. Our approach relies on introducing effective pairing degrees of freedom in spacetime, describing a pairing of forward and backward Feynman trajectories, inspired by the structure emerging in random unitary circuits. This provides a framework for applying ideas of coarse-graining to dynamical quantities in chaotic systems. We apply the approach to some translationally invariant Floquet spin chains studied in the literature. We show that a consistent line tension may be defined for the entanglement membrane, and that there are qualitative differences in this tension between generic models and ``dual-unitary'' circuits. These results allow scaling pictures for out-of-time-order correlators and for entanglement to be taken over from random circuits to non-random Floquet models. We also provide an efficient numerical algorithm for determining the entanglement line tension in 1+1D.

cond-mat.str-el↗

arXiv2020

The operator Lévy flight: light cones in chaotic long-range interacting systems

We argue that chaotic power-law interacting systems have emergent limits on information propagation, analogous to relativistic light cones, which depend on the spatial dimension $d$ and the exponent $α$ governing the decay of interactions. Using the dephasing nature of quantum chaos, we map the problem to a stochastic model with a known phase diagram. A linear light cone results for $α\ge d+1/2$. We also provide a Lévy flight (long-range random walk) interpretation of the results and show consistent numerical data for 1d long-range spin models with 200 sites.

cond-mat.stat-mech↗

arXiv2020

Diffusive scaling of Rényi entanglement entropy

Recent studies found that the diffusive transport of conserved quantities in non-integrable many-body systems has an imprint on quantum entanglement: while the von Neumann entropy of a state grows linearly in time $t$ under a global quench, all $n$th Rényi entropies with $n > 1$ grow with a diffusive scaling $\sqrt{t}$. To understand this phenomenon, we introduce an amplitude $A(t)$, which is the overlap of the time-evolution operator $U(t)$ of the entire system with the tensor product of the two evolution operators of the subsystems of a spatial bipartition. As long as $|A(t)| \ge e^{-\sqrt{Dt}}$, which we argue holds true for generic diffusive non-integrable systems, all $n$th Rényi entropies with $n >1$ (annealed-averaged over initial product states) are bounded from above by $\sqrt{t}$. We prove the following inequality for the disorder average of the amplitude, $\overline{|A(t)|} \ge e^{ - \sqrt{Dt}} $, in a local spin-$\frac{1}{2}$ random circuit with a $\text{U}(1)$ conservation law by mapping to the survival probability of a symmetric exclusion process. Furthermore, we numerically show that the typical decay behaves asymptotically, for long times, as $|A(t)| \sim e^{ - \sqrt{Dt}} $ in the same random circuit as well as in a prototypical non-integrable model with diffusive energy transport but no disorder.

cond-mat.stat-mech↗

arXiv2019

Barrier from chaos: operator entanglement dynamics of the reduced density matrix

It is believed that thermalization drives the reduced density matrix of a subsystem to approach a short-range entangled operator. If the initial state is also short-range entangled, it is possible that the reduced density matrix remains low-entangled throughout thermalization; or there could exist a barrier with high operator entanglement between the initial and thermalized reduced density matrix. In this paper, we study such dynamics in three classes of models: the rational CFTs, the random unitary circuit, and the holographic CFTs, representing systems of increasing quantum chaoticity. We show that in all three classes of models, the operator entanglement (or variant of) exhibits three phases, a linear growth phase, a plateau phase, and a decay phase. The plateau phase characterized by volume-law operator entanglement corresponds to the barrier in operator entanglement. While it is present in all three models, its persistence and exit show interesting distinctions among them. The rational CFTs have the shortest plateau phase, followed by the slowest decay phase; the holographic CFTs mark the opposite end, i.e. having the longest plateau phase followed by a discontinuous drop; and the random unitary circuit shows the intermediate behavior. We discuss the mechanisms underlying these behaviors in operator entanglement barriers, whose persistence might serve as another measure for quantum chaoticity.

hep-th↗

arXiv2019

Quantum chaos dynamics in long-range power law interaction systems

We use out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: the number of qubits $N$ on each site (the onsite Hilbert space dimension) and the power law exponent $α$. Three light cone structures of OTOC appear at $N = 1$: (1) logarithmic when $0.5<α\lesssim 0.8$, (2) sublinear power law when $0.8 \lesssim α\lesssim 1.5$ and (3) linear when $α\gtrsim 1.5$. The OTOC scales as $\exp(λt)/x^{2α} $ and $t^{2 α/ ζ} / x^{ 2 α} $ respectively beyond the light cones in the first two cases. When $α\geq 2$, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large $N$ limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for $ α\gtrsim 1.5$. This implies the locality is never fully recovered for finite $α$. Our result provides a unified physical picture for the chaos dynamics in long-range power law interaction system.

cond-mat.stat-mech↗

arXiv2019

Emergent statistical mechanics of entanglement in random unitary circuits

We map the dynamics of entanglement in random unitary circuits, with finite on-site Hilbert space dimension $q$, to an effective classical statistical mechanics, and develop general diagrammatic tools for calculations in random unitary circuits. We demonstrate explicitly the emergence of a `minimal membrane' governing entanglement growth, which in 1+1D is a directed random walk in spacetime (or a variant thereof). Using the replica trick to handle the logarithm in the definition of the $n$th Rényi entropy $S_n$, we map the calculation of the entanglement after a quench to a problem of interacting random walks. A key role is played by effective classical spins (taking values in a permutation group) which distinguish between different ways of pairing spacetime histories in the replicated system. For the second Rényi entropy, $S_2$, we are able to take the replica limit explicitly. This gives a mapping between entanglement growth and a directed polymer in a random medium at finite temperature (confirming Kardar-Parisi-Zhang (KPZ) scaling for entanglement growth in generic noisy systems). We find that the entanglement growth rate (`speed') $v_n$ depends on the Rényi index $n$, and we calculate $v_2$ and $v_3$ in an expansion in the inverse local Hilbert space dimension, $1/q$. These rates are determined by the free energy of a random walk, and of a bound state of two random walks, respectively, and include contributions of `energetic' and `entropic' origin. We give a combinatorial interpretation of the Page-like subleading corrections to the entanglement at late times and discuss the dynamics of the entanglement close to and after saturation. We briefly discuss the application of these insights to time-independent Hamiltonian dynamics.

cond-mat.stat-mech↗

arXiv2019

Operator Dynamics in Brownian Quantum Circuit

We explore the operator dynamics in a random $N$-spin model with pairwise interactions (Brownian quanum circuit). We introduce the height $h$ of an operator to characterize its spatial extent, and derive the master equation of the height probability distribution. The study of an initial simple operator with $h = 1$ (minimal nonzero height) shows that the mean height, which is proportional to the squared commutator, has an initial exponential growth. It then slows down around the scrambling time $\sim\log N$ and finally saturates to a steady state in a manner similar to the logistic function. The deviation to the logistic function is due to the large fluctuations (order $N$) in the intermediate time. Moreover, we find that the exponential growth rate (quantum Lyapunov exponent) is smaller for initial operator with $\langle h\rangle\gg 1$. Based on this observation, we propose that the chaos bound at finite temperature can be produced by an initial operator whose height distribution is biased towards higher operators. We numerically test the power law initial height distribution $1/h^α $ in a Brownian circuit with number of spin $N=10000$ and show that the Lyapunov exponent is linearly constrained by $α$ before reaching the infinite temperature value.

cond-mat.str-el↗

arXiv2018

Measuring the distance between quantum many-body wave functions

We study the distance of two wave functions under chaotic time evolution. The two initial states are differed only by a local perturbation. To be entitled "chaos" the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance $d^2(t)$ by investigating the difference of two reduced density matrices of the subsystem $A$ that is spatially separated from the local perturbation. This distance $d^2(t)$ grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates $d^2(t)$ to the square of commutator $C(t)$ (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem $A$. Among various one-dimensional spin-$\frac{1}{2}$ models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in $d^2(t)$ when the local perturbation and subsystem $A$ are well separated. This regime is absent in the spin-$\frac{1}{2}$ chain with local interaction only. After sufficiently long time evolution, $d^2(t)$ relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that $d^2(t)$ is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.

cond-mat.stat-mech↗

arXiv2018

Operator scrambling and quantum chaos

Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-$1/2$ chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite different, a simple Pauli spin operator can eventually approach a "highly entangled" operator with operator entanglement entropy taking a volume law value (close to the Page value). Meanwhile, the spectrum of the operator reduced density matrix develops a universal spectral correlation which can be characterized by the Wishart random matrix ensemble. In the second example, we further connect the 2-local model into a one dimensional chain and briefly discuss the operator scrambling there. In contrast, in the quantum linear map, although the square of commutator can increase exponentially with time, a simple operator does not scramble but performs chaotic motion in the operator basis space determined by the classical linear map. We show that once we modify the quantum linear map such that operator can mix in the operator basis, the operator entanglement entropy can grow and eventually saturate to its Page value, thus making it a truly quantum chaotic model.

cond-mat.str-el↗

arXiv2017

Bipartite Fidelity and Loschmidt Echo of Bosonic Conformal Interface

We study the quantum quench problem for a class of bosonic conformal interfaces by computing the Loschmidt echo and the bipartite fidelity. The quench can be viewed as a sudden change of boundary conditions parameterized by $θ$ when connecting two one-dimensional critical systems. They are classified by $S(θ)$ matrices associated with the current scattering processes on the interface. The resulting Loschmidt echo of the quench has long time algebraic decay $t^{-α}$, whose exponent also appears in the finite size bipartite fidelity as $L^{-\fracα{2}}$. We perform analytic and numerical calculations of the exponent $α$, and find that it has a quadratic dependence on the change of $θ$ if the prior and post quench boundary conditions are of the same type of $S$, while remaining $\frac{1}{4}$ otherwise. Possible physical realizations of these interfaces include, for instance, connecting different quantum wires (Luttinger liquids), quench of the topological phase edge states, etc. and the exponent can be detected in a X-ray edge singularity type experiment.

cond-mat.stat-mech↗

arXiv2017

Operator entanglement entropy of the time evolution operator in chaotic systems

We study the growth of the operator entanglement entropy (EE) of the time evolution operator in chaotic, many-body localized and Floquet systems. In the random field Heisenberg model we find a universal power law growth of the operator EE at weak disorder, a logarithmic growth at strong disorder, and extensive saturation values in both cases. In a Floquet spin model, the saturation value after an initial linear growth is identical to the value of a random unitary operator (the Page value). We understand these properties by mapping the operator EE to a global quench problem evolved with a similar parent-Hamiltonian in an enlarged Hilbert space with the same chaotic, MBL and Floquet properties as the original Hamiltonian. The scaling and saturation properties reflect the spreading of the state EE of the corresponding time evolution. We conclude that the EE of the evolution operator should characterize the propagation of information in these systems.

cond-mat.stat-mech↗

arXiv2016

Out-of-time-order correlations in many-body localized and thermal phases

We use the out-of-time-order (OTO) correlators to study the slow dynamics in the many-body localized (MBL) phase. We investigate OTO correlators in the effective ("l-bit") model of the MBL phase, and show that their amplitudes after disorder averaging approach their long-time limits as power-laws of time. This power-law dynamics is due to dephasing caused by interactions between the localized operators that fall off exponentially with distance. The long-time limits of the OTO correlators are determined by the overlaps of the local operators with the conserved l-bits. We demonstrate numerically our results in the effective model and three other more "realistic" spin chain models. Furthermore, we extend our calculations to the thermal phase and find that for a time-independent Hamiltonian, the OTO correlators also appear to vanish as a power law at long time, perhaps due to coupling to conserved densities. In contrast, we find that in the thermal phase of a Floquet spin model with no conserved densities the OTO correlator decays exponentially at long times.

cond-mat.str-el↗

arXiv2016

Entanglement Entropy of Local Operators in Quantum Lifshitz Theory

We study the growth of entanglement entropy(EE) of local operator excitation in the quantum Lifshitz model which has dynamic exponent z = 2. Specifically, we act a local vertex operator on the groundstate at a distance $l$ to the entanglement cut and calculate the EE as a function of time for the state's subsequent time evolution. We find that the excess EE compared with the groundstate is a monotonically increasing function which is vanishingly small before the onset at $t \sim l^2$ and eventually saturates to a constant proportional to the scaling dimension of the vertex operator. The quasi-particle picture can interpret the final saturation as the exhaustion of the quasi-particle pairs, while the diffusive nature of the time scale $t \sim l^2$ replaces the common causality constraint in CFT calculation. To further understand this property, we compute the excess EE of a small disk probe far from the excitation point and find chromatography pattern in EE generated by quasi-particles of different propagation speeds.

cond-mat.stat-mech↗

arXiv2016

Entanglement Entropy and Mutual Information of Circular Entangling Surfaces in 2 + 1-dimensional Quantum Lifshitz Model

We investigate the entanglement entropy (EE) of circular entangling cuts in the 2+1-dimensional quantum Lifshitz model, whose ground state wave function is a spatially conformal invariant state of the Rokhsar-Kivelson type, whose weight is the Gibbs weight of 2D Euclidean free boson. We show that the finite subleading corrections of EE to the area-law term as well as the mutual information are conformal invariants and calculate them for cylinder, disk-like and spherical manifolds with various spatial cuts. The subtlety due to the boson compactification in the replica trick is carefully addressed. We find that in the geometry of a punctured plane with many small holes, the constant piece of EE is proportional to the number of holes, indicating the ability of entanglement to detect topological information of the configuration. Finally, we compare the mutual information of two small distant disks with Cardy's relativistic CFT scaling proposal. We find that in the quantum Lifshitz model, the mutual information also scales at long distance with a power determined by the lowest scaling dimension local operator in the theory.

cond-mat.stat-mech↗

arXiv2015

Solitons in a continuous classical Haldane-Shastry spin chain

Motivated by Polychronakos' discovery that solitons exist in the hydrodynamic equations of continuum version of the Calogero model, we seek solitons in the classical dynamics of a continuum version of the Haldane-Shastry spin chain. We have obtained analytic multi-lump solitary wave solutions for our spin-field equation, and these solutions possess interesting topological features. We have performed numerical collision experiments showing that these solitary waves survive collisions, and thus suggest the existence of true multi-soliton solutions.

cond-mat.other↗