arXiv subjects

Paola Ruggiero

Publications and source records attributed to Paola Ruggiero.

32 records · Page 2Linked to original sources

arXiv2020

Quantum Generalized Hydrodynamics

Physical systems made of many interacting quantum particles can often be described by Euler hydrodynamic equations in the limit of long wavelengths and low frequencies. Recently such a classical hydrodynamic framework, now dubbed Generalized Hydrodynamics (GHD), was found for quantum integrable models in one spatial dimension. Despite its great predictive power, GHD, like any Euler hydrodynamic equation, misses important quantum effects, such as quantum fluctuations leading to non-zero equal-time correlations between fluid cells at different positions. Focusing on the one-dimensional gas of bosons with delta repulsion, and on states of zero entropy, for which quantum fluctuations are larger, we reconstruct such quantum effects by quantizing GHD. The resulting theory of quantum GHD can be viewed as a multi-component Luttinger liquid theory, with a small set of effective parameters that are fixed by the Thermodynamic Bethe Ansatz. It describes quantum fluctuations of truly nonequilibrium systems where conventional Luttinger liquid theory fails.

cond-mat.quant-gas↗

arXiv2019

Negativity Spectrum in the Random Singlet Phase

Entanglement features of the ground state of disordered quantum matter are often captured by an infinite randomness fixed point that, for a variety of models, is the random singlet phase. Although a copious number of studies covers bipartite entanglement in pure states, at present, less is known for mixed states and tripartite settings. Our goal is to gain insights in this direction by studying the negativity spectrum in the random singlet phase. Through the strong disorder renormalization group technique, we derive analytic formulas for the universal scaling of the disorder averaged moments of the partially transposed reduced density matrix. Our analytic predictions are checked against a numerical implementation of the strong disorder renormalization group and against exact computations for the XX spin chain (a model in which free fermion techniques apply). Importantly, our results show that the negativity and logarithmic negativity are not trivially related after the average over the disorder.

cond-mat.stat-mech↗

arXiv2019

Subsystem trace distance in low-lying states of $(1+1)$-dimensional conformal field theories

We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices $ρ_A$ and $σ_A$ is obtained from the moments $\textrm{tr} (ρ_A-σ_A)^n$ and taking the limit $n\to1$ of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length $\ell$ embedded in the low lying eigenstates of a one-dimensional critical system of length $L$, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a scale invariant universal function of $\ell/L$. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten $n$-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of $\ell/L$. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.

hep-th↗

arXiv2019

Symmetry resolved entanglement in free fermionic systems

We consider the symmetry resolved Rényi entropies in the one dimensional tight binding model, equivalent to the spin-1/2 XX chain in a magnetic field. We exploit the generalised Fisher-Hartwig conjecture to obtain the asymptotic behaviour of the entanglement entropies with a flux charge insertion at leading and subleading orders. The o(1) contributions are found to exhibit a rich structure of oscillatory behaviour. We then use these results to extract the symmetry resolved entanglement, determining exactly all the non-universal constants and logarithmic corrections to the scaling that are not accessible to the field theory approach. We also discuss how our results are generalised to a one-dimensional free fermi gas.

cond-mat.stat-mech↗

arXiv2019

Twisted and untwisted negativity spectrum of free fermions

A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion. Such criterion is based on the observation that the spectrum of the partially transposed density matrix of an entangled state contains negative eigenvalues, in turn, used to define an entanglement measure called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the operation of PT to fermionic systems remained a technical challenge until recently when a more natural definition of PT for fermions that accounts for the Fermi statistics has been put forward. In this paper, we study the many-body spectrum of the reduced density matrix of two adjacent intervals for one-dimensional free fermions after applying the fermionic PT. We show that in general there is a freedom in the definition of such operation which leads to two different definitions of PT: the resulting density matrix is Hermitian in one case, while it becomes pseudo-Hermitian in the other case. Using the path-integral formalism, we analytically compute the leading order term of the moments in both cases and derive the distribution of the corresponding eigenvalues over the complex plane. We further verify our analytical findings by checking them against numerical lattice calculations.

cond-mat.stat-mech↗

arXiv2019

Conformal field theory on top of a breathing one-dimensional gas of hard core bosons

The recent results of [J. Dubail, J.-M. Stéphan, J. Viti, P. Calabrese, Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale correlation functions of inhomogeneous critical one-dimensional quantum systems -- e.g. a gas of hard core bosons in a trapping potential -- are extended to a dynamical situation: a breathing gas in a time-dependent harmonic trap. Hard core bosons in a time-dependent harmonic potential are well known to be exactly solvable, and can thus be used as a benchmark for the approach. An extensive discussion of the approach and of its relation with classical and quantum hydrodynamics in one dimension is given, and new formulas for correlation functions, not easily obtainable by other methods, are derived. In particular, a remarkable formula for the large scale asymptotics of the bosonic $n$-particle function $\left< Ψ^\dagger (x_1,t_1) \dots Ψ^\dagger (x_n,t_n) Ψ(x_1',t_1') \dots Ψ(x_n',t_n') \right>$ is obtained. Numerical checks of the approach are carried out for the fermionic two-point function -- easier to access numerically in the microscopic model than the bosonic one -- with perfect agreement.

cond-mat.stat-mech↗

arXiv2019

Subsystem Trace Distance in Quantum Field Theory

We develop a systematic method to calculate the trace distance between two reduced density matrices in 1+1 dimensional quantum field theories. The approach exploits the path integral representation of the reduced density matrices and an ad hoc replica trick. We then extensively apply this method to the calculation of the distance between reduced density matrices of one interval of length $\ell$ in eigenstates of conformal field theories. When the interval is short, using the operator product expansion of twist operators, we obtain a universal form for the leading order in $\ell$ of the trace distance. We compute the trace distances among the reduced density matrices of several low lying states in two-dimensional free massless boson and fermion theories. We compare our analytic conformal results with numerical calculations in XX and Ising spin chains finding perfect agreement.

hep-th↗

arXiv2018

Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

Conformal field theories in curved backgrounds have been used to describe inhomogeneous one-dimensional systems, such as quantum gases in trapping potentials and non-equilibrium spin chains. This approach provided, in a elegant and simple fashion, non-trivial analytic predictions for quantities, such as the entanglement entropy, that are not accessible through other methods. Here, we generalise this approach to low-lying excited states, focusing on the entanglement and relative entropies in an inhomogeneous free-fermionic system. Our most important finding is that the universal scaling function characterising these entanglement measurements is the same as the one for homogeneous systems, but expressed in terms of a different variable. This new scaling variable is a non-trivial function of the subsystem length and system's inhomogeneity that is easily written in terms of the curved metric. We test our predictions against exact numerical calculations in the free Fermi gas trapped by a harmonic potential, finding perfect agreement.

cond-mat.stat-mech↗

arXiv2018

Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks

We reconsider the computation of the entanglement entropy of two disjoint intervals in a (1+1) dimensional conformal field theory by conformal block expansion of the 4-point correlation function of twist fields. We show that accurate results may be obtained by taking into account several terms in the operator product expansion (OPE) of twist fields and by iterating the Zamolodchikov recursion formula for each conformal block. We perform a detailed analysis for the Ising conformal field theory and for the free compactified boson. Each term in the conformal block expansion can be easily analytically continued and so this approach also provides a good approximation for the von Neumann entropy.

cond-mat.stat-mech↗

arXiv2018

Unusual area-law violation in random inhomogeneous systems

The discovery of novel entanglement patterns in quantum many-body systems is a prominent research direction in contemporary physics. Here we provide the example of a spin chain with random and inhomogeneous couplings that in the ground state exhibits a very unusual area law violation. In the clean limit, i.e., without disorder, the model is the rainbow chain and has volume law entanglement. We show that, in the presence of disorder, the entanglement entropy exhibits a power-law growth with the subsystem size, with an exponent 1/2. By employing the Strong Disorder Renormalization Group (SDRG) framework, we show that this exponent is related to the survival probability of certain random walks. The ground state of the model exhibits extended regions of short-range singlets (that we term ``bubble'' regions) as well as rare long range singlet (``rainbow'' regions). Crucially, while the probability of extended rainbow regions decays exponentially with their size, that of the bubble regions is power law. We provide strong numerical evidence for the correctness of SDRG results by exploiting the free-fermion solution of the model. Finally, we investigate the role of interactions by considering the random inhomogeneous XXZ spin chain. Within the SDRG framework and in the strong inhomogeneous limit, we show that the above area-law violation takes place only at the free-fermion point of phase diagram. This point divides two extended regions, which exhibit volume-law and area-law entanglement, respectively.

cond-mat.str-el↗

arXiv2017

Thermally correlated states in Loop Quantum Gravity

We study a class of loop-quantum-gravity states characterized by (ultra-local) thermal correlations that reproduce some features of the ultraviolet structure of the perturbative quantum field theory vacuum. In particular, they satisfy an analog of the Bisognano-Wichmann theorem. These states are peaked on the intrinsic geometry and admit a semiclassical interpretation. We study how the correlations extend on the spin-network beyond the ultra local limit.

gr-qc↗

arXiv2016

Relative Entanglement Entropies in 1+1-dimensional conformal field theories

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy $S(ρ_1 \| ρ_0)$ between two given reduced density matrices $ρ_1$ and $ρ_0$ of a quantum field theory, we employ the replica trick which relies on the path integral representation of ${\rm Tr} ( ρ_1 ρ_0^{n-1} )$ and define a set of Rényi relative entropies $S_n(ρ_1 \| ρ_0)$. We compute these quantities for integer values of the parameter $n$ and derive via the replica limit, the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator $i \partialϕ$, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.

hep-th↗

arXiv2016

Negativity spectrum of one-dimensional conformal field theories

The partial transpose $ρ_A^{T_2}$ of the reduced density matrix $ρ_A$ is the key object to quantify the entanglement in mixed states, in particular through the presence of negative eigenvalues in its spectrum. Here we derive analytically the distribution of the eigenvalues of $ρ_A^{T_2}$, that we dub negativity spectrum, in the ground sate of gapless one-dimensional systems described by a Conformal Field Theory (CFT), focusing on the case of two adjacent intervals. We show that the negativity spectrum is universal and depends only on the central charge of the CFT, similarly to the entanglement spectrum. The precise form of the negativity spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. We also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of $ρ_A^{T_2}$. We check our results against DMRG simulations for the critical Ising and Heisenberg chains, and against exact results for the harmonic chain, finding good agreement for the spectrum, but showing that the smallest eigenvalue is affected by very large scaling corrections.

cond-mat.stat-mech↗

arXiv2016

The entanglement negativity in random spin chains

We investigate the logarithmic negativity in strongly-disordered spin chains in the random-singlet phase. We focus on the spin-1/2 random Heisenberg chain and the random XX chain. We find that for two arbitrary intervals the disorder-averaged negativity and the mutual information are proportional to the number of singlets shared between the two intervals. Using the strong-disorder renormalization group (SDRG), we prove that the negativity of two adjacent intervals grows logarithmically with the intervals length. In particular, the scaling behavior is the same as in conformal field theory, but with a different prefactor. For two disjoint intervals the negativity is given by a universal simple function of the cross ratio, reflecting scale invariance. As a function of the distance of the two intervals, the negativity decays algebraically in contrast with the exponential behavior in clean models. We confirm our predictions using a numerical implementation of the SDRG method. Finally, we also implement DMRG simulations for the negativity in open spin chains. The chains accessible in the presence of strong disorder are not sufficiently long to provide a reliable confirmation of the SDRG results.

cond-mat.str-el↗