Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

We study the return probability and its imaginary ($\tau$) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy $|\Delta|< 1$. We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to $\tau^2$, which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region $|\Delta|<1$ the decay for large times $t$ is nowhere continuous as a function of anisotropy: it is either gaussian at root of unity or exponential otherwise. As an aside, we also determine that the front moves as $x_{\rm f}(t)=t\sqrt{1-\Delta^2}$, by analytic continuation of known arctic curves in the six vertex model. Exactly at $|\Delta|=1$, we find the return probability decays as $e^{-\zeta(3/2) \sqrt{t/\pi}}t^{1/2}O(1)$. It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench.Comment: 33 pages, 8 figures. v2: typos fixed, references added. v3: minor change

Full counting statistics in the Haldane-Shastry chain

We present analytical and numerical results regarding the magnetization full counting statistics (FCS) of a subsystem in the ground-state of the Haldane-Shastry chain. Exact Pfaffian expressions are derived for the cumulant generating function, as well as any observable diagonal in the spin basis. In the limit of large systems, the scaling of the FCS is found to be in agreement with the Luttinger liquid theory. The same techniques are also applied to inhomogeneous deformations of the chain. This introduces a certain amount of disorder in the system; however we show numerically that this is not sufficient to flow to the random singlet phase, that corresponds to $XXZ$ chains with uncorrelated bond disorder.Comment: 15 pages, 7 figure

Phase transition in the R\'enyi-Shannon entropy of Luttinger liquids

The R\'enyi-Shannon entropy associated to critical quantum spins chain with central charge $c=1$ is shown to have a phase transition at some value $n_c$ of the R\'enyi parameter $n$ which depends on the Luttinger parameter (or compactification radius R). Using a new replica-free formulation, the entropy is expressed as a combination of single-sheet partition functions evaluated at $n-$ dependent values of the stiffness. The transition occurs when a vertex operator becomes relevant at the boundary. Our numerical results (exact diagonalizations for the XXZ and $J_1-J_2$ models) are in agreement with the analytical predictions: above $n_c=4/R^2$ the subleading and universal contribution to the entropy is $\ln(L)(R^2-1)/(4n-4)$ for open chains, and $\ln(R)/(1-n)$ for periodic ones (R=1 at the free fermion point). The replica approach used in previous works fails to predict this transition and turns out to be correct only for $n<n_c$. From the point of view of two-dimensional Rokhsar-Kivelson states, the transition reveals a rich structure in the entanglement spectra.Comment: 4 pages, 3 figure

R\'enyi entropy of a line in two-dimensional Ising models

We consider the two-dimensional (2d) Ising model on a infinitely long cylinder and study the probabilities $p_i$ to observe a given spin configuration $i$ along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave-functions. We analyze the subleading constant to the R\'enyi entropy $R_n=1/(1-n) \ln (\sum_i p_i^n)$ and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a step-like fashion as a function of $n$, with a discontinuity at the Shannon point $n=1$. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the R\'enyi parameter are of special interest: $n=1/2$ and $n=\infty$ are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions respectively.Comment: 8 pages, 6 figures, 2 tables. To be submitted to Physical Review

A R\'enyi entropy perspective on topological order in classical toric code models

Concepts of information theory are increasingly used to characterize collective phenomena in condensed matter systems, such as the use of entanglement entropies to identify emergent topological order in interacting quantum many-body systems. Here we employ classical variants of these concepts, in particular R\'enyi entropies and their associated mutual information, to identify topological order in classical systems. Like for their quantum counterparts, the presence of topological order can be identified in such classical systems via a universal, subleading contribution to the prevalent volume and boundary laws of the classical R\'enyi entropies. We demonstrate that an additional subleading $O(1)$ contribution generically arises for all R\'enyi entropies $S^{(n)}$ with $n \geq 2$ when driving the system towards a phase transition, e.g. into a conventionally ordered phase. This additional subleading term, which we dub connectivity contribution, tracks back to partial subsystem ordering and is proportional to the number of connected parts in a given bipartition. Notably, the Levin-Wen summation scheme -- typically used to extract the topological contribution to the R\'enyi entropies -- does not fully eliminate this additional connectivity contribution in this classical context. This indicates that the distillation of topological order from R\'enyi entropies requires an additional level of scrutiny to distinguish topological from non-topological $O(1)$ contributions. This is also the case for quantum systems, for which we discuss which entropies are sensitive to these connectivity contributions. We showcase these findings by extensive numerical simulations of a classical variant of the toric code model, for which we study the stability of topological order in the presence of a magnetic field and at finite temperatures from a R\'enyi entropy perspective.Comment: 17 pages, 19 figure

Exact time evolution formulae in the XXZ spin chain with domain wall initial state

We study the time evolution of the spin-1/2 XXZ chain initialized in a domain wall state, where all spins to the left of the origin are up, all spins to its right are down. The focus is on exact formulae, which hold for arbitrary finite (real or imaginary) time. In particular, we compute the amplitudes corresponding to the process where all but $k$ spins come back to their initial orientation, as a $k-$fold contour integral. These results are obtained using a correspondence with the six vertex model, and taking a somewhat complicated Hamiltonian/Trotter-type limit. Several simple applications are studied and also discussed in a broader context.Comment: 44 pages, 5 figure

Emptiness formation probability, Toeplitz determinants, and conformal field theory

We revisit the study of the emptiness formation probability, the probability of forming a sequence of $\ell$ spins with the same ferromagnetic orientation in the ground-state of a quantum spin chain. We focus on two different examples, exhibiting strikingly different behavior: the XXZ and Ising chains. One has a conserved number of particles, the other does not. In the latter we show that the sequence of fixed spins can be viewed as an additional boundary in imaginary time. We then use conformal field theory (CFT) techniques to derive all universal terms in its scaling, and provide checks in free fermionic systems. These are based on numerical simulations or, when possible, mathematical results on the asymptotic behavior of Toeplitz and Toeplitz+Hankel determinants. A perturbed CFT analysis uncovers an interesting $\ell^{-1}\log \ell$ correction, that also appears in the closely related spin full counting statistics. The XXZ case turns out to be more challenging, as scale invariance is broken. We use a simple qualitative picture in which the ferromagnetic sequence of spins freezes all degrees of freedom inside of a certain "arctic" region, that we determine numerically. We also provide numerical evidence for the existence of universal logarithmic terms, generated by the massless field theory living outside of the arctic region.Comment: 34 pages, 11 figures. v2: additional discussions, typos fixed. To appear in J. Stat. Mec

R\'enyi entanglement entropies in quantum dimer models : from criticality to topological order

Thanks to Pfaffian techniques, we study the R\'enyi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar-Kivelson wave functions constructed from a dimer model on the triangular lattice. By including a fugacity $t$ on some suitable bonds, one interpolates between the triangular lattice (t=1) and the square lattice (t=0). The wave function is known to be a massive $\mathbb Z_2$ topological liquid for $t>0$ whereas it is a gapless critical state at t=0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder, and the disk-like setup proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)]. In the cylinder case, the entropies contain an extensive term -- proportional to the length of the boundary -- and a universal sub-leading constant $s_n(t)$. Fitting these cylinder data (up to a perimeter of L=32 sites) provides $s_n$ with a very high numerical accuracy ($10^{-9}$ at t=1 and $10^{-6}$ at $t=0.5$). In the topological $\mathbb{Z}_2$ liquid phase we find $s_n(t>0)=-\ln 2$, independent of the fugacity $t$ and the R\'enyi parameter $n$. At t=0 we recover a previously known result, $s_n(t=0)=-(1/2)\ln(n)/(n-1)$ for $n1$. In the disk-like geometry -- designed to get rid of the boundary contributions -- we find an entropy $s^{\rm KP}_n(t>0)=-\ln 2$ in the whole massive phase whatever $n>0$, in agreement with the result of Flammia {\it et al.} [Phys. Rev. Lett. 103, 261601 (2009)]. Some results for the gapless limit $R^{\rm KP}_n(t\to 0)$ are discussed.Comment: 33 pages, 17 figures, minor correction

Inhomogeneous quenches in a fermionic chain: exact results

We consider the non-equilibrium physics induced by joining together two tight binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be the same or different. We present a number of exact results for the correlation functions in the non-interacting case. We present a short-time expansion, which can sometimes be fully resummed, and which reproduces the so-called `light cone' effect or wavefront behavior of the correlators. For large times, we show how all interesting physical regimes may be obtained by stationary phase approximation techniques. In particular, we derive semiclassical formulas in the case when both time and positions are large, and show that these are exact in the thermodynamic limit. We present subleading corrections to the large-time behavior, including the corrections near the edges of the wavefront. We also provide results for the return probability or Loschmidt echo. In the maximally inhomogeneous limit, we prove that it is exactly gaussian at all times. The effects of interactions on the Loschmidt echo are also discussed.Comment: 5 pages+14 pages supplementary material+9 figure

Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidt echo, and light-cone effects

We study a particular type of local quench in a generic quantum critical one-dimensional system, using conformal field theory (CFT) techniques, and providing numerical checks of the results in free fermion systems. The system is initially cut into two subsystems $A$ and $B$ which are glued together at time $t=0$. We study the entanglement entropy (EE) between the two parts $A$ and $B$, using previous results by Calabrese and Cardy, and further extending them. We also study in detail the (logarithmic) Loschmidt echo (LLE). For finite-size systems both quantities turn out to be (almost) periodic in the scaling limit, and exhibit striking light-cone effects. While these two quantities behave similarly immediately after the quench---namely as $c/3 \log t$ for the EE and $c/4 \log t$ for the LLE---, we observe some discrepancy once the excitations emitted by the quench bounce on the boundary and evolve within the same subsystem $A$ (or $B$). The decay of the EE is then non-universal, as noticed by Eisler and Peschel. On the contrary, we find that the evolution of the LLE is less sensitive than the EE to non-universal details of the model, and is still accurately described by our CFT prediction. To further probe these light-cone effects, we also introduce a variant of the Loschmidt echo specifically constructed to detect the excitations emitted just after the quench.Comment: 28 pages, 20 figures. v3: published versio