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Replica manipulation of the ground state in one-dimensional quantum spin systems

裕太 熊野

Published Mar 24, 2015• 1 citation
Publisher

Abstract

In the last decade it has become evident that information-theoretic quantities, which quantify a quantum entanglement encoded in a quantum state, are extremely useful to analyze a state of mane-body systems.Celebrated examples are the entanglement entropy and the Shannon entropy, which extract universal information of the underlying field theory of the system.Furthermore, a significant concept, the entanglement spectrum, has been proposed recently.It contains more complete information than the entanglement entropy.They have achieved a considerable success in characterizing exotic quantum phases that are beyond conventional descriptions.Inspired by these developments, we propose a new state, which we name a Rényified state, as follows: for a given quantum state |Ψ⟩ = ∑ i ψ i |i⟩ and the basis states |i⟩, the Rényified state is defined by raising the wave-function coefficients ψ i to the power of n (and normalized).The Shannon entropy can be obtained from the Rényified state.Moreover, the latter contains more information than the former.We study the Rényified state starting from a Tomonaga-Luttinger liquids (TLL), which is an important universality class of one-dimensional quantum systems.A TLL is characterized by a TLL parameter K, and described by the free boson field theory with a central charge c = 1 in the conformal field theory (CFT) context.We will show, using numerical calculations, that the Rényified state behaves as a TLL described by a modified TLL parameter K = K/n for small n.For larger n, the Rényified state behaves in a different way: the longitudinal correlation has long-range order while the transverse one remains algebraic.Based on the numerical results, we apply a replica field theory formulation to an analysis of the Rényified state.It suggests that the Rényified state is also a TLL described by a modified TLL parameter K = K/n, when n < 4K.The field theoretical analysis indicates that the TLL description breaks down at n = 4K, which is related to a phase transition in the Rényi-Shannon entropy.Beyond the transition, n > 4K, it suggests that the Rényified state is no longer a TLL since the longitudinal correlation has long-range order while the transverse one remains i algebraic.This exceptional behavior is unlikely to be realized in the ground state of a Hamiltonian with only short-range interactions.This indicates that the Rényified state beyond the transition belongs to a new class of exotic quantum phase.We explain an origin of the exotic behavior by constructing a particular conformal invariant boundary state of a two-component massless free boson.The relationship between the Rényified state and the Rényi-Shannon entropy is also elucidated in boundary CFT formalism.

DOI
10.15083/00008056

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In the last decade it has become evident that information-theoretic quantities, which quantify a quantum entanglement encoded in a quantum state, are extremely useful to analyze a state of mane-body systems.Celebrated examples are the entanglement entropy and the Shannon entropy, which extract universal information of the underlying field theory of the system.Furthermore, a significant concept, the entanglement spectrum, has been proposed recently.It contains more complete information than the entanglement entropy.They have achieved a considerable success in characterizing exotic quantum phases that are beyond conventional descriptions.Inspired by these developments, we propose a new state, which we name a Rényified state, as follows: for a given quantum state |Ψ⟩ = ∑ i ψ i |i⟩ and the basis states |i⟩, the Rényified state is defined by raising the wave-function coefficients ψ i to the power of n (and normalized).The Shannon entropy can be obtained from the Rényified state.Moreover, the latter contains more information than the former.We study the Rényified state starting from a Tomonaga-Luttinger liquids (TLL), which is an important universality class of one-dimensional quantum systems.A TLL is characterized by a TLL parameter K, and described by the free boson field theory with a central charge c = 1 in the conformal field theory (CFT) context.We will show, using numerical calculations, that the Rényified state behaves as a TLL described by a modified TLL parameter K = K/n for small n.For larger n, the Rényified state behaves in a different way: the longitudinal correlation has long-range order while the transverse one remains algebraic.Based on the numerical results, we apply a replica field theory formulation to an analysis of the Rényified state.It suggests that the Rényified state is also a TLL described by a modified TLL parameter K = K/n, when n < 4K.The field theoretical analysis indicates that the TLL description breaks down at n = 4K, which is related to a phase transition in the Rényi-Shannon entropy.Beyond the transition, n > 4K, it suggests that the Rényified state is no longer a TLL since the longitudinal correlation has long-range order while the transverse one remains i algebraic.This exceptional behavior is unlikely to be realized in the ground state of a Hamiltonian with only short-range interactions.This indicates that the Rényified state beyond the transition belongs to a new class of exotic quantum phase.We explain an origin of the exotic behavior by constructing a particular conformal invariant boundary state of a two-component massless free boson.The relationship between the Rényified state and the Rényi-Shannon entropy is also elucidated in boundary CFT formalism.
The R\'enyi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge c=1c=1c=1, we show that it exhibits a phase transition at some value nc{n}_{c}nc​ of the R\'enyi parameter nnn which depends on the Luttinger parameter RRR. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and ${J}_{1}\ensuremath{-}{J}_{2}$ spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for n<ncn<{n}_{c}n<nc​. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.
Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, captures aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer Rényi entropy has revealed universal features of nonstabilizerness qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic Rényi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. We show that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations. The MRE reveals common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary transition through bulk-induced boundary renormalization-group flows. As a concrete example, we present a CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters K=1/3K=1/3K=1/3 and K=3K=3K=3. Our field-theoretical predictions are confirmed by numerical calculations. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.
Entanglement in quantum many-body systems can exhibit universal phenomena governed by long-distance properties. We study universality and phase transitions of the entanglement inherent to open many-body systems, namely, the entanglement between a system of interest and its environment. Specifically, we consider the Tomonaga-Luttinger liquid (TLL) under a local measurement and analyze its unconditioned nonunitary evolution, where the measurement outcomes are averaged over. We quantify the system-environment entanglement by the R\'enyi entropy of the post-measurement density matrix, whose size-independent term encodes the universal low-energy physics. We develop a field-theoretical description to relate the universal term to the effective ground-state degeneracy known as the ggg function in a boundary conformal field theory, and use the renormalization group method to determine its value. We show that the universal contribution is determined by the TLL parameter KKK and can exhibit singularity signifying an entanglement phase transition. Surprisingly, in certain cases the size-independent contribution can increase as a function of the measurement strength in contrast to what is na\"{\i}vely expected from the ggg-theorem. We argue that this unconventional behavior could be attributed to the dangerously irrelevant term which has been found in studies of the resistively shunted Josephson junction. We also check these results by numerical calculations in the spin-12\frac{1}{2}21​ XXZ chain subject to a site-resolved measurement. Possible experimental realization in ultracold gases, which requires no postselections, is discussed.
We consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the R\'enyi entanglement entropy Sn{S}_{n}Sn​ between the two chains. Here the entanglement cut is introduced between the chains, not along the perpendicular direction, as has been done in previous studies of one-dimensional systems. The limit $n\ensuremath{\rightarrow}1$ corresponds to the von Neumann entanglement entropy. The system is effectively described by two-component bosonic field theory with different TLL parameters in the symmetric and antisymmetric channels as far as the coupled system remains in a gapless phase. We argue that in this system, Sn{S}_{n}Sn​ is a linear function of the length of the chains (boundary law) followed by a universal subleading constant ${\ensuremath{\gamma}}_{n}$ determined by the ratio of the two TLL parameters. The formulas of ${\ensuremath{\gamma}}_{n}$ for integer $n\ensuremath{\geqslant}2$ are derived using (a) ground-state wave functionals of TLL's and (b) boundary conformal field theory, which lead to the same result. These predictions are checked in a numerical diagonalization analysis of a hard-core bosonic model on a ladder. Although the analytic continuation of ${\ensuremath{\gamma}}_{n}$ to $n\ensuremath{\rightarrow}1$ turns out to be a difficult problem, our numerical result suggests that the subleading constant in the von Neumann entropy is also universal. Our results may provide useful characterization of inherently anisotropic quantum phases such as the sliding Luttinger liquid phase via qualitatively different behaviors of the entanglement entropy with the entanglement partitions along different directions.
We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.
The eect of electronic many-body interactions in one-dimensional (1D) systems is drastically enhanced compared to higher dimensions due to the restricted phasespace, often leading to strong electronic correlations. It can be shown that due to a breakdown of the Fermi liquid theory, the ground state at low temperatures becomes a Tomonaga-Luttinger liquid (TLL). In a TLL, the strength of electronic correlations is characterized by the Luttinger parameter K, ranging between K = 0 and K = 1, where K = 1 describes non-interacting electrons. Various systems, such as single-walled carbon nanotubes, 1D atomic metal chains, and MoS 2 mirror twin boundaries, have been reported to host TLLs, in which spin-degeneracy is intact.