There are no clean samples in nature. Therefore, when we come to discuss the entanglement properties of quantum materials, the effects of disorder must be taken into account. This question is of particular interest for highly entangled phases, such as quantum spin liquids, which lie outside the Landau paradigm for classifying phases of matter. In this work, we explore what experimentally accessible measures, in the form of concurrence, residual tangle, and quantum Fisher information can teach us about the entanglement in the presence of disorder. As a representative example, we consider the Tomonaga-Luttinger liquids (TLL) and disorder-driven random singlet (RS) phases found in antiferromagnetic quantum spin chains. Using quantum Fisher information and residual tangle, we demonstrate that both TLL and RS phases exhibit multipartite entanglement. In the case of the RS phase, we attribute this to entanglement localized below a crossover length scale. We further show that the order of disorder average matters in calculating measures like concurrence and that this can lead to false inferences when interpreting experiments. Nonetheless, correctly interpreted, these witnesses provide useful information about the effects of disorder. We explore how information about the central charge of the TLL can be extracted from the low-temperature behavior of concurrence and conjecture that this analysis can be extended to the effective central charge of the RS phase. Finally, we establish how RS and TLL phases can be distinguished through the growth of multipartite entanglement, as witnessed by the equal-time structure factor. These results establish that, used carefully, experiments based on entanglement witnesses can provide important information about quantum spin systems in the presence of disorder.
There are no clean samples in nature. Therefore, when we come to discuss the entanglement properties of quantum materials, the effects of disorder must be taken into account. This question is of particular interest for highly entangled phases, such as quantum spin liquids, which lie outside the Landau paradigm for classifying phases of matter. In this work, we explore what experimentally accessible measures, in the form of concurrence, residual tangle, and quantum Fisher information can teach us about the entanglement in the presence of disorder. As a representative example, we consider the Tomonaga-Luttinger liquids (TLL) and disorder-driven random singlet (RS) phases found in antiferromagnetic quantum spin chains. Using quantum Fisher information and residual tangle, we demonstrate that both TLL and RS phases exhibit multipartite entanglement. In the case of the RS phase, we attribute this to entanglement localized below a crossover length scale. We further show that the order of disorder average matters in calculating measures like concurrence and that this can lead to false inferences when interpreting experiments. Nonetheless, correctly interpreted, these witnesses provide useful information about the effects of disorder. We explore how information about the central charge of the TLL can be extracted from the low-temperature behavior of concurrence and conjecture that this analysis can be extended to the effective central charge of the RS phase. Finally, we establish how RS and TLL phases can be distinguished through the growth of multipartite entanglement, as witnessed by the equal-time structure factor. These results establish that, used carefully, experiments based on entanglement witnesses can provide important information about quantum spin systems in the presence of disorder.
In 1935, as part of a broader debate about the interpretation of quantum mechanics, Schrödinger introduced the term “verschränkung” (entanglement) to describe the way in which mutually-exclusive experimental outcomes could be woven together in a single quantum mechanical state. Ninety years later, entanglement is no longer a philosophical curiosity but a technological resource, central to the pursuit of quantum advantage sought in the entanglement of a few hundred qubits. In the context of condensed matter physics, the most celebrated example of entanglement arises in quantum spin liquids (QSL), exotic magnetic phases where, at comparable temperatures, billions of spins can be entangled. The sharpest distinctions between conventional magnetic phases, QSL, and states driven by disorder, are expected to be found in their entanglement properties. Yet exactly what these distinctions are, or how they might be used in experiments, remains poorly understood. In this thesis, we explore how material-based imperfections such as bond disorder and finite temperature affect experimentally accessible entanglement-based diagnostics in models of quantum spin liquids (QSLs). We ask: (1) Can spin liquid states be reliably distinguished from disorder-driven phases using such diagnostics? (2) What can these experimentally accessible entanglement witnesses reveal about the microscopic structure of entanglement in QSLs? In particular, we examine the robustness of these quantum correlations under realistic conditions where thermal fluctuations are unavoidable. In addressing (1), we find that measures like concurrence become unreliable in disordered systems. In contrast, the finite temperature scaling of multipartite diagnostics such as the quantum Fisher information (QFI) provides a more effective means of distinguishing QSLs from competing disorder-dominated phases. For (2), we investigate how the finite temperature behavior of multipartite entanglement measures obtained from semidefinite programming (SDP) methods supports a two step picture for how a high temperature paramagnet crosses over into a QSL ground state. In particular, these results provide a conceptual framework that places quantitative bounds on the temperature window over which candidate materials can be expected to exhibit the characteristic physics of the QSL ground state.
At the theoretical level, quantum spin liquids are distinguished from other phases of matter by their entanglement properties. However, since the usual measure of entanglement, entanglement entropy, cannot accessed in experiment, indentifying quantum spin liquids in candidate materials remains an acute problem. Here we show other, experimentally-accessible, measures of entanglement can be used to distinguish a quantum spin liquid from a competing disorder-driven "random singlet" phase, in a model of a disordered antiferromagnet on a triangular lattice. The application of these results to the triangular-lattice systems YbZnGaO4, YbZn2GaO5 and KYbSe2 is discussed.
The lack of methods to experimentally detect and quantify entanglement in quantum matter impedes our ability to identify materials hosting highly entangled phases, such as quantum spin liquids. We thus investigate the feasibility of using inelastic neutron scattering (INS) to implement a model-independent measurement protocol for entanglement based on three entanglement witnesses: one-tangle, two-tangle, and quantum Fisher information (QFI). We perform high-resolution INS measurements on Cs_{2}CoCl_{4}, a close realization of the S=1/2 transverse-field XXZ spin chain, where we can control entanglement using the magnetic field, and compare with density-matrix renormalization group calculations for validation. The three witnesses allow us to infer entanglement properties and make deductions about the quantum state in the material. We find QFI to be a particularly robust experimental probe of entanglement, whereas the one and two-tangles require more careful analysis. Our results lay the foundation for a general entanglement detection protocol for quantum spin systems.
Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\ensuremath{\alpha}$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\ensuremath{\alpha}>\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\ensuremath{\alpha}$, corresponding to a critical behavior with an effective central charge c=ln(2), independent of $\ensuremath{\alpha}$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\ensuremath{\alpha}$. For a wide range of partition size l, the EE fits a critical behavior with a central charge close to c=1, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\ensuremath{\alpha}=2$. Only for small $l\ensuremath{\ll}L$, in a range which increases with the number of spins N, we find deviations which are rather consistent with the strong disorder fixed point central charge c=ln(2). Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\ensuremath{\alpha}<2$, whereas for larger $\ensuremath{\alpha}$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.
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 g 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 K 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 g-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-21 XXZ chain subject to a site-resolved measurement. Possible experimental realization in ultracold gases, which requires no postselections, is discussed.
Multipartite entanglement in the 1-D spin-21 Heisenberg Antiferromagnet
Multipartite entanglement refers to the simultaneous entanglement between multiple subsystems of a many-body quantum system. While multipartite entanglement can be difficult to quantify analytically, it is known that it can be witnessed through the Quantum Fisher information (QFI), a quantity that can also be related to dynamical Kubo response functions. In this work, we first show that the finite temperature QFI can generally be expressed in terms of a static structure factor of the system, plus a correction that vanishes as T→0. We argue that this implies that the static structure factor witnesses multipartite entanglement near quantum critical points at temperatures below a characteristic energy scale that is determined by universal properties, up to a non-universal amplitude. Therefore, in systems with a known static structure factor, we can deduce finite temperature scaling of multipartite entanglement and low temperature entanglement depth without knowledge of the full dynamical response function of the system. This is particularly useful to study 1D quantum critical systems in which sub-power-law divergences can dominate entanglement growth, where the conventional scaling theory of the QFI breaks down. The 1D spin-21 antiferromagnetic Heisenberg model is an important example of such a system, and we show that multipartite entanglement in the Heisenberg chain diverges non-trivially as ∼log(1/T)3/2. We verify these predictions with calculations of the QFI using conformal field theory and matrix product state simulations. Finally we discuss the implications of our results for experiments to probe entanglement in quantum materials, comparing to neutron scattering data in KCuF3, a material well-described by the Heisenberg chain.