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Journal of Physics Conference Series

Linear higher derivative field theories, entanglement entropy and quantum phase transition

Suman Ghosh, S. Shankaranarayanan

Published Dec 13, 2012• 0 citations
Publisher

Abstract

Quantum systems described by linear, higher order spatial derivatives of the field exhibit a new kind of phase transition as the coupling strength of the higher order terms varies. This essentially happens because a fundamental change in the ratio of ground state energy eigenvalues, between linear and non-linear regime, takes place which leads to excitation of higher modes. The new phase has inverse relation with area.

DOI
10.1088/1742-6596/405/1/012016

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Quantum systems described by linear, higher order spatial derivatives of the field exhibit a new kind of phase transition as the coupling strength of the higher order terms varies. This essentially happens because a fundamental change in the ratio of ground state energy eigenvalues, between linear and non-linear regime, takes place which leads to excitation of higher modes. The new phase has inverse relation with area.
We show that the variations of the ground state entanglement in linear, higher spatial derivative field theories at zero temperature have signatures of phase transition. Around the critical point, when the dispersion relation changes from linear to nonlinear, there is a fundamental change in the reduced density matrix leading to a change in the scaling of entanglement entropy. We suggest possible explanations involving both kinematical and dynamical effects. We discuss the implication of our work for 2D condensed matter systems, black-hole entropy, and models of quantum gravity.
In a bipartite set-up, the vacuum state of a free Bosonic scalar field is entangled in real space and satisfies the area-law- entanglement entropy scales linearly with area of the boundary between the two partitions. In this work, we show that the area law is violated in two spatial dimensional model Hamiltonian having dynamical critical exponent z = 3. The model physically corresponds to next-to-next-to-next nearest neighbour coupling terms on a lattice. The result reported here is the first of its kind of violation of area law in Bosonic systems in higher dimensions and signals the evidence of a quantum phase transition. We provide evidence for quantum phase transition both numerically and analytically using quantum Information tools like entanglement spectra, quantum fidelity, and gap in the energy spectra. We identify the cause for this transition due to the accumulation of large number of angular zero modes around the critical point which catalyses the change in the ground state wave function due to the next-to-next-to-next nearest neighbor coupling. Lastly, using Hubbard-Stratanovich transformation, we show that the effective Bosonic Hamiltonian can be obtained from an interacting fermionic theory and provide possible implications for condensed matter systems.
We study the new class of solutions in linearized open string field theory (OSFT) involving higher-spin modes. Unlike the elementary OSFT solutions (on-shell vertex operators) that, acting on a vacuum, define wavefunctions of pure states (e.g. a tachyon), the solutions that we describe correspond to the reduced density matrices which eigenvalues describe the entanglement between higher-spin modes with different spin values. We compute the entanglement entropy on these OSFT solutions, and the answer is expressed in terms of converging series in inverse weighted partition numbers. In the case of DDD-dimensional bosonic string theory, the entanglement entropy of spin 111 subsystem and the system of all the spin values is given by Dlog⁡λ0+Dλ0∑N=3∞∣β(N)∣λ(N)log⁡(λ(N)∣β(N)∣)D{\log{λ_0}}+{D\over{λ_0}}\sum_{N=3}^\infty{{|β(N)|}\over{λ(N)}} {\log{({{λ(N)}\over{|β(N)|}})}}Dlogλ0​+λ0​D​∑N=3∞​λ(N)∣β(N)∣​log(∣β(N)∣λ(N)​), where λ(N)λ(N)λ(N) is the weighted number of partitions of NNN, β(N)=(N−1)ζ(3)−ζ(2)(N−1)4β(N)={{(N-1)ζ(3)-ζ(2)}\over{(N-1)^4}}β(N)=(N−1)4(N−1)ζ(3)−ζ(2)​ and λ0=∑N=1∞β(N)λ(N)λ_0=\sum_{N=1}^{\infty}{{β(N)}\over{λ(N)}}λ0​=∑N=1∞​λ(N)β(N)​ (ζζζ is Riemann's zeta-function). The first term, Dlog⁡λ0D{\log{λ_0}}Dlogλ0​, represents the entanglement swapping between string vacuum and string excitations. We generalize this result to obtain the entanglement for a subsystem of a given spin sss in a given space-time dimension. We also discuss how open string field theory may be used to study the entanglement of systems other than higher spin excitations in string theory.
We study the momentum space entanglement between different energy modes of interacting scalar fields propagating in general (D+1D+1D+1)-dimensional flat space-time. As opposed to some of the recent works [V. Balasubramanian et al., Phys. Rev. D 86, 045014 (2012)], we use a Lorentz invariant normalized ground state to obtain the momentum space entanglement entropy. We show that the Lorenz invariant definition removes the spurious power-law behavior obtained in the earlier works. More specifically, we show that the cubic interacting scalar field in (1+11+11+1) dimensions leads to logarithmic divergence of the entanglement entropy and is consistent with the results from real space entanglement calculations. We study the effects of the introduction of the Lorentz violating higher derivative terms in the presence of a nonlinear self-interacting scalar field potential and show that the divergence structure of the entanglement entropy is improved in the presence of spatial higher derivative terms.
The entanglement entropy measures quantum correlations and it can be seen as the uncertainty on a quantum state. In one spatial dimension, the entanglement entropy scales as the boundary that divides two subsystems, so an area law has been proposed. However, the entanglement entropy diverges logarithmically at conformally invariant critical points, so the area law does not hold. The purpose of the work is to find a way to get more information about a critical state. The ground state of the Heisenberg XXZ model at criticality is analyzed by means of critical Ising eigenstates. Two ways of analysis are followed: a basis made of Ising eigenstates is built up and used to represent the XXZ ground state, then the Shannon entropy in the new basis is computed; the adiabatic evolution from the Ising ground state to the XXZ ground state. The result is that the Shannon entropy in the Ising basis scales linearly with the length of the system, while a phase transition is encountered during the adiabatic evolution. The conclusion is that there is no net gain in information after the procedure and possibly it is related to the fact the two systems stand in different phases.
Quantum fluctuations are usually regarded as a background effect that perturbs classical order. In this work we propose that quantum fluctuations themselves can undergo a phase transition, a phenomenon we term the Quantum Fluctuation Phase Transition (QFPT). We develop a theoretical framework that treats the amplitude of quantum fluctuations as an order parameter, allowing a direct mapping between the fluctuation strength and the free‑energy landscape. Using a transverse‑field Ising model with tunable long‑range couplings, we derive a mean‑field free energy that exhibits a discontinuity in the fluctuation amplitude at a critical field strength. We further analyze the critical scaling, demonstrating that the QFPT belongs to a new universality class distinct from conventional quantum critical points. The theory predicts observable signatures in the low‑temperature heat capacity, dynamic structure factor, and entanglement entropy, and suggests that extreme‑cold atom or solid‑state systems with controllable interaction ranges could realize QFPTs. This novel mechanism for quantum phase change offers a route to discovering new quantum phases and could have implications for quantum technologies that rely on fluctuation‑driven properties.