Unbroken Phase Research Articles

Truncation-dependent PT phase transition for the edge states of a two-dimensional non-Hermitian system

We consider a bulk system supporting parity and time reversal ($PT$) symmetry, and investigate how the $PT$ phase transition of edge states is influenced by different truncations of the system. As an example, we study a two-dimensional $PT$-symmetric Su--Schrieffer--Heeger lattice with non-Hermitian onsite potentials. We find that when the truncation preserves certain symmetries of the bulk lattice, the edge states can remain in the $PT$-unbroken phase when the non-Hermitian onsite potentials are less than a nonzero critical value. On the other hand, when the truncation removes such symmetries, edge states with complex eigen-energies are observed for infinitesimal non-Hermitian onsite potentials. We develop an analytic theory to account for such behaviors. Our results are important in the manipulation of the gain and loss behaviors of edge states in non-Hermitian systems, with potential applications in the study of topological lasers, quantum sensors, and unidirectional invisibility.

Discrete scale-invariant boson-fermion duality in one dimension

We introduce models of one-dimensional $n(\geq3)$-body problems that undergo phase transition from a continuous scale-invariant phase to a discrete scale-invariant phase. In this paper, we focus on identical spinless particles that interact only through two-body contacts. Without assuming any particular cluster-decomposition property, we first classify all possible scale-invariant two-body contact interactions that respect unitarity, permutation invariance, and translation invariance in one dimension. We then present a criterion for the breakdown of continuous scale invariance to discrete scale invariance. Under the assumption that the criterion is met, we solve the many-body Schr\"{o}dinger equation exactly; we obtain the exact $n$-body bound-state spectrum as well as the exact $n$-body S-matrix elements for arbitrary $n\geq3$, all of which enjoy discrete scale invariance or log-periodicity. Thanks to the boson-fermion duality, these results can be applied equally well to both bosons and fermions. Finally, we demonstrate how the criterion is met in the case of $n=3$; we determine the exact phase diagram for the scale-invariance breaking in the three-body problem of identical bosons and fermions. The zero-temperature transition from the unbroken phase to the broken phase is the Berezinskii-Kosterlitz-Thouless-like transition discussed in the literature.

Efficient simulation of the dynamics of an n -dimensional PT -symmetric system with a local-operations-and-classical-communication protocol based on an embedding scheme

$\mathcal{PT}$-symmetric quantum theory has attracted extensive attention in recent years because of its novel properties, applications, and even controversial discussions. Many difficulties in applications stem from the unknown of how to simulate the dynamics of $\mathcal{PT}$-symmetric systems in conventional quantum systems. In this work, by clarifying some common confusion in the simulation of $\mathcal{PT}$-symmetric systems, we are able to naturally extend the application scope of the original embedding simulation scheme from pure states to arbitrary mixed states. Based on the above groundwork, we further propose a local-operations-and-classical-communication (LOCC) protocol scheme to simulate the dynamics of a $\mathcal{PT}$-symmetric system. Our LOCC protocol scheme, which needs one ancillary qubit only, can be applied to simulate any arbitrary finite-dimensional $\mathcal{PT}$-symmetric system in the $\mathcal{PT}$-unbroken phase and has several advantages over the original embedding scheme. The success probability is increased to ${\ensuremath{\lambda}}_{min}$ (${\ensuremath{\lambda}}_{min}>1$) times in general, even approaching $100%$ in some special cases, and has a concise bound that makes it easier to estimate the necessary experimental resources in advance. Furthermore, our LOCC protocol has less dependence on but more flexibility in the selection of the metric operator and more adaptability in practical applications, and it is more consistent with the physical intuition. Finally, the physical or philosophical meaning behind the embedding scheme and the LOCC protocol is discussed.

PT phase transition in open quantum systems with Lindblad dynamics

We investigate parity-time ($\mathcal{PT}$) phase transitions in open quantum systems and discuss a criterion of Liouvillian $\mathcal{PT}$ symmetry proposed recently by Huber \textit{et al}. [J. Huber \textit{et al}., SciPost Phys. $\textbf{9}$, 52 (2020)]. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of $\mathcal{PT}$ symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the $\mathcal{PT}$ symmetry breaking point for an open 2-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a $\mathcal{PT}$ unbroken phase, some eigenvalues are pure imaginary numbers while in a $\mathcal{PT}$ broken phase, all the eigenvalues are real. From this result, it is analytically shown for an open quantum system including quantum jumps that the dynamics in the long time limit changes from an oscillatory to an overdamped behavior at the proposed $\mathcal{PT}$ symmetry breaking point. Furthermore, we show a direct relation between the criterion of Huber \textit{et al}. of Liouvillian $\mathcal{PT}$ symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian $\mathcal{PT}$ symmetry.

Two-loop rational terms for spontaneously broken theories

Rational counterterms are a key ingredient for the automation of loop calculations through numerical methods. Building on the recently established properties of rational terms of UV origin at two loops, in this paper we present a systematic method for the determination of rational counterterms within spontaneously broken theories. In particular we introduce a generalised vev-expansion approach that makes it possible to obtain the rational counterterms of UV origin for a spontaneously broken theory by means of calculations in the unbroken phase. The drastic simplifications that result from the underlying symmetry open the door to the efficient determination of rational counterterms for the full Standard Model at two loops. The renormalisation-scheme dependence is analysed in detail, and we show that rational counterterms need to be determined only once and for all in a generic renormalisation scheme for the symmetric phase and, a posteriori, they can be easily adapted to a wide range of physical renormalisation schemes for the spontaneously broken phase. As a first application we determine the full set of mathcal{O}left({alpha}_{mathrm{S}}^2right) rational counterterms of UV origin for the full Standard Model, i.e. for all superficially UV-divergent two-loop vertex functions involving combinations of gluons, quarks, electroweak vector bosons and scalar bosons.

Unidirectional transparency in epsilon-near-zero based rectangular waveguides induced by parity-time symmetry

Epsilon-near-zero (ENZ) metamaterials present exciting features such as an almost infinite effective wavelength and phase velocity. However, a major challenge when using such artificial media is to achieve impedance matching due to their vanishing permittivity. To address this, it has recently been shown that wave propagation within ENZ media combined with parity-time (PT) symmetry concepts can exhibit asymmetric reflection and enhanced transmission. Here, we exploit this technique by studying a PT-symmetric ENZ structure consisting of a rectangular waveguide operating near the cutoff frequency of the dominant TE10 mode (emulating an ENZ metamaterial) and placed between balanced thin loss and gain blocks. Theoretical studies based on the transmission line theory, eigenvalue problems, and full-wave numerical simulations are performed. It is shown how unbroken and broken PT-symmetric phases can exist with exceptional points in between, demonstrating asymmetric reflection and enhanced transmission while preserving the ENZ behavior of the system.

Detection of an unbroken phase of a non-Hermitian system via a Hermitian factorization surface

In the traditional quantum theory, one-dimensional quantum spin models possess a factorization surface where the ground states are fully separable having vanishing bipartite as well as multipartite entanglement. We report that in the non-Hermitian counterpart of these models, these factorization surfaces either can predict the exceptional points where the unbroken-to-broken transition occurs or can guarantee the reality of the spectrum, thereby proposing a procedure to reveal the unbroken phase. We first analytically demonstrate it for the nearest-neighbour rotation-time RT-symmetric XY model with uniform and alternating transverse magnetic fields, referred to as the iATXY model. Exact diagonalization techniques are then employed to establish this fact for the RT-symmetric XYZ model with short- and long-range interactions as well as for the long-ranged iATXY model. Moreover, we show that although the factorization surface prescribes the unbroken phase of the non-Hermitian model, the bipartite nearest-neighbour entanglement at the exceptional point is nonvanishing.

Dynamics of spontaneous symmetry breaking in a space-time crystal

We present the theory of spontaneous symmetry breaking (SSB) of discrete time translations as recently realized in the space-time crystals of an atomic Bose-Einstein condensate. The non-equilibrium physics related to such a driven-dissipative system is discussed in both the Langevin as well as the Fokker-Planck formulation. We consider a semi-classical and a fully quantum approach, depending on the dissipation being either frequency independent or linearly dependent on frequency, respectively. For both cases, the Langevin equation and Fokker-Planck equation are derived, and the resulting equilibrium distribution is studied. We also study the time evolution of the space-time crystal and focus in particular on its formation and the associated dynamics of the spontaneous breaking of a Z2 symmetry out of the symmetry unbroken phase, i.e., the equilibrium Bose-Einstein condensate before the periodic drive is turned on. Finally, we compare our results with experiments and conclude that our theory provides a solid foundation for the observations.

PT-symmetry and supersymmetry: interconnection of broken and unbroken phases

The broken and unbroken phases of P T and supersymmetry in optical systems are explored for a complex refractive index profile in the form of a Scarf potential, under the framework of supersymmetric quantum mechanics. The transition from unbroken to the broken phases of P T -symmetry, with the merger of eigenfunctions near the exceptional point is found to arise from two distinct realizations of the potential, originating from the underlying supersymmetry. Interestingly, in P T -symmetric phase, spontaneous breaking of supersymmetry occurs in a parametric domain, possessing non-trivial shape invariances, under reparametrization to yield the corresponding energy spectra. One also observes a parametric bifurcation behaviour in this domain. Unlike the real Scraf potential, in P T -symmetric phase, a connection between complex isospecrtal superpotentials and modified Korteweg-de Vries equation occurs, only with certain restrictive parametric conditions. In the broken P T -symmetry phase, supersymmetry is found to be intact in the entire parameter domain yielding the complex energy spectra, with zero-width resonance occurring at integral values of a potential parameter.

Examples of PT Phase Transition : QM to QFT

Parity Time Reversal (PT) phase transition is a typical characteristic of most of the PT symmetric non-Hermitian (NH) systems. Depending on the theory, a particular system and spacetime dimensionality PT phase transition has various interesting features. In this article we review some of our works on PT phase transitions in quantum mechanics (QM) as well as in Quantum Field theory (QFT). We demonstrate typical characteristics of PT phase transition with the help of several analytically solved examples. In one dimensional QM, we consider examples with exactly as well as quasi exactly solvable (QES) models to capture essential features of PT phase transition. The discrete symmetries have rich structures in higher dimensions which are used to explore the PT phase transition in higher dimensional systems. We consider anisotropic SHOs in two and three dimensions to realize some connection between the symmetry of original hermitian Hamiltonian and the unbroken phase of the NH system. We consider the 2+1 dimensional massless Dirac particle in the external magnetic field with PT symmetric non-Hermitian spin-orbit interaction in the background of the Dirac oscillator potential to show the PT phase transition in a relativistic system. A small mass gap, consistent with the other approaches and experimental observations is generated only in the unbroken phase of the system. Finally we develop the NH formulation in an SU(N) gauge field theoretic model by using the natural but unconventional Hermiticity properties of the ghost fields. Deconfinement to confinement phase transition has been realized as PT phase transition in such a non-hermitian model.

Static synthetic gauge field control of double optomechanically induced transparency in a closed-contour interaction scheme

We study theoretically an optical cavity and a parity-time ($\mathcal{PT}$)-symmetric pair of mechanical resonators, where all oscillators are pairwise coupled, forming an optomechanical system with a closed-contour interaction. Due to the presence of both gain and feedback, we explore its stability and the root loci over a wide coupling range. Under the red-sideband pumping and for the so-called $\mathcal{PT}$-unbroken phase it displays a double optomechanically induced transparency (OMIT) for an experimentally realizable parameter set. We show that both the transmission amplitude and the group delay can be continuously steered from the lower transmission window to the upper one by the loop coupling phase which breaks the time-reversal symmetry and introduces a static synthetic gauge field. In the $\mathcal{PT}$-unbroken phase both the gain-bandwidth and delay-bandwidth products remain constant over the full range of the controlling phase. Tunability in transmission and bandwidth still prevails in the $\mathcal{PT}$-broken phase, albeit over a reduced range. In essence, we suggest a simple scheme that grants coupling phase-dependent control of the single and double OMIT phenomena within an effective $\mathcal{PT}$-symmetric optomechanical system.

Metastable two-component solitons near an exceptional point

We consider a two-dimensional nonlinear waveguide with distributed gain and losses. The optical potential describing the system consists of an unperturbed complex potential depending only on one transverse coordinate, i.e., corresponding to a planar waveguide, and a small non-separable perturbation depending on both transverse coordinates. It is assumed that the spectrum of the unperturbed planar waveguide features an exceptional point (EP), while the perturbation drives the system into the unbroken phase. Slightly below the EP, the waveguide sustains two-component envelope solitons. We derive one-dimensional equations for the slowly varying envelopes of the components and show their stable propagation. When both traverse directions are taken into account within the framework of the original model, the obtained two-component bright solitons become metastable and persist over remarkably long propagation distances.

On Bell’s Inequality in PT-Symmetric Quantum Systems

Bell’s inequality is investigated in parity-time (PT) symmetric quantum mechanics, using a recently developed form of the inequality by Maccone, with two PT-qubits in the unbroken phase with real energy spectrum. It is shown that the inequality produces a bound that is consistent with the standard quantum mechanics even after using Hilbert space equipped with CPT inner product and therefore, the entanglement has identical structure with standard quantum mechanics. Consequently, the no-signaling principle for a two-qubit system in PT-symmetric quantum theory is preserved.

Eigenstate entanglement entropy in a PT -invariant non-Hermitian system

Much has been learned about universal properties of the eigenstate entanglement entropy for one-dimensional lattice models, which is described by a Hermitian Hamiltonian. While very less of it has been understood for non-Hermitian systems. In the present work we study a non-Hermitian, non-interacting model of fermions which is invariant under combined $PT$ transformation. Our models show a phase transition from $PT$ unbroken phase to broken phase as we tune the hermiticity breaking parameter. Entanglement entropy of such systems can be defined in two different ways, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates which form a complete set of bi-orthonormal eigenstates. We demonstrate that the entanglement entropy of the ground state and also of the typical excited states show some unique features in both of these phases of the system. Most strikingly, entanglement entropy obtained taking a combination of both left and right eigenstates shows an exponential divergence with system size at the transition point. While in the $PT$-unbroken phase, the entanglement entropy obtained from only the right (or equivalently left) eigenstates shows identical behavior as of an equivalent Hermitian system which is connected to the non-Hermitian system by a similarity transformation.

Bound states of WIMP dark matter in Higgs-portal models. Part I. Cross-sections and transition rates

We investigate the role of the Higgs doublet in the thermal decoupling of multi-TeV dark matter coupled to the Weak interactions of the Standard Model and the Higgs. The Higgs doublet can mediate a long-range force that affects the annihilation processes and binds dark matter into bound states. More importantly, the emission of a Higgs doublet by a pair of dark matter particles can give rise to extremely rapid monopole bound-state formation processes and bound-to-bound transitions. We compute these effects in the unbroken electroweak phase. To this end, we consider the simplest renormalisable fermionic model, consisting of a singlet and a doublet under SUL(2) that are stabilised by a ℤ2 symmetry, in the regime where the two multiplets coannihilate. In a companion paper, we use the results to show that the formation of metastable bound states via Higgs-doublet emission and their decay decrease the relic density very significantly.

Finding Short-Range Parity-Time Phase-Transition Points with a Neural NetworkSupported by the National Natural Science Foundation of China (Grant Nos. 11535004, 11975167, 11761161001, 11375086, 11565010, 11881240623 and 11961141003), the National Key R&D Program of China (Grant Nos. 2018YFA0404403 and 2016YFE0129300), the Science and Technology Development Fund of Macau (Grant No. 008/2017/AFJ), and the Fundamental Research Funds for the Central Universities (Grant

The non-Hermitian PT-symmetric system can live in either unbroken or broken PT-symmetric phase. The separation point of the unbroken and broken PT-symmetric phases is called the PT-phase-transition point. Conventionally, given an arbitrary non-Hermitian PT-symmetric Hamiltonian, one has to solve the corresponding Schrödinger equation explicitly in order to determine which phase it is actually in. Here, we propose to use artificial neural network (ANN) to determine the PT-phase-transition points for non-Hermitian PT-symmetric systems with short-range potentials. The numerical results given by ANN agree well with the literature, which shows the reliability of our new method.

Renormalization group improvement of the effective potential: an EFT approach

We apply effective field theory (EFT) methods to compute the renormalization group improved effective potential for theories with a large mass hierarchy. Our method allows one to compute the effective potential in a systematic expansion in powers of the mass ratio, as well as to sum large logarithms of mass ratios using renormalization group evolution. The effective potential is the sum of one-particle irreducible diagrams (1PI) but information about which diagrams are 1PI is lost after matching to the EFT, since heavy lines get shrunk to a point. We therefore introduce a tadpole condition in place of the 1PI condition, and use the renormalization group improved value of the tadpole in computing the effective potential. We explain why the effective potential computed using an EFT is not the same as the effective potential of the EFT. We illustrate our method using the O(N) model, a theory of two scalars in the unbroken and broken phases, and the Higgs-Yukawa model. Our leading-log result, obtained by integrating the one-loop β-functions, correctly reproduces the log-squared term in explicit two-loop calculations. Our method does not have a Goldstone boson infrared divergence problem.

Self-dualities and renormalization dependence of the phase diagram in 3d O(N) vector models

In the classically unbroken phase, 3d O(N) symmetric ϕ4 vector models admit two equivalent descriptions connected by a strong-weak duality closely related to the one found by Chang and Magruder long ago. We determine the exact analytic renormalization dependence of the critical couplings in the weak and strong branches as a function of the renormalization scheme (parametrized by κ) and for any N. It is shown that for κ = κ∗ the two fixed points merge and then, for κ < κ∗, they move into the complex plane in complex conjugate pairs, making the phase transition no longer visible from the classically unbroken phase. Similar considerations apply in 2d for the N = 1 ϕ4 theory, where the role of classically broken and unbroken phases is inverted. We verify all these considerations by computing the perturbative series of the 3d O(N) models for the vacuum energy and for the mass gap up to order eight, and Borel resumming the series. In particular, we provide numerical evidence for the self-duality and verify that in renormalization schemes where the critical couplings are complex the theory is gapped. As a by-product of our analysis, we show how the non-perturbative mass gap at large N in 2d can be seen as the analytic continuation of the perturbative one in the classically unbroken phase.

Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank r on U(1) we explicitly calculate beta functions and find equivalence with those of O(N) models but with an effective dimension flowing from r − 1 to zero. In the r − 1 dimensional regime, the equivalence to O(N) models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.

Parity-time phase transition in photonic crystals with C_{6v} symmetry

We investigate the parity-time (PT) phase transition in photonic crystals with C_{6v} symmetry, with balanced gain and loss on dielectric rods in the triangular lattice. A two-level non-Hermitian model that incorporates the gain and loss in the tight-binding approximation was employed to describe the dispersion of the PT symmetric system. In the unbroken PT phase, the double Dirac cone feature associated with the C_{6v} symmetry is preserved, with a frequency shift of second order due to the presence of gain and loss. The helical edge states with real eigenfrequencies can exist in the common band gap for two topologically distinct lattices. In the broken PT phase, the non-Hermitian perturbation deforms the dispersion by merging the frequency bands into complex conjugate pairs and forming the exceptional contours that feature the PT phase transition. In this situation, the band gap closes and the edge states are mixed with the bulk states.