⚛️ quantum physics

Constructing Non-Hermitian Theories with Tunable Exceptional Points and Controlled State Purification

This paper establishes momentum-space deformation as a universal design principle for engineering non-Hermitian many-body systems with tunable exceptional points that induce exponential eigenvector coalescences, enable controlled state purification with distinct size-dependent regimes, and facilitate the systematic construction of diverse non-Hermitian quantum matter.

Original authors: Soumya Kanti Pal, Rupak Majumder, Shamik Gupta

Published 2026-08-06

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CC BY 4.0

Original authors: Soumya Kanti Pal, Rupak Majumder, Shamik Gupta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of physics aren't just about solid, unchanging objects, but about systems that are constantly interacting with their surroundings—like a spinning top that slowly loses energy to the air, or a radio signal that fades as it travels through a storm. In the quantum world, where particles behave like waves, scientists have discovered a special kind of "open system" called non-Hermitian physics. Think of these systems as a dance floor where the dancers (particles) aren't just moving to the music; they are also borrowing energy from the room or losing it to the floor. Usually, this leads to chaos or the system dying out. However, there is a magical sweet spot in these systems called an "Exceptional Point" (EP). You can think of an EP as a cosmic traffic jam where two different dancers, who were previously doing their own unique moves, suddenly merge into a single, indistinguishable entity. At this exact moment, the system's behavior changes dramatically, offering new ways to control quantum information, sense tiny changes, or even clean up messy data. Scientists have been trying to figure out how to build these "traffic jams" on purpose, especially in systems with many particles, to see what strange new powers they might unlock.

Now, imagine you have a giant, complex machine made of thousands of tiny gears (a many-body quantum system), and you want to create one of these magical "traffic jams" (an Exceptional Point) exactly where you want it. In a new paper, researchers Soumya Kanti Pal, Rupak Majumder, and Shamik Gupta from the Tata Institute of Fundamental Research in India have come up with a clever blueprint for doing just that. They discovered a general design principle called "momentum-space deformation." To understand this, picture the machine's gears not as physical objects, but as waves moving in different directions. The team found that by slightly "stretching" or "twisting" these waves in a specific mathematical way (deforming them), they could force the system to create an Exceptional Point. It's like tuning a guitar string: if you tighten it just right, the note changes. Here, by tuning the "twist" of the waves, they can make the system hit that special point where particles merge.

The most surprising thing they found is what happens when you create just one of these traffic jams in a single part of the system. Instead of just affecting that one spot, it causes a "domino effect" that spreads exponentially through the entire machine. If you have a system with many particles, a single merged pair in one section causes millions of other pairs to merge in the rest of the system. It's as if one person in a stadium standing up causes a wave of standing people to ripple out, but in this case, the "standing up" is particles losing their individual identities and becoming one. This "exponential proliferation" means that a tiny change in one area can completely reshape the behavior of the whole quantum system.

The paper also reveals a fascinating trick for cleaning up "messy" quantum states. In the quantum world, information can get scrambled or "mixed up," like a deck of cards that has been shuffled too many times. The researchers showed that by tuning their system to these Exceptional Points, they can act like a magical filter that automatically sorts the cards back into a perfect order, turning a messy mix into a pure, clear state. However, there is a catch that depends on the size of the system, specifically whether it has an even or odd number of particles. If the system has an odd number of particles, it's impossible to clean it up completely; it will always remain a little bit messy. But if the system has an even number, it can be purified perfectly. This creates a fundamental difference between odd and even-sized systems, suggesting that the size of your quantum machine matters more than you might think.

Furthermore, the authors provided a "reverse-engineering" guide. Instead of starting with a complex machine and trying to guess where the traffic jams will form, you can now start with the traffic jam you want and work backward to design the machine that creates it. They showed how to build systems with short-range connections (neighbors only) or long-range connections (neighbors far apart), and even systems where the rules aren't the same in both directions (nonreciprocal). Finally, they offered a practical recipe for how to build these systems in a real lab using "Lindblad embedding," which is a way of describing how a system interacts with its environment to create these effects without needing magic.

In summary, this paper doesn't just find a new phenomenon; it gives scientists a universal toolkit. They proved that by deforming the momentum of particles, you can reliably create Exceptional Points, which in turn can clean up quantum states and control how information flows. They showed that while a single point of merging can trigger a massive chain reaction, the ability to fully clean up the system depends on a simple rule: even numbers work, odd numbers don't. This work suggests that we are moving closer to being able to design and control complex quantum machines with the precision of an engineer, opening the door to better sensors, more powerful quantum computers, and new ways to handle information in a noisy world.

Technical Summary: Constructing Non-Hermitian Theories with Tunable Exceptional Points and Controlled State Purification

Problem Statement
Non-Hermitian quantum mechanics has emerged as a framework for describing open and driven systems, characterized by exceptional points (EPs)—spectral singularities where eigenvalues and eigenvectors simultaneously coalesce. While EPs have been realized experimentally in various platforms and studied in specific non-Hermitian Hamiltonians, a general, systematic method to engineer EP structures across extensive classes of many-body Hamiltonians remains an open challenge. Specifically, there is a need to understand how EPs in momentum-space blocks translate to the many-body spectrum and how they can be utilized for controlling quantum state dynamics, such as the purification of mixed states.

Methodology
The authors introduce momentum-space deformation as a general design principle for engineering non-Hermitian many-body systems. The methodology proceeds as follows:

Framework Construction: The study focuses on quadratic Bogoliubov–de Gennes (BdG) Hamiltonians for free fermions on a periodic lattice. These Hamiltonians are decomposed into independent momentum-space blocks ( $H_k$ ) via Fourier transformation.
Non-Hermitian Deformation: An arbitrary non-Hermitian Hamiltonian ( $H^{nH}_k$ ) is constructed by deforming a Hermitian base ( $H^{(1)}_k$ ) with a non-Hermitian term ( $H^{(2)}_k$ ), expressed as $H^{nH}_k = H^{(1)}_k - iH^{(2)}_k$ . The deformation is parametrized using Pauli matrices, allowing for systematic tuning of deformation strengths ( $\gamma_\alpha$ ) along specific directions ( $x, y, z$ ).
EP Condition Derivation: The authors derive universal, microscopic-independent criteria for the existence of EPs in a single momentum block. These conditions require the deformation vector $\vec{b}$ to be orthogonal to the Hermitian vector $\vec{a}$ and of equal magnitude ( $\vec{a} \cdot \vec{b} = 0$ and $|\vec{a}| = |\vec{b}|$ ).
Symmetry Analysis: The role of Parity-Time ($PT$) symmetry is analyzed. The authors demonstrate that $PT$ symmetry imposes specific constraints on the vectors $\vec{a}$ and $\vec{b}$ , ensuring that every momentum block can host an EP when the deformation strength is tuned to a critical value.
Many-Body Mapping: The paper maps the single-block EP conditions to the full many-body spectrum, analyzing the coalescence of many-body eigenvectors.
Dynamical Simulation: The time evolution of mixed quantum states under these non-Hermitian Hamiltonians is calculated to investigate state purification. The purity of the system is tracked as a function of deformation strength and system size.
Real-Space Reconstruction and Embedding: The authors perform inverse Fourier transforms to determine the real-space structure of the deformed Hamiltonians (short-range vs. long-range, reciprocal vs. nonreciprocal). Finally, they provide an explicit Lindblad embedding to realize these effective non-Hermitian Hamiltonians within open quantum systems.

Key Contributions and Results

Universal EP Criteria: The paper establishes that for a single momentum block to host an EP, the deformation vector must lie on a specific cone relative to the Hermitian part of the Hamiltonian. Crucially, the momentum location ( $k_c$ ) of the EP is determined solely by the intrinsic structure of the Hermitian Hamiltonian, while the critical deformation strength is tunable.
Exponential Proliferation of Coalescences: A central finding is that a single EP in one momentum block induces an exponential proliferation of pairwise eigenvector coalescences throughout the many-body Hilbert space. Specifically, if block $k_c$ is at an EP, all many-body states differing only in the occupation of $k_c$ coalesce, resulting in $2^{L/2-1}$ (even parity) or $2^{L/2-2}$ (odd parity) coalescences.
PT-Symmetric Engineering: The authors show that $PT$ symmetry is a powerful mechanism for EP engineering. In a $PT$-symmetric system, every momentum block can host an EP at a specific critical deformation strength, completely characterizing the many-body $PT$-symmetry-breaking transition.
Controlled State Purification: The study identifies EPs as a universal mechanism for purifying arbitrary mixed quantum states. Three distinct purification regimes are uncovered based on the deformation strength ( $\gamma_y$ $γ_{y}$ ):
1. $\gamma_y < \gamma^{(1)}_c$ : All blocks are in the $PT$-symmetry-preserved (SP) phase; purity oscillates persistently.
2. $\gamma^{(1)}_c < \gamma_y < \gamma^{(2)}_c$ : A subset of blocks is in the symmetry-broken (SB) phase; the system undergoes partial purification.
3. $\gamma_y > \gamma^{(2)}_c$ : All blocks are in the SB phase; the system achieves complete purification ( $P(t) \to 1$ ).
Odd-Even System Size Dichotomy: In the thermodynamic limit, a fundamental dichotomy is revealed. For even system sizes ( $L$ ), complete purification is attainable in the even-parity sector, while the odd-parity sector (containing unpaired modes $k=0, \pi$ ) caps the asymptotic purity at 0.25. For odd system sizes, both parity sectors contain a single unpaired mode, yielding an asymptotic purity of 0.5. This renders the thermodynamic limits of odd and even systems fundamentally inequivalent.
Reverse-Engineering Protocol: The framework allows for the systematic generation of non-Hermitian Hamiltonians with specific real-space properties. The classification relies on the parity of the deformation function $F_\alpha(k)$ $F_{α} (k)$ combined with the antisymmetry of the operator $\hat{S}_\alpha(k)$ $\hat{S}_{α} (k)$ (for $\alpha=x,y$ $α = x, y$ ) in momentum space:
- Short-range: Achieved when the deformation function $F_\alpha(k)$ is symmetric ( $F_\alpha(k) = F_\alpha(-k)$ ). In this case, the resulting real-space Hamiltonian involves short-range couplings ( $\sim \hat{c}_i \hat{c}_{i+r}$ with $r \ll L$ ).
- Long-range: Achieved when the deformation function $F_\alpha(k)$ is antisymmetric ( $F_\alpha(k) = -F_\alpha(-k)$ ). Since the Fourier transform of the sign function decays algebraically, this leads to intrinsically long-range couplings.
- Nonreciprocal: Achieved by simultaneous deformations along different axes (e.g., $x$ and $y$ ), creating asymmetric hopping amplitudes.
Lindblad Embedding: An explicit construction of jump operators is provided, demonstrating how these effective non-Hermitian Hamiltonians can emerge from the "no-click" limit of Markovian open quantum dynamics.

Significance
The paper claims to establish momentum-space deformation as a unified route for exceptional-point engineering and the controlled design of many-body non-Hermitian quantum systems. By providing a general framework that connects momentum-space deformations to real-space interactions and many-body spectral properties, the work offers a systematic protocol for:

Designing non-Hermitian matter with tunable EP structures.
Utilizing EPs for the deterministic purification of mixed quantum states.
Generating diverse classes of non-Hermitian Hamiltonians (short/long-range, reciprocal/nonreciprocal) with explicit experimental pathways via Lindblad embedding.

The authors position these results as a bridge between abstract non-Hermitian spectral theory and practical many-body physics, potentially stimulating further research into correlated non-Hermitian phases, dissipative control, and quantum information processing.

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