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Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field

E. Medina-Guerra, I. V. Gornyi, Yuval Gefen

arXiv:2502.07775v3quant-phcond-mat.mes-hallcond-mat.stat-mechcond-mat.str-el

TL;DR

The paper asks whether Krylov spread can reveal structure in a non-Hermitian Ising chain that conventional correlations miss. It analytically connects spread to correlations in the non-Hermitian Bogoliubov vacuum and finds three dynamical phases in the gapped regime, while correlations decay exponentially in the gapped phase and algebraically in the gapless phase.

  • Problem

    Krylov spread is largely unexplored for non-Hermitian dynamics, whose conventional correlation functions do not reveal all dynamical phases.

    Source: §I · p. 2

  • Method

    The authors analyze an integrable non-Hermitian Ising chain initialized in the all-spin-down state, deriving its correlations and Krylov spread analytically.

    Source: §VI · p. 16

  • Results

    The Krylov spread reveals three dynamical phases in the gapped spectral regime, while z-spin correlations decay exponentially with oscillations in the gapped phase and algebraically with oscillations in the gapless phase.

    Source: §I · p. 3, §VI · pp. 16–17

  • Takeaways & Limitations

    Krylov spread supplies dynamical phase information beyond the static correlation-function classification of the non-Hermitian Ising chain.

    Source: §I · p. 2, §VI · p. 16

Abstract

from arXiv · show

Krylov complexity measures the spread of an evolved state in a natural basis, induced by the generator of the dynamics and the initial state. Here, we study the spread in Hilbert space of the state of an Ising chain subject to a complex-valued transverse magnetic field, initialized in a trivial product state with all spins pointing down. We demonstrate that Krylov spread reveals structural features of many-body systems that remain hidden in correlation functions that are traditionally employed to determine the phase diagram. When the imaginary part of the spectrum of the non-Hermitian Hamiltonian is gapped, the system state asymptotically approaches the non-Hermitian Bogoliubov vacuum for this Hamiltonian. We find that the spread of this evolution unravels three different dynamical phases based on how the spread reaches its infinite-time value. Furthermore, we establish a connection between the Krylov spread and the static correlation function for the z-components of spins in the underlying non-Hermitian Bogoliubov vacuum, providing a full analytical characterization of correlations across the phase diagram. Specifically, for a gapped imaginary spectrum in a finite magnetic field, we find that the correlation function exhibits an oscillatory behavior that decays exponentially in space. Conversely, for a gapless imaginary spectrum, the correlation function displays an oscillatory behavior with an amplitude that decays algebraically in space; the underlying power law depends on the manifestation of two exceptional points within this phase.

I. INTRODUCTION

The paper studies Krylov spread in an integrable non-Hermitian Ising chain with a complex transverse field, showing that it exposes dynamical phases beyond traditional spectral and correlation-function classifications.

I. INTRODUCTION

  • Krylov spread is introduced as a natural measure of how an evolved state explores a basis generated from the initial state and dynamics.The Krylov basis has been used to study chaos, integrability, topology, dynamical transitions, and quantum speed limits.Source: §I · p. 2
  • The static z-spin correlation function is algebraic and oscillatory in the gapless phase but exponentially decaying and oscillatory in the gapped phase.In the gapless phase, the power-law exponent changes when two exceptional points emerge.Source: §I · pp. 2–3
  • The spread is analytically connected to the z-spin correlation function and undergoes a third-order transition at the boundary between gapless and gapped imaginary spectra.In the oscillatory gapless regime, the first derivative of the spread with respect to h diverges logarithmically as h approaches J at γ=0.Source: §I · p. 3
  • The paper identifies previously unnoticed dynamical phases in the gapped region through the time dependence of Krylov spread fidelity.Three characteristic timescales arise from the interplay between the real field and hopping-driven unitary dynamics and the imaginary field's dissipative dynamics.Source: §I · pp. 2–3

II. MODEL: SPECTRUM AND NON-HERMITIAN VACUUM

  • The non-Hermitian Ising chain has a complex transverse field h+iγ/4 and can be diagonalized using Jordan-Wigner and non-Hermitian Bogoliubov transformations.The initial state is the all-spin-down product state, and the right vacuum is a generalized coherent state of the Jordan-Wigner vacuum.Source: §II · p. 3

A. Spectrum of H

  • The spectrum separates the h-γ plane into gapless and gapped phases according to whether the imaginary part Γ(k) vanishes, with additional distinctions involving the real part E(k).The convention Im Γ(k) ≤ 0 makes the state with the largest imaginary eigenvalue the slowest-decaying state.Source: A. Spectrum of H · pp. 4–5, Figure 1 · p. 5

B. Non-Hermitian Bogoliubov vacuum

The right non-Hermitian Bogoliubov vacuum is constructed from the Jordan-Wigner vacuum using a generalized coherent-state representation and non-Hermitian fermionic operators.

B. Non-Hermitian Bogoliubov vacuum

  • The right vacuum |Ω⟩ is defined as the state annihilated by every non-Hermitian quasiparticle operator η_k.It can be represented as a generalized coherent state in the coset space SU(2)⊗N/U(1)⊗N.Source: B. Non-Hermitian Bogoliubov vacuum · p. 6
  • The operators η_k are constructed from the Jordan-Wigner fermions so that their annihilation conditions reproduce the non-Hermitian vacuum.The corresponding excited states are generated by non-Hermitian quasiparticle pairs acting on |Ω⟩.Source: B. Non-Hermitian Bogoliubov vacuum · p. 6

C. Time evolution of JW vacuum |0⟩

The evolved Jordan–Wigner vacuum reaches the non-Hermitian Bogoliubov vacuum in the gapped phase, whereas gapless evolution retains a two-quasiparticle component. The section then sets up the asymptotic analysis of the z-spin correlation function.

C. Time evolution of JW vacuum |0⟩

III. CORRELATION FUNCTIONS

  • The correlation analysis evaluates the large-distance z-spin correlator Czz(x) in the non-Hermitian Bogoliubov vacuum across gapped and gapless regimes.Source: §III · p. 8

A. Correlation function in the gapped phase

In the gapped phase, the z-spin correlation function is obtained asymptotically from complex-plane branch points and exhibits behavior controlled by the correlation length. For nonzero real magnetic field, the asymptotics is oscillatory and exponentially decaying, while the approach to criticality produces a diverging correlation length and method-specific prefactor issues.

A. Correlation function in the gapped phase

B. Correlation function in the gapless phase

The gapless-phase analysis derives oscillatory algebraic correlations, including a stronger power-law suppression at the critical curve, and compares them with the Hermitian limit. It also establishes how Krylov spread is defined, computed, and connected to correlations and phase transitions.

B. Correlation function in the gapless phase

  • In the oscillatory gapless phase, Czz(x) has leading algebraic oscillations, with subdominant exponentially decaying terms.The asymptotic contractions include ⟨A0Ax⟩∼O(x^-2 sin qx), while the exponential terms retain correlation length ξ.Source: B. Correlation function in the gapless… · p. 10
  • At γ=γc(h), the leading decay changes from x^-2 to x^-4 because two exceptional points emerge at ±q.For h=0 and γ=γc(0)=4J, the auxiliary functions become constant; the critical-line result was not previously calculated for Czz(x).Source: B. Correlation function in the gapless… · p. 11

C. Hermitian limiting case

  • In the Hermitian limit, the non-Hermitian vacuum need not equal the true Hermitian ground state, so γ→0 can yield different correlations from setting γ=0 initially.For finite γ, long-time evolution reveals the non-Hermitian vacuum even when γ is arbitrarily small, with gapless excitations at ±q.Source: C. Hermitian limiting case · p. 11
  • For h>J as γ→0, the leading correlation term is oscillatory and algebraic rather than exponentially decaying, unlike the standard Hermitian ferromagnetic phase.The disordered Hermitian and critical-Ising limits coincide with correlations in |GS⟩, whereas the oscillatory phase does not.Source: Figure 2 · p. 10, C. Hermitian limiting case · p. 11

A. Generalities

  • Krylov spread is the mean Krylov-basis distance of the evolved state and can be computed from repeated Hamiltonian action, with the JW vacuum exploiting an su(2)⊗N structure.The study compares unitary evolution to |Ω⟩ with non-Hermitian evolution toward the stationary state; the spreads coincide in the gapped phase and relate to the same-site contraction ⟨BnAn⟩.Source: A. Generalities · pp. 11–12

B. Krylov spread via unitary dynamics

  • The unitary spread density is continuous across γc(h), constant for h≤J at γ→0, increasing for h>J, and approaches unity as h→∞.Its first h-derivative diverges logarithmically as h→J+ in the Hermitian limit, witnessing a second-order transition; dissipation produces a third-order transition at fixed 0<γ<4J.Source: B. Krylov spread via unitary dynamics · p. 13

V. DYNAMICAL PHASE TRANSITION AND KRYLOV SPREAD

The non-Hermitian Krylov construction evolves the JW vacuum with a normalized state and a Hessenberg representation, then compares its long-time spread with the unitary result. In the gapped phase, the infinite-time spread agrees with the unitary spread, while three dynamical phases are identified through distinct relaxation times.

V. DYNAMICAL PHASE TRANSITION AND KRYLOV SPREAD

  • The non-Hermitian Krylov spread is built from the span of H^n|0⟩, its Gram-Schmidt Krylov basis, and normalized time-dependent Krylov coefficients.The su(2) structure permits thermodynamic-limit Krylov vectors analogous to the unitary construction, with Rx(k) replacing α+(k).Source: §V · p. 14
  • In the gapped phase, the infinite-time spread equals the unitary-evolution spread, while time averaging before the thermodynamic limit yields the same unitary result.This connects the stationary non-Hermitian evolution to the spread obtained by directly mapping |0⟩ to |Ω⟩.Source: §V · p. 14
  • A Krylov spread fidelity defines the stationary time-dependent spread density and reveals three dynamical phases in the gapped-spectrum regime.Each phase is associated with a characteristic time describing how the spread reaches its infinite-time value.Source: §V · p. 14

A. Time-dependent Krylov spread in the gapped phase and Krylov fidelity

Krylov spread fidelity refines the gapped region into three dynamical phases according to how the spread approaches its stationary value, while gapless dynamics has no corresponding characteristic time. The spread also connects to spin correlations and distinguishes long-time behavior governed by modes beyond those with vanishing real spectral part.

A. Time-dependent Krylov spread in the gapped phase and Krylov fidelity

  • The long-time Krylov dynamics is governed by modes at k = 0, π, and k ≈¯k rather than by the modes ±q where the real spectrum changes gap character.This explains why the dynamical phase boundaries do not simply follow the real-spectrum gap line h = J.Source: A. Time-dependent Krylov spread in the… · p. 15

B. Time-dependent Krylov density in the gapless phase

  • In the gapless phase, the time-averaged non-Hermitian spread density coincides with the spread density generated by corresponding unitary dynamics.For initial states with the required nonzero overlaps, the long-time spread behavior is universal within the gapless or gapped stationary-state structure.Source: B. Time-dependent Krylov density in… · pp. 15–16

VI. SUMMARY AND DISCUSSION

  • The infinite-time Krylov spread is related to the same-site spin correlation function, and its spread density shows a third-order transition across the gapless-to-gapped boundary.Near the critical field, the second γ derivative of the spread density diverges as (hc −h)^−1/2 when approached from the left.Source: §VI · p. 16
  • The spin correlation function decays as x^-3 exp(−x/ξ) times an oscillatory factor in the gapped phase, but algebraically as x^-n in the gapless phase.The gapless exponent is n = 2 without exceptional points and n = 4 when two exceptional points occur.Source: §VI · p. 16

A. Time-dependent Krylov spread in the gapped phase and Krylov fidelity

VI. SUMMARY AND DISCUSSION

  • Krylov spread exposes hidden dynamical transitions that conventional time-dependent correlations or entanglement measures may not readily distinguish.This extends Krylov spread beyond its established use as a probe of chaos, integrability, and known dynamical transitions.Source: §VI · pp. 16–17

Appendix A: The Bogoliubov vacuum as a generalized coherent state

The appendix identifies the non-Hermitian Bogoliubov vacuum with a generalized coherent state and derives the stationary-state limits reached from the Jordan-Wigner vacuum. A gapped imaginary spectrum yields the vacuum, whereas non-decaying ±q modes add a two-excitation component in the gapless case.

Appendix A: The Bogoliubov vacuum as a generalized coherent state

  • The non-Hermitian Bogoliubov vacuum coincides with a generalized coherent state of the Jordan-Wigner vacuum.The equivalence follows by applying su(2) normal ordering and the modewise lowering-operator identities.Source: Appendix A · p. 18

Appendix B: Stationary state

  • When the imaginary spectrum is gapped, the evolved Jordan-Wigner vacuum reaches the Bogoliubov right vacuum in the stationary-state regime t ≫t⋆.The stationary-state onset is defined using bounds on decaying mode contributions and t⋆ = maxk tk over relevant modes.Source: Appendix B · pp. 19–20
  • When Γ(±q) = 0, the stationary state instead has the leading structure A(t)|Ω⟩ + A∗(t)|q, −q⟩.The non-decaying modes prevent direct reduction to the Bogoliubov vacuum alone.Source: Appendix B · p. 20

Appendix C: Calculation of the spin-spin correlation function

The appendix derives the large-distance behavior of Czz(x) in the gapped phase using contour deformations, branch-point expansions, and the Darboux method.

1. Gapped phase

  • The gapped-phase contractions are represented as contour integrals, with the contours and branch cuts chosen to isolate their asymptotic contributions.The relevant contours evaluate ⟨A0Ax⟩ and ⟨A0B±x⟩; the gapped phase corresponds to γ > γc(h).Source: §1 · p. 21, Figure 5 · p. 23
  • The Darboux analysis deforms the contours around external branch cuts, where the branch point nearest the unit circle supplies the dominant contribution as x →∞.The integrands are expanded around the dominant branch point and integrated term-by-term.Source: §1 · p. 22
  • The resulting contractions determine the asymptotic spin correlation through Czz(x) = −⟨B0Ax⟩⟨B0A−x⟩−⟨A0Ax⟩2.The gapped-phase correlation length is ξ = (2 ln |z1|)−1.Source: §1 · p. 24

2. Gapless phase

In the gapless phase, the appendix evaluates Czz(x) by expanding contraction integrands near special momenta and regulating the resulting integrals.

2. Gapless phase

  • The gapless-phase calculation uses exponential regulators and local expansions near k = 0, π, and q = arccos(h/J) to obtain algebraic asymptotics.The analysis treats both γ = γc(h ≥0) and γ < γc(h), with x even.Source: §2 · pp. 24–25
  • The regulated contraction analysis produces the asymptotic correlation formulas summarized in Eqs. (78)–(80).The leading behavior includes oscillatory factors and powers of x determined by the local expansions.Source: §2 · pp. 25–26
  • For γ < γc(h), the points k = 0 and π do not contribute to the leading terms, leaving k = q as the relevant point for the expansion.The same selection of k = q applies to the ⟨B0A±x⟩ contractions.Source: §2 · pp. 24–26

3. Limiting Hermitian case

The limiting Hermitian case is obtained by taking γ →0 at the level of the non-Hermitian Hamiltonian, then evaluating correlations separately for h > J, h = J, and h < J.

3. Limiting Hermitian case

  • Taking γ →0 in the non-Hermitian Hamiltonian fixes the spectrum’s sign convention and yields a state that is an excited state of the true Hermitian ground state.The resulting excitations have |k| > π/2.Source: §3 · p. 27
  • For h > J and h = J, Darboux analysis or direct integration yields the limiting asymptotic correlation expressions summarized in Eq. (81).The h > J case uses contour deformation around an external branch cut, while h = J is calculated analytically.Source: §3 · pp. 27–28
  • For h < J, only the point k = q contributes at leading order, while exponentially decaying terms remain subdominant.This also yields Eq. (81) in the γ →0 limit.Source: §3 · p. 28

4. Correlation function in the true Hermitian ground state

The appendix separately computes the spin correlation function in the true Hermitian ground state, setting γ = 0 before imposing any spectral sign convention.

4. Correlation function in the true Hermitian ground state

  • The true Hermitian-ground-state calculation treats the ferromagnetic, paramagnetic, and critical regimes using Darboux methods or direct integration.The corresponding asymptotic results are given in Eq. (C68), with a separate expression at h = J.Source: §4 · p. 29

a. Critical point

  • At the critical point h = J, the Green’s functions simplify and allow the correlation function to be evaluated exactly.The resulting exact expression is reported after the analytic continuation of the integral.Source: a. Critical point · p. 29

b. Paramagnetic and ferromagnetic phases

The correlation function is analyzed asymptotically in the paramagnetic and ferromagnetic phases using complex-plane contour deformation and the Darboux method.

b. Paramagnetic and ferromagnetic phases

Appendix D: Stationary state in the thermodynamic limit: Krylov fidelity approach

Appendix D derives the long-time Krylov fidelity by separating endpoint and bulk momentum contributions and applying asymptotic analysis to each.

Appendix D: Stationary state in the thermodynamic limit: Krylov fidelity approach

  • Near k = 0 and k = π, the asymptotic expansions use quadratic approximations for 1 + |τ_k|^2 and the limiting behavior f_k → 0 and J_k(t) → −1.Source: Appendix D · p. 31
  • Endpoint neighborhoods near k = 0 and k = π provide boundary contributions, while the remaining momentum interval contributes through a localized, rapidly suppressed wave packet.Source: Appendix D · pp. 31–32
  • The bulk contribution I_b,2(t) requires complex-mode continuation and steepest-descent analysis around modes satisfying the stationary conditions.Source: Appendix D · pp. 32–33
  • The long-time fidelity has four contributions with distinct dissipation rates, and comparing them yields three dynamical regions in the gapped regime.Source: Appendix D · p. 33
  • The stationary-state time is defined by requiring the fidelity to fall below ε, after which the spread complexity approaches its infinite-time value.Source: Appendix D · pp. 31, 33
  • The three regional decay scales are associated with γ, 4…, and γ_Y, and the corresponding times produce Eqs. (107)–(109).Source: Appendix D · p. 33
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