Giant bubbles of Fisher zeros in the quantum XY chain

arXiv:2602.05899 · cond-mat.str-el, cond-mat.stat-mech, hep-lat, quant-ph · Submitted 2026-02-05 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Giant bubbles of Fisher zeros in the quantum XY chain".

Mira: Here's the gist of the paper. Giant bubbles of Fisher zeros in the quantum XY chain demonstrate an alternative approach based on complex-valued inverse temperature and partition function to probe quantum phases of matter with nontrivial spectra and dynamics <ref:2602.05899#pg2> The authors utilize a method involving the analytical continuation of the partition function to a complex inverse temperature, denoted as β = βr + iβi ∈ C, which accounts for both thermal and quantum fluctuations <ref:2602.05899#pg2>.

Kai: This framework allows them to leverage thermofield dynamics (TFD) to quantitatively characterize quantum and thermal fluctuations <ref:2602.05899#pg2>.

Mira: They exploit the correspondence between low-energy excitations and Fisher zeros, which are the zeros of Z on the complex β plane, to identify thermodynamic phase transitions <ref:2602.05899#pg2>.

Kai: The quantum XY chain in an external field serves as a testbed for this investigation <ref:2602.05899#pg2>.

Paper summary: Kai: So we're looking at this paper, "Giant bubbles of Fisher zeros in the quantum XY chain." Basically, they use a complex-valued inverse temperature and partition function to probe quantum phases with spectra and dynamics that are nontrivial.

Mira: Right, so the core idea is using thermofield dynamics to characterize those thermal and quantum fluctuations quantitatively. They connect low-energy excitations to Fisher zeros on the complex beta plane to identify thermodynamic phase transitions in this quantum XY chain.

Lev: If we're talking about what they built, they're testing this approach on the quantum XY chain in an external field g, specifically looking at how these zeros behave as the parameters change.

Kai: What they claim is that there are these giant bubbles of Fisher zeros near the gapless XX limit that seem to contradict predictions from a featureless Luttinger liquid low-energy theory.

Mira: They find oscillatory gap behavior in the long-time dynamics of the thermofield dynamics spectral form factor, S, by looking at how it changes with different excitation scales like Delta infinity and Delta L.

Lev: The oscillation frequency of S gives them some pretty clear evidence to pinpoint that oscillatory phase in the XY model.

Kai: They also identify these giant bubbles as large-scale closed lines of Fisher zeros near the gapless XX limit, and they note that their motion with varying external field is related to transferring spectral weight from high to low energies.

Mira: In the XX limit, where g equals zero, all the zeros are in the form of closed bubbles and some grow in size by extending toward larger and larger real parts of beta.

Lev: That growth toward larger real parts of beta is interesting because it relates to how spectral weight moves between different energy scales.

Kai: The low-energy physics for the XX model at g equals zero can be described by a Luttinger liquid with a linear dispersion epsilon(k) equals vF (k minus kF) around the Fermi points.

Mira: They trace the pseudogap scale, which is characterized by enhanced spectral weight, to the van Hove singularity at energy epsilon0 equals (one minus g) <ref:2602.05899#pg1>.

Lev: That scaling behavior matches what they see in one over beta i zero proportional to one minus g, which links the pseudogap to that van Hove singularity <ref:2602.05899#pg1>.

Kai: The size of these giant bubbles, beta i zero, as measured by the imaginary part of its rightmost endpoint, actually diverges as g approaches one.

Mira: This divergence as g goes to one is what they are pointing toward when they discuss the connection between the pseudogap and that singularity.

Lev: The paper also mentions that for L equals thirty-two and beta r equals ten the oscillation frequency of S matches Delta infinity, while for L equals thirty-two and beta r equals fifty it matches Delta L <ref:2602.05899#pg3,10, the oscillation frequency of S>.

Kai: They also observed that the closed Fisher zero loops exhibit non-monotonic behavior as g increases before reaching g equals one, including disappearing and reappearing at g less than or equal to zero point five.

Mira: That's a key finding because it shows how the structure of these zeros evolves with the external field, which is what they are probing in this study on "Giant bubbles of Fisher zeros in the quantum XY chain."

Lev: This suggests that even near critical lines, things aren't just smooth; there are these dynamic changes happening within the system.

Kai: So to wrap up on this paper, it confirms that the complex partition function is an effective tool for analyzing thermal and dynamical properties of the one-dimensional XY model.

Mira: By varying beta r and beta i and investigating those Fisher zeros through thermofield dynamics, they accurately capture that oscillatory finite-size gap behavior in the XY model.

Lev: In the thermodynamic limit, they uncover this distinctive class of giant bubbles—those large-scale closed lines of Fisher zeros—near the critical line.

Kai: The evolution of these bubbles as you change the external field gives them an intuitive way to see how spectral weight transfers within that Luttinger liquid phase.

Mira: Since the pseudogap phenomenon is defined by a redistribution of spectral weight, they conjecture that Fisher zeros and thermofield dynamics might offer a new perspective for exploring those complex many-body physics related to unconventional gap behaviors.

Conclusion: Kai: So we're wrapping up on this paper about these giant bubbles of Fisher zeros in the quantum XY chain, and I want to talk about what that actually means for our experimental side of things with superconducting qubits.

Mira: Yeah, we’re looking at how these complex partition functions help us see those hard-to-pin-down quantum phases. The authors are using this method to probe the spectrum and dynamics of the system.

Kai: It seems like they're using some pretty advanced math—complex inverse temperature and thermofield dynamics—to find these features. I mean, for us building hardware, what does finding a "giant bubble" actually tell me about the physics we're trying to control?

Mira: Well, these giant bubbles are essentially large-scale closed lines of Fisher zeros that appear near the gapless XX limit. The authors claim this structure seems to contradict what we expect from a simple featureless Luttinger liquid description at low energies.

Kai: So it suggests there’s some kind of hidden complexity there, something more than the basic low-energy model predicts? And how does that relate to the spectral weight transfer they mentioned earlier?

Mira: Exactly. The motion of these bubbles as you change the external field is related to moving spectral weight from high energies down to lower energies. It’s a dynamic process in a way that's really interesting for understanding these many-body systems.

Kai: That makes sense because spectral weight redistribution is what defines the pseudogap phenomenon in cuprates, right? So, what are the authors suggesting this tells us about those unconventional gap behaviors we see in materials science?

Mira: They’re suggesting that Fisher zeros and thermofield dynamics give us a new way to look at these gaps. It opens up avenues for understanding how spectral weight moves around in strongly correlated systems.

Kai: That's a big picture thing. It sounds like this approach could be extended to even more complicated two-dimensional models, which is something I’m interested in exploring with my own quantum hardware simulations down the line.

Key Laboratory of Polar Materials and Devices (MOE), School of Physics, East China Normal University, Shanghai 200241, China · Department of Physics and Astronomy, University of California, Riverside, California 92521, USA · Department of Physics and Astronomy, George Mason University, Fairfax Virginia 22030 · State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University Beijing 100084 China · Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences Beijing 100190 China · School of Physical Sciences, University of Chinese Academy of Sciences Beijing 100049 China

cond-mat.str-el, cond-mat.stat-mech, hep-lat, quant-ph

Submitted: 2026-02-05

Updated: 2026-10-29

Comments: 6+4 pages, 4+1 figures

Journal ref: Phys. Rev. B 113, L241117 (2026)

DOI: 10.1103/pwcc-75fn

License: http://creativecommons.org/publicdomain/zero/1.0/

Importance score: 93/100

The gist: Giant bubbles of Fisher zeros in the quantum XY chain demonstrate an alternative approach based on complex-valued inverse temperature and partition function to probe quantum phases of matter with

Key concepts

Complex Inverse Temperature ($eta = eta_r + ieta_i$)
This method extends standard thermal physics by introducing an imaginary component to the inverse temperature. This allows researchers to simultaneously account for both thermal fluctuations (real part, $eta_r$) and quantum fluctuations (imaginary part, $eta_i$), providing a richer framework for studying quantum systems.
Fisher Zeros
These are specific points where the partition function ($Z$) of the system equals zero when plotted on a complex plane. The locations and patterns of these zeros reveal crucial information about the underlying thermodynamic phase transitions and excitations within the quantum system.
Giant Bubbles
These are large-scale, closed lines formed by Fisher zeros that appear near the gapless XX limit. Their size and motion as an external field changes serve as a characteristic energy scale that helps map out how spectral weight moves between different energy levels in the system.

Terminology

Summary

Giant bubbles of Fisher zeros in the quantum XY chain demonstrate an alternative approach based on complex-valued inverse temperature and partition function to probe quantum phases of matter with nontrivial spectra and dynamics <ref:2602.05899#pg2>

Methodology for Probing Quantum Phases

The authors utilize a method involving the analytical continuation of the partition function to a complex inverse temperature, denoted as β = βr + iβi ∈ C, which accounts for both thermal and quantum fluctuations <ref:2602.05899#pg2>. This framework allows them to leverage thermofield dynamics (TFD) to quantitatively characterize quantum and thermal fluctuations <ref:2602.05899#pg2>. They exploit the correspondence between low-energy excitations and Fisher zeros, which are the zeros of Z on the complex β plane, to identify thermodynamic phase transitions <ref:2602.05899#pg2>. The quantum XY chain in an external field serves as a testbed for this investigation <ref:2602.05899#pg2>.

Characterization of Oscillatory Behavior

The oscillatory gap behavior in the quantum XY model manifests as oscillations in the long-time dynamics of the TFD spectral form factor <ref:2602.05899#pg2>. The analysis involves examining the normalized spectral form factor S = Z(β = βr +iβi)/Z(β = βr)2 and its contributions from different excitation scales, such as ∆∞ and ∆L. By choosing appropriate values for L, βr, and βi, the researchers can access both the thermodynamic-limit and finite-size gap structures <ref:2602.05899#pg2>. The oscillation frequency of S provides unambiguous evidence to identify the oscillatory phase in the XY model <ref:2602.05899#pg2>.

Identification of Giant Fisher Bubbles

The study identifies giant bubbles, defined as large-scale closed lines of Fisher zeros near the gapless XX limit <ref:2602.05899#pg2>. These giant bubbles provide a characteristic energy scale that seems to contradict the predictions of the low energy theory of a featureless Luttinger liquid <ref:2602.05899#pg2>. The motion of these giant bubbles with varying external field is related to the transfer of spectral weight from high to low energies <ref:2602.05899#pg2>. In the XX limit (γ = 0), all zeros are in the form of closed bubbles, and some grow in size by extending toward larger and larger βr <ref:2602.05899#pg2>.

Connection to Low-Energy Physics

The low-energy physics of the XX model at γ = 0 can be effectively described by a Luttinger liquid with a linear dispersion ϵ(k) = vF (k − kF) around the Fermi points <ref:2602.05899#pg2>. The pseudogap scale, which is characterized by enhanced spectral weight, is traced to the van Hove singularity at energy ϵ0 = (1 − g) <ref:2602.05899#pg2>. This scaling behavior agrees with 1/βi0 ∝ (1 − g) observed from Fig. 4(b), which links the pseudogap to the van Hove singularity <ref:2602.05899#pg2>. The size of these giant bubbles, βi0, as measured by the imaginary part of its rightmost endpoint, diverges as g → 1 <ref:2602.05899#pg2>.

Broader Implications and Outlook

The deep connection between Fisher zeros, dynamics, and excitations opens up promising avenues for understanding the unconventional gap behaviors in strongly correlated many-body systems <ref:2602.05899#pg2>. The approach outlined here can be extended to 2D strongly interacting models for which the complexvalued partition functions cannot be solved exactly but can be evaluated to high precision by tensor network algorithms <ref:2602.05899#pg2>. The investigation suggests that Fisher zeros and thermofield dynamics may offer a new perspective for exploring the intricate many-body physics related to unconventional gap behaviors <ref:2602.05899#pg2>.

Summary and Outlook

Our detailed analysis confirms that complex partition function offers an effective tool for analyzing the thermal and dynamical properties of the 1D XY model <ref:2602.05899#pg2>. By varying βr and βi and investigating the Fisher zeros and thermofield dynamics (through S), we accurately capture the oscillatory finite-size gap behavior <ref:2602.05899#pg2>. In the thermodynamic limit, we uncover a distinctive class of giant bubbles (large-scale closed lines of Fisher zeros) near the critical line <ref:2602.05899#pg2>. We further demonstrate that the evolution of these bubbles with varying external field offers an intuitive yet sensitive probe for the transfer of spectral weight within the Luttinger liquid phase <ref:2602.05899#pg2>. Since the pseudogap phenomenon is characterized by a redistribution of the spectral weight, we conjecture that Fisher zeros and thermofield dynamics may offer a new perspective for exploring the intricate many-body physics related to unconventional gap behaviors <ref:2602.05899#pg2>.

Acknowledgments

We thank Jiansong Pan, Fan Yang, and Xuefeng Zhang for helpful discussions <ref:2602.05899#pg2>. H.Z. is supported by the National Natural Science Foundation of China (Grant No. 12274126) and Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics <ref:2602.05899#pg2>. E.Z. acknowledges the support from NSF Grant PHY-206419, and AFOSR Grant FA9550-23-1-0598 <ref:2602.05899#pg2>.

References

[1] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179 (2015) <ref:2602.05899#pg2>

[2] T. Timusk and B. Statt, The pseudogap in hightemperature superconductors: an experimental survey, Reports on Progress in Physics 62, 61 (1999) <ref:2602.05899#pg2>

[3] J. Tallon and J. Loram, The doping dependence of t∗– what is the real high-tc phase diagram?, Physica C: Superconductivity 349, 53 (2001) <ref:2602.05899#pg2>

[4] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mott insulator: Physics of high-temperature superconductivity, Reviews of Modern Physics 78, 17 (2006) <ref:2602.05899#pg2>

[5] X. G. Wen and P. A. Lee, Theory of underdoped cuprates, Physical Review Letters 76, 503 (1996) <ref:2602.05899#pg2>

[6] C. M. Varma, Theory of the pseudogap state of the cuprates, Phys. Rev. B 73, 155113 (2006) <ref:2602.05899#pg2>

[7] K.-Y. Yang, T. M. Rice, and F.-C. Zhang, Phenomenological theory of the pseudogap state, Phys. Rev. B 73, 174501 (2006) <ref:2602.05899#pg2>

[8] M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollw¨ock, S. R. White, and S. Zhang (Simons Collaboration on the Many-Electron Problem), Absence of superconductivity in the pure two-dimensional hubbard model, Phys. Rev. X 10, 031016 (2020) <ref:2602.05899#pg2>

[9] H. Xu, C.-M. Chung, M. Qin, U. Schollw¨ock, S. R. White, and S. Zhang, Coexistence of superconductivity with partially filled stripes in the hubbard model, Science 384, 10 <ref:2602.05899#pg2>

[10] M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annual Review of Condensed Matter Physics 14, 335 (2023) <ref:2602.

Improvements for AI systems

  1. Precision characterization of quantum critical points in strongly correlated materials can be achieved by analyzing Fisher zeros, which provide a characteristic energy scale that seems to contradict the predictions of the low energy theory of a featureless Luttinger liquid. This allows for identifying unconventional gap behaviors beyond simple Fermi liquid descriptions.

  2. Development of dynamical probes using complexified partition functions allows AI systems to monitor the transfer of spectral weight from high to low energies as an external field is varied, providing a sensitive tool for understanding the pseudogap phenomenon.

  3. Advanced numerical algorithms based on tensor network methods can be used to compute the complex Z reliably for lattice spin models, enabling high-precision evaluation of quantum many-body systems even when exact solutions are unavailable.

  4. AI systems can be trained to interpret patterns in Fisher zero configurations—such as giant bubbles of Fisher zeros—to diagnose the system's phase, as these structures change predictably with varying external field and temperature.

  5. The framework can be extended to 2D strongly interacting models by using tensor network algorithms to evaluate [the complex partition functions] to high precision, allowing for the study of phenomena where traditional exact solutions are impossible.

Abstract

We demonstrate an alternative approach based on a complex-valued inverse temperature and partition function to probe quantum phases of matter with nontrivial spectra and dynamics. It leverages thermofield dynamics (TFD) to quantitatively characterize quantum and thermal fluctuations, and exploit the correspondence between low-energy excitations and Fisher zeros. Using the quantum XY chain in an external field as a testbed, we show that the oscillatory gap behavior manifests as oscillations in the long-time dynamics of the TFD spectral form factor. We also identify giant bubbles, i.e. large-scale closed lines, of Fisher zeros near the gapless XX limit. They provide a characteristic energy scale that seems to contradict the predictions of the low energy theory of a featureless Luttinger liquid. We identify this energy scale and relate the motion of these giant bubbles with varying external field to the transfer of spectral weight from high to low energies. The deep connection between Fisher zeros, dynamics, and excitations opens up promising avenues for understanding the unconventional gap behaviors in strongly correlated many-body systems.

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