🔬 condensed matter

Interacting quantum criticality with U(1) symmetry breaking in 1+1 dimensions

Contrary to the Hohenberg-Mermin-Wagner theorem, this paper demonstrates that 1D lattice spin chains can exhibit spontaneous U(1) symmetry breaking and an interacting quantum critical point characterized by critical exponents $\nu \approx 0.60$ and $z \approx 1.986$ , revealing a mechanism for such breaking distinct from frustration-free models.

Original authors: Mutsumi Shimomura, Kohei Kawabata

Published 2026-10-06

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CC BY 4.0

Original authors: Mutsumi Shimomura, Kohei Kawabata

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

In the quantum world, the rules of order are often stricter than in our everyday experience. For decades, physicists have relied on a fundamental principle known as the Hohenberg-Mermin-Wagner theorem to understand how matter behaves at the smallest scales. This rule states that in one-dimensional systems—think of a single, endless line of atoms—continuous symmetries cannot be broken in the ground state. To visualize this, imagine a line of compass needles that are free to point in any direction. In a two-dimensional sheet or a three-dimensional block, these needles can all align to point north, creating a stable, ordered state. However, in a single line, the constant, jittery quantum fluctuations are so strong that they prevent the needles from ever settling into a single, unified direction. The system remains disordered, or at best, only loosely correlated over short distances. This expectation has long been the bedrock for understanding quantum chains, suggesting that true, long-range order is impossible in one dimension without special, artificial constraints.

A team of researchers at the University of Tokyo has now challenged this long-held expectation. By studying specific models of quantum spin chains, they have demonstrated that continuous symmetry can indeed be broken in a one-dimensional ground state, but only under a very specific set of conditions involving complex interactions. Their work reveals a new type of quantum critical point—a precise tipping point where a material changes its fundamental nature—that behaves differently from the standard theories physicists have used for years. They found that when the interactions between the spins are tuned just right, the system spontaneously develops a preferred direction, defying the usual quantum jitter that keeps it disordered. This discovery is significant because it shows that the mechanisms allowing order to emerge in one dimension are more diverse than previously thought, moving beyond simple, frustration-free models to include more complex, interacting scenarios.

The researchers focused on two distinct types of quantum chains: one made of spin-1 particles and another of spin-1/2 particles. In these systems, the "spins" act like tiny magnets that can point in various directions. The team used powerful computer simulations, specifically a method called density-matrix renormalization group, to model the behavior of these chains as they adjusted the strength of the interactions between the particles. They were looking for a critical point where the system transitions from a disordered state to one with long-range order. To locate this point, they employed a technique called subsystem scaling, which analyzes how fluctuations in a small section of the chain behave as the total size of the chain grows. This allowed them to pinpoint the exact moment the transition occurs, identifying a critical coupling value of approximately -0.826 for the spin-1 chain.

Once they found this critical point, the team measured how the system responded to changes in size and energy. They calculated a key number known as the correlation-length critical exponent, which describes how quickly correlations between distant parts of the chain decay. Their measurements yielded a value of 0.60, which is distinctly different from the value of 0.5 predicted by older, simpler theories. This difference is crucial because it proves that the critical behavior is not just a standard, free-flowing phenomenon but is driven by strong interactions between the particles. Furthermore, they determined the dynamical critical exponent, a number that describes how energy and time scale together in the system. They found this value to be approximately 1.986. This number is slightly less than 2, a small but significant deviation that has profound implications.

This specific value for the dynamical exponent acts as a fingerprint for the type of physics at play. The researchers showed that a value less than 2 is incompatible with a class of models known as frustration-free systems, which are often used to construct quantum states because they are mathematically simpler. Frustration-free models require this exponent to be at least 2. By finding a value below this threshold, the team effectively ruled out the possibility that their observed symmetry breaking was a result of these simpler, frustration-free mechanisms. Instead, their results point to a more complex, interacting theory where the particles influence each other in ways that cannot be simplified away. This confirms that the spontaneous breaking of symmetry in their one-dimensional chains arises from a unique, interacting quantum criticality.

To ensure their findings were robust, the researchers repeated the entire analysis on a different type of chain, the spin-1/2 model, which has a different microscopic structure. Despite the differences in the particles and the specific interactions, they found the same critical behavior. The correlation-length exponent for this second model was 0.61, consistent with the first, and the system again showed clear signs of spontaneous symmetry breaking. This agreement between two different microscopic setups suggests that they both belong to the same universal class of behavior, governed by the same underlying physics. It demonstrates that this phenomenon is not an accident of a specific model but a general feature of interacting quantum systems in one dimension.

The implications of this work extend beyond just these specific chains. The researchers compared their results to theoretical predictions made using perturbative renormalization group methods, which are mathematical tools used to approximate the behavior of complex systems. Their numerical findings aligned closely with these theoretical predictions, providing strong evidence that the proposed interacting field theory correctly describes what is happening in the lattice models. They also noted that while their results match some previous theoretical estimates, they differ from other recent studies that suggested different values for the critical exponents. This discrepancy highlights the importance of direct numerical verification, as theoretical approximations can sometimes miss subtle effects that only appear in a full, non-perturbative treatment of the system.

Ultimately, this study provides a concrete example of how continuous symmetry can be broken in a one-dimensional quantum ground state, a feat that was previously thought to be forbidden by standard theorems. The researchers have shown that by introducing specific, interacting terms into the Hamiltonian—the mathematical description of the system's energy—it is possible to stabilize an ordered state against the disruptive quantum fluctuations. The key to this stability is the presence of an interacting critical point where the scaling of space and time is slightly anisotropic, meaning they do not scale in the same way. This finding expands our understanding of quantum phase transitions and suggests that there are multiple pathways for order to emerge in low-dimensional systems, not just the ones we have known about for decades.

The work also serves as a reminder that even in the most constrained environments, like a single line of atoms, nature can find surprising ways to organize itself. By carefully tuning the interactions, the researchers were able to guide the system into a state where the quantum noise is tamed enough to allow a global direction to emerge. This does not mean the Hohenberg-Mermin-Wagner theorem is wrong; rather, it clarifies the boundaries of its application. The theorem holds for systems with sufficiently short-range interactions and specific assumptions, but as this study shows, those assumptions can be bypassed in more complex, interacting scenarios. The result is a richer, more nuanced picture of quantum matter, where the interplay of symmetry, dimension, and interaction creates a landscape of possibilities far broader than previously imagined.

In the broader context of condensed matter physics, these findings offer a new benchmark for understanding quantum criticality. The precise values measured for the critical exponents provide a target for future theoretical work and experimental realizations. If similar behavior can be observed in physical materials or engineered quantum simulators, it would validate the existence of this interacting universality class in the real world. For now, the study stands as a rigorous demonstration that the rules of quantum order are not as rigid as once believed, opening the door to new ways of thinking about how matter organizes itself at the most fundamental level. The path forward involves exploring whether other types of interactions can produce similar effects and how these interacting critical points might influence the properties of materials in higher dimensions.

Technical Summary: Interacting Quantum Criticality with U(1) Symmetry Breaking in 1+1 Dimensions

Problem Statement
The Hohenberg-Mermin-Wagner theorem generally precludes the spontaneous breaking of continuous symmetries in one-dimensional quantum ground states, suggesting that fluctuations destroy long-range order. While exceptions exist in frustration-free systems (e.g., the ferromagnetic Heisenberg chain) where the order parameter commutes with the Hamiltonian, the mechanisms for continuous symmetry breaking in more general, interacting 1D systems remain unclear. Specifically, Nahum [23] proposed a field-theoretic scenario for an interacting quantum critical point where U(1) spin-rotation symmetry is spontaneously broken in 1+1 dimensions. This scenario involves a relevant quartic interaction that destabilizes the Gaussian fixed point, leading to non-Gaussian critical exponents and a dynamical exponent $z < 2$ . However, perturbative renormalization-group (RG) calculations for this theory are uncontrolled in $d=1$ , and rigorous numerical verification of the microscopic realization and its universal properties was lacking.

Methodology
The authors investigate two microscopic lattice models using numerical techniques, primarily the Density Matrix Renormalization Group (DMRG) and the Variational Uniform Matrix Product State (VUMPS) algorithm:

Spin-1 Chain: A model proposed by Nahum [23] with Hamiltonian $H_1 = -J \sum S_i \cdot S_{i+1} + g_1 \sum (S^z_i)^2 + g_2 \sum (S^z_i)^2(S^z_{i+1})^2$ .
Spin-1/2 Chain: A model with four-site longitudinal interactions, $H_{1/2} = -J \sum S_i \cdot S_{i+1} + g_1 \sum S^z_i S^z_{i+1} + g_4 \sum S^z_i S^z_{i+1} S^z_{i+2} S^z_{i+3}$ .

To locate the critical point and determine exponents without relying on an order parameter that is conserved globally (which would vanish in the symmetric sector), the authors employ subsystem Binder scaling. They analyze the fluctuations of the longitudinal magnetization within a half-chain block ( $M_{z,\text{half}}$ ) to define a Binder ratio $R_{\text{half}}$ .

Critical Coupling and $\nu$ : The crossing points of $R_{\text{half}}$ for different system sizes $L$ are used to locate the critical coupling $g_{1c}$ . Finite-size scaling of these crossings yields the correlation-length critical exponent $\nu$ .
Symmetry Breaking: Spontaneous U(1) symmetry breaking is verified by calculating the squared transverse magnetization density $m^2_\perp$ in the $M_z=0$ sector. Both finite-size scaling (extrapolating $L \to \infty$ ) and finite-entanglement scaling (extrapolating bond dimension $\chi \to \infty$ via VUMPS) are used to confirm a non-zero thermodynamic limit.
Dynamical Exponent $z$ : The lowest neutral excitation gap $\Delta_1$ is calculated for various $L$ . The scaling $\Delta_1 \propto L^{-z}$ is used to extract the dynamical critical exponent.
Scaling Dimensions: Longitudinal spin correlations are analyzed using VUMPS to check consistency with the scaling dimension $\Delta_\psi$ derived from $z$ .

Key Contributions and Results

Numerical Evidence for Interacting Criticality:
The study provides direct numerical evidence for the interacting quantum critical point proposed by Nahum. For the spin-1 chain, the correlation-length exponent is determined to be $\nu = 0.60 \pm 0.02$ . This value is distinct from the Gaussian fixed point value ( $\nu_G = 0.5$ ) but consistent with perturbative RG predictions ( $\nu \approx 0.6258$ at $O(\epsilon^2)$ ).
Spontaneous U(1) Symmetry Breaking:
Contrary to the expectation that continuous symmetries remain unbroken in 1D, the authors demonstrate spontaneous breaking of U(1) spin-rotation symmetry at the critical point. The squared transverse magnetization density extrapolates to a finite positive value in the thermodynamic limit:
- Spin-1: $m^2_\perp \approx 0.947 \pm 0.003$ .
- Spin-1/2: $m^2_\perp \approx 0.212 \pm 0.003$ .
  This confirms that the critical state supports long-range order in the transverse direction.
Dynamical Critical Exponent $z < 2$ :
The dynamical critical exponent is determined to be $z = 1.986 \pm 0.005$ . This represents a small but statistically significant deviation from the Gaussian value $z_G = 2$ and is consistent with the perturbative prediction $z \approx 1.992$ .
Universality and Model Independence:
The authors construct a spin-1/2 chain with four-site interactions and show that it flows to the same universality class as the spin-1 model. The estimated critical exponent $\nu = 0.61 \pm 0.02$ for the spin-1/2 chain is consistent with the spin-1 result, supporting a common interacting universality class despite different local Hilbert spaces and microscopic anisotropies.
Scaling Dimension Consistency:
Using the relation $\Delta_\psi = (3-z)/2$ , the estimated $z$ implies a scaling dimension $\Delta_\psi \approx 0.507$ . Finite-entanglement scaling of longitudinal correlations shows that this value provides a better data collapse than the Gaussian value $\Delta_\psi = 0.5$ , although the precision is limited by finite bond dimensions.

Significance and Implications
The paper claims that these results establish a mechanism for continuous symmetry breaking in one-dimensional quantum ground states that is distinct from frustration-free constructions.

Obstruction to Frustration-Free Realizations: Rigorous bounds for frustration-free Hamiltonians require $z \geq 2$ . The observed value $z \approx 1.986 < 2$ implies that the critical theory realized in these models cannot be described by a frustration-free Hamiltonian. This distinguishes the present interacting transition from previously known examples of 1D symmetry breaking.
Validation of Field Theory: The numerical results validate the existence of the interacting fixed point proposed in field theory, confirming that the relevant quartic interaction destabilizes the Gaussian fixed point even in $d=1$ .
Universality Class: The agreement between the spin-1 and spin-1/2 models, as well as the consistency with perturbative RG and classical Monte Carlo simulations of related nonequilibrium systems (Lifshitz-like theories), supports the identification of a robust universality class characterized by non-Gaussian exponents and anisotropic scaling ( $z \neq 2$ ).

The authors conclude that continuous symmetry breaking in 1D is not exhausted by frustration-free constructions, revealing diverse mechanisms for circumventing the Hohenberg-Mermin-Wagner theorem in quantum many-body systems.

Technical Summary: Interacting Quantum Criticality with U(1) Symmetry Breaking in 1+1 Dimensions

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