Phase structure of (3+1)-dimensional dense two-color QCD at $T=0$ in the strong coupling limit with the tensor renormalization group

We investigate the phase structure of the (3+1)-dimensional strong coupling two-color QCD at zero temperature ($T=0$) with finite chemical potential using the tensor renormalization group method. The chiral and diquark condensates and the quark number density are evaluated as a function of the chemical potential. We further determine the critical exponents associated with the diquark condensate, which suggest consistency with the predictions of mean-field theory.

💡 Research Summary

The authors study the zero‑temperature phase diagram of (3+1)‑dimensional two‑color QCD (QC₂D) in the strong‑coupling limit using the tensor renormalization group (TRG). Because QC₂D remains free of the sign problem even at non‑zero chemical potential μ, it serves as a testbed for dense QCD where conventional Monte‑Carlo methods fail. The fermionic sector is discretized with Kogut‑Susskind staggered quarks; an explicit U(1)_V‑breaking source λ is added to probe diquark condensation. In the g→∞ limit the gauge action disappears, leaving only link variables in the hopping terms. By introducing auxiliary Grassmann fields ζ_ν and ξ_ν the authors rewrite the partition function as a Grassmann tensor network. After performing the SU(2) link integration analytically via Weingarten calculus, the remaining degrees of freedom are encoded in an eight‑index local tensor T that carries fermion occupation numbers (0 or 1) for each direction and color indices (1 or 2). The full partition function becomes a Grassmann trace over the product of these tensors.

Numerically the authors employ an anisotropic TRG algorithm with a bond dimension D. They systematically increase D up to 55 and verify convergence of the thermodynamic potential f=ln Z/V on a lattice as large as 1024⁴ (V=1024⁴). The relative deviation δf falls below 10⁻⁴ at D≈55, indicating that the truncation error is negligible for the observables of interest. Multi‑GPU parallelization is used to handle the enormous initial tensor size (2¹⁶ components).

Physical observables are obtained by finite‑difference derivatives of ln Z: the chiral condensate ⟨χ̄χ⟩=∂ln Z/∂m, the quark number density ⟨n⟩=∂ln Z/∂μ, and the diquark condensate ⟨χχ⟩=∂ln Z/∂λ|_{λ→0}. To avoid numerical instability in the λ→0 limit, the λ‑dependence of the free energy is fitted to f(m,μ,λ)=b₁λ²+b₂|λ|+f(m,μ,0); the coefficient b₂ directly yields ⟨χχ⟩. The authors scan μ for several values of the bare mass m=1.0 and source λ, adjusting the λ range according to μ to keep the fit reliable.

The results display three distinct regimes. For μ below a lower critical value μ_low^c≈1.09 (the Silver‑Blaze region) the chiral condensate remains essentially constant, the quark number density is zero, and the diquark condensate vanishes. When μ exceeds μ_low^c, a diquark condensate develops, signalling a superfluid phase; simultaneously ⟨χ̄χ⟩ decreases and ⟨n⟩/2 rises, eventually saturating at ⟨n⟩/2=1 for μ above an upper critical value μ_up^c≈1.19. These features match the mean‑field (MF) predictions and the 1/d expansion results of Nishida and collaborators. Finite‑volume checks show negligible dependence for V≥2¹⁶, confirming that the 1024⁴ lattice effectively reaches the thermodynamic limit at T=0.

Critical behavior near the onset of diquark condensation is analyzed by fitting ⟨χχ⟩∝(μ−μ_low^c)^{β_m}. The fit yields β_m=0.514(27) and μ_low^c=1.0950(7), in excellent agreement with the MF exponent β=½ and the MF estimate μ_low^c≈1.0913. Although the paper mentions the exponent δ, its numerical value is not explicitly reported; the authors anticipate consistency with the MF value δ=3. Thus the transition appears to be a second‑order one belonging to the mean‑field universality class.

In summary, this work demonstrates that the Grassmann‑anisotropic TRG can handle a (3+1)‑dimensional gauge theory with fermions at finite density, overcoming the sign problem without stochastic sampling. The methodology successfully reproduces known analytical results, determines critical exponents, and scales to unprecedented lattice sizes. The study provides a concrete proof‑of‑concept that TRG techniques may be extended to realistic three‑color QCD at finite density, where conventional Monte‑Carlo methods are currently inapplicable. Future directions include exploring finite‑temperature effects, moving away from the strict strong‑coupling limit, and incorporating more realistic fermion actions.